University of Southern California Dissertations and Theses

Forward-backward stochastic differential equations with discontinuous coefficient and regime switching term structure model

(USC Thesis Other)

Forward-backward stochastic differential equations with discontinuous coefficient and regime switching term structure model

PDF

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content FORWARD-BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH DISCONTINUOUS COEFFICIENT AND REGIME SWITCHING TERM STRUCTURE MODEL by Jianfu Chen A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2011 Copyright 2011 Jianfu Chen Dedication To my family ii Acknowledgments This dissertation would not have been possible without the guidance and the help of many people who extended their valuable assistance to me during my doctoral study. I would like to express my sincere gratitude to all of them for helping me and inspiring me. Firstandforemost,myutmostgratitudetomyadvisor,ProfessorJinMa. With his immense knowledge, enthusiasm, inspiration and patient, he helped to make this research fun. Throughout my time working with him, Professor Ma provided good teaching, sound advice, encouragement, and lots of great ideas. I would have been lost without him. It has been an honor to be his student. I would like to thank Professors Sergey Lototsky, Remigijus Mikulevicius, Fer- nando Zapatero and Jianfeng Zhang for serving on my committee and for all the insightful comments and advices they provided me. Iamdeeply gratefultoProfessorDoyoonKimforhelpingmewithhisexpertise in the PDE area, and his great efforts to explain things clearly and simply. iii I would also like to thank Dr. Bin Hong at Union Bank for offering me the opportunity to work on the research project from which the idea of this thesis was started, the insight he shared and the continuous encouragement he gave. My time at USC was made enjoyable in large part due to the many friends that became a part of my life. I am very thankful to Ally Chan, Bridget Hu and Michelle Ye for being there with me through the good times and bad. I thank my mathbuddies, particularlyHaoyuan Liu, MinzhaoTan, HuanhuanWang, Xinyang Wang, Shanshan Xu and Jie Du, for your camaraderie, caring and support. Last but not least, I cannot overstate my gratitude to my family, my parent Weilian Chen and Jingru Huang, and my sister Suyin Chen. They raised me, taught me, supported me, motivated me and loved me. They spared no effort to provide the best possible environment for me to grow up, and pave the way for a privileged education. I would not be where I am now without them. To them I dedicate this thesis. iv Table of Contents Dedication ii Acknowledgments iii List of Figures vii Abstract viii Chapter 1: Introduction 1 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of the Term Structure Models . . . . . . . . . . . . . . . . . 5 1.3 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Chapter 2: Preliminaries 10 2.1 Probability Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Solution to Forward-backward SDEs . . . . . . . . . . . . . . . . . 11 Chapter 3: FBSDEs with Discontinuous Coefficient 14 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.2 Solving the PDE: Approximation and Convergence . . . . . . . . . 18 3.2.1 Approximating PDE(0) . . . . . . . . . . . . . . . . . . . . 19 3.2.2 Convergence of u ǫ . . . . . . . . . . . . . . . . . . . . . . . . 26 3.3 Solving the FBSDEs: A Weak Solution . . . . . . . . . . . . . . . . 34 3.3.1 Regularity of ˆ u . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.3.2 Solution to the FBSDEs . . . . . . . . . . . . . . . . . . . . 40 Chapter 4: Regime Switching Term Structure Model 48 4.1 Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.2 A weak solution to the FBSDE . . . . . . . . . . . . . . . . . . . . 52 4.2.1 Solving the PDE . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2.2 Solving the FBSDEs . . . . . . . . . . . . . . . . . . . . . . 59 4.2.3 Extension to Other Model . . . . . . . . . . . . . . . . . . . 65 v Chapter 5: Numerical Experiments 67 5.1 Regime Switching Hull-White Model . . . . . . . . . . . . . . . . . 68 5.1.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.1.2 The Crank-Nicolson Method . . . . . . . . . . . . . . . . . . 69 5.1.3 The Layer Method . . . . . . . . . . . . . . . . . . . . . . . 73 5.2 Regime Switching Black-Karasinski Model . . . . . . . . . . . . . . 87 5.2.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 5.2.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . 88 Bibliography 94 vi List of Figures 5.1 HW Model: Plot of u(·,x) (C-N Method) . . . . . . . . . . . . . . . 72 5.2 HW Model: 3-D Plot of ¯ u ǫ (t,x) - (1) . . . . . . . . . . . . . . . . . 78 5.3 HW Model: Plot of ¯ u ǫ (t,·) . . . . . . . . . . . . . . . . . . . . . . . 79 5.4 HW Model: Plot of ¯ u ǫ (t,·) . . . . . . . . . . . . . . . . . . . . . . . 80 5.5 HW Model: 3-D Plot of Δu . . . . . . . . . . . . . . . . . . . . . . 82 5.6 HW Model: 3-D Plot of ¯ u ǫ (t,x) - (2) . . . . . . . . . . . . . . . . . 82 5.7 HW Model: The Short Rate(r t ) Trajectory - (1) . . . . . . . . . . . 84 5.8 HW Model: The Bond Price ( Y t ) Trajectory - (1) . . . . . . . . . . 84 5.9 HW Model: The Short Rate(r t ) Trajectory - (2) . . . . . . . . . . . 85 5.10 HW Model: The Short Rate(r t ) Trajectory - (3) . . . . . . . . . . . 86 5.11 HW Model: The Bond Price(Y t ) Trajectory - (2) . . . . . . . . . . . 86 5.12 BK Model: 3-D Plot of ¯ u ǫ (t,m) - (1) . . . . . . . . . . . . . . . . . 90 5.13 BK Model: Plot of ¯ u ǫ (t,·) . . . . . . . . . . . . . . . . . . . . . . . 91 5.14 BK Model: 3-D Plot of ¯ u ǫ (t,m) - (2) . . . . . . . . . . . . . . . . . 93 5.15 BK Model: Regime Switching of the Short Rate . . . . . . . . . . . 93 vii Abstract In this dissertation, we propose a regime switch term structure model built as forward-backward stochastic differential equations. We first generalize the model and study the forward-backward SDEs with discontinuous coefficients. Generally speaking, we adopt the tactics of the Four-Step scheme as our strategy of finding the solution. First, we find a solution v to the PDE associated with the FBSDEs. Then we use this v to decouple the FBSDEs, and solve each equation individually. A weak solution was successfully found. We then returned to the term structure model. We apply the framework of analysis from the general model to the regime switch term structure model, with some minor modifications. A solution, slightly weaker than the general case, was found for the term structure model, as well as the explicit relationship between the short rate and the long term bond price, all ofwhich confirmed our model is fairlywell-grounded. Numerical experiments were also conducted to show the validity of our theory. Results are rather satisfactory, providing us with strong empirical support. viii Chapter 1 Introduction 1.1 Motivation Term structure of interest rates describe how interest rate change through time. Ever since the introduction of Vasicek Model in 1977, term structure models have been studied extensively. For the past 3 decades, hundreds of models were devel- oped; and because of the market development and the use of new calibration techniques, the modeling of term structure is still evolving rapidly. Studied by many academic researchers and practitioners, term structure modeling represents and remains to be one of the most fascinating yet challenging topics in the area of mathematical finance research. The ideaofthisthesis grew fromaresearch project. Duringthefallsemester of 2007, I was very fortunate to have the chance to work at Union Bank (previously Union BankofCalifornia)onaproject aboutvalidatingthe term structure models in a financial service system called Quantitative Risk Management(QRM). Part of the project was to calibrate and investigate its newest feature: Regime Shift model. QRM’s regime shift model claims that, depending on the rate level at any 1 point in time relative tosome preset threshold level L, short rate switches between regimes. In a low rate environment, short rate is closer to a normal model. But in a high rate environment, a log-normal model might be more suitable. The switch of regime is triggered by the short rate itself; once the short rate crosses the level L, a different model would be admitted to portray the short rate dynamics. According to the above claim, the switching is endogenous. It’s a known fact that short rate has relatively large volatility. Whatever model one uses, impact of the stochastic term can not be ignored. Consider the moment when the short rate just had a switch. Due to the stochastic effect, short rate might make an instant switch to the other regime, and possibly another switch right after the previous one. So what could happen is that short rate keeps going back and forth between the regimes frequently, oscillating around the threshold level. This suggests that rates generated by the model could be quite unstable. Because of this shortcoming, we propose that, rather than an endogenous model, one should consider an exogenous model. By that we mean the switch should be caused by an external variable instead of the short rate itself. This external variable should have tight connection to the short rate, and it should serve as the role of stabilizing the model. One variable naturally comes to mind, the long rate. It’s well known that short rate and long rate are closely related to each other. Studies on the short rate and long rate could be seen in litera- tures such as Expectations Hypothesis, Brennan and Schwartz [10], and Schaefer 2 and Schwartz [42]. More importantly, we believe that long rate would be a more appropriate regime switching indicator. Idiosyncratic shocks such as short-term adjustments and policy errors are not usually what characterizes a regime. How- ever they do have a large impact on short rate. During the course of these events, short rate might experience a great deal of changes. But these impacts are only temporally, they tend to average out over time. So it’s unwise to assume that short rate goes through a regime switch just because short rate jumps or drops over a certain range. Also notice that, long rate often is nearly unaffected during these events. On the other hand, systematic shocks such as business cycle and monetary policy are more likely to define a regime. They are the main determi- nant of changes in long rate. When there are considerable changes occurred to the longrate, onewould expect the aggregateeconomy togothroughsome substantial transition, and thus the change of regime. Building on these understandings, we propose a two-factor term structure model, using forward-backward stochastic differential equations:          X t =x+ Z t 0 [b(Y s )−βX s ]ds+ Z t 0 σdW s , Y t =E n g(X T )+ Z T t 1−e ζ(Xs) Y s ds F t o . (1.1) Inourmodel,weconsidertheshortraterepresentedbyaforwardSDE;andinplace of the long rate, we employ the price of the long-term bond which is formulated as abackward SDE.Thelong-termbondpricetriggerstheswitch oftheregime; when 3 the bond price moves pass a certain threshold, short rate switches its dynamics to the according regime. Detailed discussion of the term structure model would be given in Chapter 4. In fact, it’s not new to model term structure using forward-backward SDEs. Duffie,MaandYong[16]proposedatwo-factormodelregardingshortrateandcon- sol rate using forward-backward SDEs. Compared to their model which assumes all the coefficients are nice and smooth, our novelty is: drift of the forward equa- tion is not continuous, and the discontinuity lies in the backward component Y, the solution itself. Studies on forward-backward SDEs with discontinuous coeffi- cients could be seen in, for example Delarue and Guatteri [14], where they allowed the drift and backward driver to be discontinuous in X. However, all the exist- ing literatures require the coefficients to be Lipschitz continuous with respect to Y. It should be emphasized that discontinuity in variable Y is fundamentally dif- ferent from problem with discontinuity in variable X. Just imagine an ordinary differential equation dx dt =f(t,x) where f is not continuous with respect to x, the equation is unsolvable. Before we start detailed discussion of the model, let’s take a look back at the history of the development of term structure model. It helps to get a better understanding of the model and gain some economic insight. 4 1.2 Review of the Term Structure Models Up until now, development of term structure models roughly went through 4 phases: equilibrium models, no-arbitrage models, forward-rate models and models accounting for volatility smile. Typical equilibrium models are Vasicek [44] and Cox, Ingersoll and Ross [12]: dr t =k(θ−r t )dt+σdW t , [Vasicek] dr t =k(θ−r t )dt+σ √ r t dW t , [CIR] where r t is the instantaneous short rate; k, θ and σ are constants. Equilibrium modelsstartwithassumptionsaboutvariablesundereconomicequilibrium(volatil- ity, market price of risk, etc.) and derive a process for the short rate, r. They then explore what the process for r implies about bond prices and option prices. Models of this kind are analytical simple. However, due to the stationary nature of the coefficients, equilibrium models lack of the ability to recover an observed arbitrary yield curve. 5 As a solution to the problem of recovering yield curve came the no-arbitrage models. Example are Hull and White [24], Black, Derman and Toy [7], and Black and Karasinski [8]: dr t =k(θ t −r t )dt+σdW t , [HW] dlnr t =θ t dt+σdW t , [BDT] dlnr t =k(θ t −lnr t )dt+σdW t , [BK] Comparedtotheequilibriummodels,no-arbitragemodelspossessthemodification of making the mean reversion level become time-dependent. Introduction of this extra degree of freedom makes possible fitting an arbitrary yield curve. Most models introduced in this phase were driven by the short rate as seen from above. Implementation of the model often includes a recombining computational lattice, with a Markovian process for the short rate, and with a finite-difference-implied PDE/SDE. This approach turned out to be considerably instable, and that leads to the development of new term structure models. The third-generation models were in general intrinsically non-Markovian. Pro- totype was given by the Heath, Jarrow and Morton [21] model, which took a alternative approach by formulating the non-instantaneous forward rate instead of the short rate: df(t,T)=−σ T t Z T t σ T u du + X k=1,n σ T t k (λ k t dt+dW k t ), [HJM] 6 where f(t,T) is the forward rate, σ T t is the volatility of f(t,T), and λ k is the market price of the risk associated with the k-th Brownian motion W k . The HJM framework imposes no structure other than the requirement of no-arbitrageon the dynamics ofthetermstructure. AndwithHJM,yieldcurve couldalwaysberecov- ered by using the forward rates as the fundamental building blocks. Based on the HJM approach, another popular model introduced was the LIBOR market model (alsoknown astheBGMmodel[9]), whichimproved theHJMmodelbyrecovering exactly the prices of caplets as produced by the Black model. The elegance and generality won these models wide acceptance, however, the progressively marked volatility smiles began to appear. At the beginning, volatility smiles were treated as a relatively small ‘perturba- tion’ of a basically correct approach, but following the market turmoil of 1998, it becameincreasinglydifficulttoignore. Asaresult,aseriesofapproacheswere(and are being) introduced in order to account for this phenomenon. The most popular among them is the constant elasticity of variance LIBOR model by Anderson and Andreasen [1], which assumes the volatility of the short rate is given by σ =Cr γ , where γ representing the elasticity is a parameter calibrated to the market. 7 Most recently, more and more empirical evidence suggests that there exists structural shift in interest rate process. We refer the reader to the work of Hamil- ton [20], Ang and Bekaert [2], Bali [6], Driffill, Kenc and Sola [15] etc. for the empirical studies. As a result, “regime switching” models started to attract peo- ple’sattention. Someusebusinesscycleasregimeclassification, whileothersmight define regime by interest rate level. Either way, the important feature of regime switching model remains to be its accommodation to the interactions between regimes and dynamics of the interest rate. A typical approach to model regime switch, as seen in Landen [31], Bansal and Zhou [5], Evans [18], Wu and Zeng [45], and Dai, Singleton, and Yang [13], is to incorporate a hidden Markov process as a state variable into the short rate dynamics. For example, r t+1 −r t =k s t+1 (θ s t+1 −r t )+σ s t+1 √ r t u t+1 , [BZ] where s t is a Markov process with given transitional probability, u t+1 ∼ N(0,1) is the noise. Introduction of the regime-dependence in these papers enriches the flexibility of the model and therefore leads to a higher capacity to fit empirical data. 8 1.3 Outline of the Thesis The main goal of this thesis is to find a solution to the proposed model and studytheintricaterelationbetweentheshortrateandbondprice(equivalently the long rate). This dissertation is organized as follows. In Chapter 2, we introduce the probability set-up, some notations and recall the definitions of strong and weak solutions to forward-backward SDEs. Chapter 3 is devoted to the study of forward-backward SDEs with discontinuous coefficient, generalized from the proposed model. Our strategy is to apply the Four-Step scheme: find a solution to the associated PDE first, then use the solution to the PDE to decouple the forward-backward SDEs and solve them separately. We give our main results about the existence of a weak solution. In Chapter 4, we return to the regime switching term structure model. With some modification, we apply the same framework of analysis as the general model, and show the existence of solution to (1.1), confirming the the soundness and reasonability of the model. Chapter 5 focuses on numerical experiments. We perform numerical analysis on two regime switching models with different short rate models. Outstanding results from both modelsshowthevalidityofourtheoryandprovideuswithsolidempiricalsupport. 9 Chapter 2 Preliminaries 2.1 Probability Set-up Let [0,T] be a finite time interval and let (Ω,F,P,F,W) be a standard set-up: (Ω,F,P) is a complete probability space; F,{F t } t∈[0,T] is a filtration satisfying the usual hypotheses; and W is an{F} t −Brownian motion. Particularly, we say the standard set-up is Brownian ifF t =F W t , the natural filtration generated by the Brownian motion W, augmented by all the P−null sets ofF and satisfying the usual hypotheses. 2.2 Notations Throughout the paper, we used the following notations. • R T ={(t,x) :t∈ [0,T],x∈R}; • Q T =(0,T)×Ω, where Ω is bounded domain inR; • The space C 1,2 (R T ) is the set of allR-valued functions f(t,u) such that f is C 1 in t and C 2 in x; 10 • The Sovolev space W 1,0 2 (Q T ) is the Banach space consisting of all elements of L 2 (Q T ) having a finite norm kuk W 1,0 2 (Q T ) = kuk 2 L 2 (Q T ) +ku x k 2 L 2 (Q T ) 1/2 ; • The Sovolev space W 1,2 2 (Q T ) is the Banach space consisting of all elements of L 2 (Q T ) having a finite norm kuk W 1,2 2 (Q T ) = kuk 2 L 2 (Q T ) +ku t k 2 L 2 (Q T ) +ku x k 2 L 2 (Q T ) +ku xx k 2 L 2 (Q T ) 1/2 . 2.3 Solution to Forward-backward SDEs Consider the following type of forward-backward stochastic differential equations:          X t =x+ Z t 0 b(s,X s ,Y s )ds+ Z t 0 σ(s,X s ,Y s )dW s , Y t =E n g(X T )− Z T t h(s,X s ,Y s )ds F t o , (2.1) where x∈R, and f :[0,T]×C([0,T],R)×R→R, f =b,σ,h, g :C([0,T],R)→R 11 are progressively measurable functions. We adopt the same definitions of solutions as in Antonelli and Ma [3], Ma, Zhang, and Zheng [35]. Definition 2.1. For a given standard set-up (Ω,F,P,F,W), a pair of process (X,Y), defined on the set-up, is called a strong solution to (2.1) if (1) (X,Y) areF t −adapted continuous process, such that E n sup t∈[0,T] |X t | 2 + sup t∈[0,T] |Y t | 2 + o <∞; (2) (X,Y) satisfies (2.1) P almost surely. Definition 2.2. A pair of process (X,Y) along with a standard set-up (Ω,F,P,F,W) on which X, Y are defined, is called a weak solution to (2.1) if (1) the process X is continuous, Y is C` adl` ag, and both areF t −adapted; (2) denoting f t =f(t,X t ,Y t ) for f =b,σ,h, it holds that P n Z T 0 (|b t |+|σ t | 2 +|h t |)ds+|g(X T )|<∞ o ; (3) (X,Y) satisfies (2.1) P almost surely. 12 If the set-up is Brownian, (2.1) could be rewritten as          X t =x+ Z t 0 b(s,X s ,Y s )ds+ Z t 0 σ(s,X s ,Y s )dW s , Y t =g(X T )− Z T t h(s,X s ,Y s )ds− Z T t Z s dW s . (2.2) Solution to (2.2) are similarly given as followed. Definition 2.3. For a given standard set-up (Ω,F,P,F,W), a triplet of process (X,Y,Z), defined on the set-up, is called a strong solution to (2.2) if (1) (X,Y,Z) areF t −adapted process, and X, Y are continuous, such that E n sup t∈[0,T] |X t | 2 + sup t∈[0,T] |Y t | 2 + Z T t |Z t | 2 dt o <∞; (2) (X,Y,Z) satisfies (2.2) P almost surely. Definition 2.4. A triplet of process (X,Y,Z) along with a standard set-up (Ω,F,P,F,W) on which X, Y, Z are defined, is called a weak solution to (2.2) if (1) (X,Y,Z) areF t −adapted process, and X, Y are continuous; (2) denoting f t =f(t,X t ,Y t ) for f =b,σ,h, it holds that P n Z T 0 (|b t |+|σ t | 2 +|h t |+|Z t | 2 )ds+|g(X T )|<∞ o ; (3) (X,Y,Z) satisfies (2.2) P almost surely. 13 Chapter 3 FBSDEs with Discontinuous Coefficient 3.1 Introduction Webeginbygeneralizingthemodelproblem(1.1). Considerthefollowingforward- backward SDEs:          X t =x+ Z t 0 b(s,X s ,Y s )ds+ Z t 0 σ(s,X s ,Y s )dW s , Y t =g(X T )− Z T t h(s,X s ,Y s )ds− Z T t Z s dW s , (3.1) with b(t,x,y) =          b 1 (t,x), y≤α, b 2 (t,x), y >α, where α is some predetermined constant. 14 For a given T ∈R + , we assume the coefficients b : [0,T]×R×R→R, σ : [0,T]×R×R→R h : [0,T]×R×R→R, g :R→R satisfytheusualmeasurabilitycondition. Andwealsomakethefollowingstanding assumptions throughout this dissertation. (H1) The drift b 1 (t,x) and b 2 (t,x) are bounded and uniformly Lipschitz continu- ous in (t,x) ; (H2) The volatility σ(t,x,y) is bounded and non-degenerate in the sense that there exist constants 0< σ < ¯ σ such that σ≤σ(x,y)≤ ¯ σ; Also, σ(t,x,y), σ x (t,x,y) and σ y (t,x,y) are all uniformly Lipschitz continu- ous in (t,x,y); (H3) The coefficient h(t,x,y) is bounded, and uniformly Lipschitz continuous in (t,x,y); (H4) The terminal function g(x) is bounded and smooth. Note that, for the purpose of illustration simplicity, we assume b(t,x,y) only has one jump in y in the model above . One could easily extend the model to a 15 finite number of jumps in y, and the results would still be true under the same arguments. Rewriting (3.1) in another form, one could translate the model as        dX t =b 1 (t,X t )dt+σ(t,X t ,Y t )dW t , when Y t ≤α, dX t =b 2 (t,X t )dt+σ(t,X t ,Y t )dW t , when Y t >α. Depending on the level of Y t , dynamics of X t follows a different pattern. Change of dynamics reflects what we called regime switch in (1.1). Our objective is to find a triplet (X,Y,Z) that solves (3.1). Take acloser lookatthemodel. Withineach regime, coefficients oftheforward equation satisfy the requirement of boundness and Lipschitz continuity, which indicates that the forward-backward SDEs have a strong solution when in between switches. Intuitively, if one defines a function piece by piece with each piece being the strong solution of the forward-backward SDEs for the according regime, this functioncouldbeasolutionto(3.1). Butthefactis, theswitch timeisastochastic stopping time, which is not easy to predict and identify. In addition to that, two adjacent stopping times might be inseparable. As a result, the above suggested solution probably is unfeasible for our model. However, we have a feeling that there still should be a solution to (3.1). We adopt one of the popular approaches to solve coupled forward-backward SDEs: the Four-Step scheme proposed by Ma, Protter and Yong [33]. Recall the 16 Four-Step scheme. Suppose (X,Y,Z) is a solution to (3.1), and suppose X, Y satisfy the relationship that Y t =v(t,X t ), for some function v(·,·) to be determined. If this v is smooth enough, for instance v∈ C 1,2 , apply Ito’s formula on v(t,X t ) and one will get dY t =dv(t,X t ) = v t (t,X t )+ 1 2 v xx (t,X t )σ 2 (t,X t ,v(t,X t ))+v x (t,X t )b(t,X t ,v(t,X t )) dt +v x (t,X t )σ(t,X t ,v(t,X t ))dW t . On the other hand, Y satisfies the SDE in (3.1): dY t =h(t,X t ,Y t )ds+Z t dW t . Comparing the two equations, it should only be true that the following holds:                  v t (t,X t )+ 1 2 v xx (t,X t )σ 2 (t,X t ,v(t,X t ))+v x (t,X t )b(t,X t ,v(t,X t )) =h(t,X t ,v(t,X t )), v(T,X T ) =g(X T ). (3.2) And so, solving the forward-backward SDEs is now turned into finding a solution to a quasi-linear parabolic PDE. Once solution to the above PDE was found, one 17 could exploit the relationship that Y t =v(t,X t ) and substitute Y t with v(t,X t ) in the forward equation, which leads to a decoupled forward-backward SDEs. Con- sequently, X and Y could be solved individually. With these heuristic arguments in mind, we are set to find the solution to(3.1) by searching for the function v that solves (3.2). 3.2 Solving the PDE: Approximation and Con- vergence Wefirsttransform(3.2)intoamoreoftenseenforwardequationbylettingu(t,x)= v(T −t,x). The first target to tackle is to find a solution to the following PDE:        u t − 1 2 σ 2 (u)u xx −b(u)u x +h(u)=0, u(0,x)=g(x). (3.3) Note that we abuse the notation a bit in the above formula to reduce the length of the formula. We emphasize the dependence of the coefficients in u, but in fact we have f(u)=f(T −t,x,u(t,x)), for f =b,σ,h. Equation(3.3),calleditPDE(0),isnowatypicalCauchyproblemofaquasi-linear equation. 18 3.2.1 Approximating PDE(0) Notice that in (3.3), for fixed t and x, the mapping b(T −t,x,·) :y∈R→b(T −t,x,y) is not continuous. Classical PDE theory requires the coefficients of the equation to be continuous. To get around this issue, let m ǫ (x) =          a ǫ exp ǫ 2 x 2 −ǫ 2 , |x| <ǫ, 0, |x|≥ǫ, and b ǫ (x,y)= Z b(x,z)m ǫ (y−z)dz. We approximate b(x,y) with its standard mollification b ǫ (x,y) and turn to study the following PDE, calling it PDE(ǫ):        u t − 1 2 σ 2 (u)u xx −b ǫ (u)u x +h(u)= 0, u(0,x)=g(x). (3.4) With regard to PDE(ǫ), we introduce the following theorem. 19 Theorem 3.1. Under the assumption (H1)-(H4), there exists a classical solution to PDE(ǫ). Further more, the solution is bounded in R T ; for any bounded Q T ⊂ R T , its derivatives are also bounded, and the bounds are independent of ǫ. To show the above claim, we state another theorem which is essentially a sim- plified version of Theorem V-8.1 from Ladyzenskaja et al [30]. Theorem 3.2. Consider the following second order quasi-linear PDE:          u t − n X i=1 ∂ ∂x i a i (t,x,u,D x u)+a 0 (t,x,u,D x u)= 0, (t,y)∈ (0,T]×R n u(0,x)= Φ(x). x∈R n (3.5) Suppose that the following conditions hold: (1) The function Φ(x) is smooth inR n , and Φ(x) is bounded; (2) For any t∈ (0,T] and arbitrary x, u, and p, the following inequalities hold: X i,j ∂a i (t,x,u,p) ∂p j ξ i ξ j ≥ 0, (3.6) u n a 0 (t,x,u,p)− X i ∂a i ∂u p i + ∂a i ∂x i o p=0 ≥−b 1 u 2 −b 2 , (3.7) for some constants b 1 ,b 2 ≥ 0; (3) For any |x| ≤ N, |u| ≤ M N , and i,j = 1,··· ,n, the function a 0 (t,x,u,p) a i (t,x,u,p) are continuous; a i (t,x,u,p) are differentiable with respect to x,u 20 and p; and there exists constants 0 <ν < μ, such that the following inequal- ities are fulfilled: ν|ξ| 2 ≤ X i,j ∂a i (t,x,u,p) ∂p j ξ i ξ j ≤ μ|ξ| 2 , ξ∈R n (3.8) and X i |a i |+ ∂a i ∂u (1+|p|)+ X i,j ∂a i ∂x j +|a 0 |≤μ(1+|p|) 2 ; (3.9) (4) For any |x| ≤ N, |u| ≤ M N , and |p| ≤ M ′ N , the functions a 0 , a i and the partial derivatives ∂a i ∂x j , ∂a i ∂u , ∂a i ∂p j are all Lipschitz continuous with respect to (x,u,p). Then the Cauchy problem (3.5) has at least one classical solution u(x,t) in R T . Furthermore, the solution is bounded in R T and its derivatives are also bounded for any bounded Q T ∈R T . Proof. First of all, rewrite PDE(ǫ) in the divergence form, and we have        u t − ∂ ∂x n 1 2 σ 2 u x o +u x σσ x +σσ u u x −b ǫ +h = 0, u(0,x)=g(x). (3.10) 21 Note that we omit the independent variables to simplify the notation. Compare (3.10) to (3.5), we define a 1 (t,x,u,p)= 1 2 σ 2 (T −t,x,u)p, a 0 (t,x,u,p)=[σ(T −t,x,u)σ x (T −t,x,u)−b ǫ (T −t,x,u)]p +σ(T −t,x,u)σ u (T −t,x,u)p 2 +h(T −t,x,u). To show that PDE(ǫ) has a solution, one just need to check thata 1 and a 0 defined as above satisfy all the conditions stated in Theorem 3.2. (1) We first check condition (2). For arbitrary x, u, and p, ∂a 1 (t,x,u,p) ∂p ξ 2 = 1 2 σ 2 ξ 2 ≥ 0. Inequality (3.6) clearly holds. Also, u n a 0 (t,x,u,p)− ∂a 1 ∂u p+ ∂a 1 ∂x o p=0 =u n [σσ x +σσ u p−b ǫ ]p+h−σσ u p 2 −σσ x p o p=0 =uh. 22 By (H3), h is bounded. So there exists some constant c ≥ 0, such that |h|≤ c. Take b 1 ≥ c 2 /4 and b 2 ≥ c 2 /4 as an example, one can easily check that b 1 u 2 +uh+b 2 ≥ 0. Hence (3.7) holds. (2) Under the assumptions (H1) - (H4), it’s trival that a 0 (t,x,u,p), a 1 (t,x,u,p) are continuous; and a 1 (t,x,u,p) are differentiable with respect to x, u and p. Since ∂a 1 (t,x,u,p) ∂p = 1 2 σ 2 , by (H2), one could find the constants μ≥ν > 0, such that (3.8) is true. For instance, take μ = ¯ σ and ν =σ. As for (3.9), |a 1 |+ ∂a 1 ∂u (1+|p|)+ ∂a 1 ∂x +|a 0 | ≤ 1 2 σ 2 |p|+|σσ u p| (1+|p|)+2|σσ x p|+|σσ u ||p| 2 +|b ǫ p|+|h| ≤ 1 2 σ 2 +2|σσ u | |p| 2 + 1 2 σ 2 +|σσ u |+2|σσ x |+|b ǫ | |p|+|h| ≤c 1 |p 2 |+c 2 |p|+c 3 . Thanks to the assumption that b ǫ , σ, h, σ x and σ u are all bounded. Here, c 1 , c 2 and c 3 are some generic non-negative constants, which could be varied from line to line. Thus, as long as we take μ large enough, (3.9) holds. 23 (3) Consider a 1 (t,x,u,p), we have a 1 = 1 2 σ 2 p, ∂a 1 ∂x =σσ x p, ∂a 1 ∂u =σσ u p, ∂a 1 ∂p = 1 2 σ 2 . For (x 1 ,u 1 ,p 1 )6=(x 2 ,u 2 ,p 2 ), |a 1 (t,x 1 ,u 1 ,p 1 )−a 1 (t,x 2 ,u 2 ,p 2 )| ≤|p 1 ||σ(x 1 ,u 1 )+σ(x 2 ,u 2 )||σ(x 1 ,u 1 )−σ(x 2 ,u 2 )|+|σ 2 (x 2 ,u 2 )||p 1 −p 2 | ≤c 1 |σ(x 1 ,u 1 )−σ(x 2 ,u 2 )|+c 2 |p 1 −p 2 |. And ∂ ∂x a 1 (t,x 1 ,u 1 ,p 1 )− ∂ ∂x a 1 (t,x 2 ,u 2 ,p 2 ) ≤|σ x (x 1 ,u 1 )p 2 ||σ(x 1 ,u 1 )−σ(x 2 ,u 2 )|+|σ(x 2 ,u 2 )p 1 ||σ x (x 1 ,u 1 )−σ x (x 2 ,u 2 )| +|σ(x 2 ,u 2 )σ x (x 2 ,u 2 )||p 1 −p 2 | ≤c 1 |σ(x 1 ,u 1 )−σ(x 2 ,u 2 )|+c 2 |σ x (x 1 ,u 1 )−σ x (x 2 ,u 2 )|+c 3 |p 1 −p 2 |. Same argument could be applied to ∂a 1 ∂u and ∂a 1 ∂p .One can easily see that a 1 , and all its derivatives are Lipschitz continuous with respect to (t,x,u,p). 24 Similarly for a 0 (t,x,u,p), |a 0 (t,x 1 ,u 1 ,p 1 )−a 0 (t,x 2 ,u 2 ,p 2 )| ≤|σ(x 1 ,u 1 )σ x (x 1 ,u 1 )p 1 −σ(x 2 ,u 2 )σ x (x 2 ,u 2 )p 2 | +|σ(x 1 ,u 1 )σ u (x 1 ,u 1 )p 2 1 −σ(x 2 ,u 2 )σ u (x 2 ,u 2 )p 2 2 | +|b ǫ (x 1 ,u 1 )p 1 −b ǫ (x 2 ,u 2 )p 2 |+|h(x 1 ,u 1 )−h(x 2 ,u 2 )| ≤c 1 |σ(x 1 ,u 1 )−σ(x 2 ,u 2 )|+c 2 |σ x (x 1 ,u 1 )−σ x (x 2 ,u 2 )| +c 3 |σ u (x 1 ,u 1 )−σ u (x 2 ,u 2 )|+c 4 |p 1 −p 2 | +c 5 |b ǫ (x 1 ,u 1 )−b ǫ (x 2 ,u 2 )|+|h(x 1 ,u 1 )−h(x 2 ,u 2 )|. Notice that b ǫ is the mollifier of b, so it’s Lipschitz continuous. Because of the Lipschitz continuity assumptions for σ and h , a 0 (t,x,u,p) are indeed Lipschitz continuous with respect to (x,u,p). And it’s assumed that the function g(x) is bounded, and smooth. At last, one could see that, all the conditions needed in Theorem 3.2 are satisfied. Therefore, by Theorem 3.2, PDE(ǫ) has a classical solution under assumption (H1) - (H4); the solution is bounded in R T , and for any bounded Q T ⊂ R T , its derivatives are also bounded. Remark. In [30], estimates for the function and its derivatives are also given. And the bounds depend only on constants from the assumptions (H1) - (H4) , and 25 the requirement of the Theorem 3.2. More specifically, suppose u ǫ is a solution to PDE(ǫ),we have: |u ǫ |≤M(b 1 ,b 2 ,M g ), for (t,x)∈R T |u ǫ x |≤M 1 (M,μ,ν,M g ,Q T ), for (t,x)∈Q T |u ǫ t |≤M 2 (M,M 1 ,μ,ν,M L ,M g ,Q T ), for (t,x)∈Q T Here, b 1 , b 2 are constants from (3.7); M g is the bound for the function g and its derivative g x due to the assumption that g is bounded and smooth; μ and ν are constants from (3.8) and (3.9); Q T is a bounded domain in R T ; M L is the bound for the coefficients and their derivatives thanks to the assumptions of uniformly Lipschitzcontinuity. Onecanseefromabove,bothM,M 1 andM 2 areindependent of ǫ. Therefore, mollification does not change the bounds. We emphasis this point as one will see it is the foundation for the following discussion. 3.2.2 Convergence of u ǫ For a fixed ǫ, by Theorem 3.1, there exists a classical solution u ǫ which solves PDE(ǫ). Let{ǫ n } ∞ n=1 be a sequence such that lim n→∞ ǫ n = 0. For each ǫ n , we can find a classical solution to PDE(ǫ n ). And thus a series of functions is obtained u ǫ 1 (t,x),u ǫ 2 (t,x),u ǫ 3 (t,x),··· . 26 As ǫ goes to zero, PDE(ǫ) converges to PDE(0). If there exists a limit for the series{u ǫn }, this limit couldpossibly beasolution to PDE(0). Followingthislead, we start by finding a limit for the series. The famous Arzela-Ascoli theorem serves as our main device here. Theorem 3.3 (Arzela-Ascoli). If a sequence {f n } ∞ 1 in C(X) (X is a compact metric space) is uniformly bounded and equicontinuous, then it has a uniformly convergent subsequence. Denote Q n T = [−n,n]×[0,T], u n (t,x) =u ǫn (t,x). In Q 1 T = [−1,1]×[0,T], as we pointed out above, |u n |≤ M (abuse of notation). Thus, {u n } is uniformly bounded in Q 1 T . Meanwhile, notice that the derivative ∂ ∂x u n is also bounded by M 1 (Q 1 T ), which implies {u n } is equicontinuous in Q 1 T as well. By Arzela-Ascoli theorem, there exists a uniformly convergent subsequence {u n k } ∞ 1 . Because{u n k } ∞ 1 is a subsequence, derivative of the subsequence ∂ ∂x u n k is also bounded by the same constant M 1 (Q 1 T ). Meanwhile, u n k is a classical solution satisfying PDE(ǫ n k ). Provided that ∂ ∂x u n k and ∂ ∂t u n k are both bounded, ∂ ∂x 2 u n k is bounded as well. Using the Arzela-Ascoli theorem once again, we could find yet another subsequence {u 1,n } ∞ 1 of{u n k } ∞ k=1 such that the{ ∂ ∂x u 1,n } ∞ 1 also converges. Next, consider Q 2 T = [−2,2]× [0,T]. In Q 2 T the sequence {u 1,n } ∞ 1 are again bounded by M, and the derivatives are now bounded by a new constant M 1 (Q 2 T ). 27 Usingthesameargumentasabove,thereexistsasubsequence{u 2,n } ∞ 1 of{u 1,n } ∞ 1 in Q 2 T such that{u 2,n } ∞ 1 converges uniformly, andthe derivative{ ∂ ∂x u 2,n } ∞ 1 converges aswell. Notethat{u 2,n } ∞ 1 isasubsequence of{u 1,n } ∞ 1 ,so{u 2,n } ∞ 1 tooisuniformly convergent in Q 1 T . Proceeding in this fashion, we obtain acollection ofsubsequence ofthe original sequence: u 11 ,u 12 ,u 13 ,··· u 21 ,u 22 ,u 23 ,··· u 31 ,u 32 ,u 33 ,··· ··············· where the sequence in the k th row are uniformly convergent in Q k T with{ ∂ ∂x u k,n } ∞ 1 also being convergent. At the same time, each row is a subsequence of the one above it. Finally, consider the diagonal sequence {u n,n } ∞ 1 . From the analysis above, one can see that {u n,n } ∞ 1 is a subsequence of the original sequence {u n } ∞ 1 , and it converges uniformly in R T . Denote ˆ u(t,x) = lim n→∞ u n,n (t,x) 28 as the limit function. The limit of a uniformly convergent sequence of continuous functions remains to be continuous. Thus, ˆ u(t,x) is continuous. And as one could see above, we also have ˆ u x (t,x) = lim n→∞ ∂ ∂x u n,n (t,x). Now that the limit function ˆ u is found, one remaining question awaits to be answered. Would this ˆ u be a solution to PDE(0)? Since b ǫ (t,x,y) only converges to b(t,x,y) almost everywhere, chance that ˆ u being a classical solution is rather limited. One might have better luck finding ˆ u as a solution in a weaker sense. The first possibility we considered is ˆ u being a viscosity solution. Given that u ǫ is a classical solution, it’s automatically a viscosity solution. If PDE(ǫ) converges to PDE(0) uniformly, by the stability of viscosity solution, ˆ u is a viscosity solution as well. Unfortunately, once again because of the discontinuity, PDE(ǫ) does not converge to PDE(0) uniformly, as b ǫ (t,x,α) does not converge to b(t,x,α). Attempt to show viscosity solution ends up being a disappointment. After lots of efforts, it turns out that we have the following theorem. Theorem 3.4. ˆ u(t,x) is a distributional solution to PDE(0). Proof. Rewrite PDE(0) in divergence form: u t − ∂ ∂x n 1 2 σ 2 (u)u x o +u x σ(u)σ x (u)+σ(u)σ u (u)u x −b(u) +h(u)= 0. (3.11) 29 We want to prove that Z R ˆ u(r,x)ϕ(r,x)dx t s + Z (s,t)×R h − ˆ uϕ r + 1 2 σ 2 (ˆ u)ˆ u x ϕ x + ˆ u x σ(ˆ u)σ x (ˆ u)+σ(ˆ u)σ u (ˆ u)ˆ u x −b(ˆ u) ϕ+h(ˆ u)ϕ i dxdr =0, (3.12) for any ϕ∈C ∞ ((0,T)×R) such that ϕ(t,·)∈C ∞ 0 (R) for all t∈ (0,T]. First, let’s consider the diagonal sequence {u n,n } ∞ 1 obtained from above. u k,k (t,x) is a classical solution to PDE(ǫ k,k ) in R T , thus u k t − 1 2 (σ k ) 2 u k xx −b k (x,u k )u k x +h k = 0. Here we simplified the notation by denoting u k =u k,k (t,x), b k (x,y)=b ǫ k,k (T −t,x,y), σ k =σ(T −t,x,u k ), h k =h(T −t,x,u k ). For ϕ∈ C ∞ ((0,T)×R), multiply ϕ to both sides of the above equation, and use integration by part, we get Z R u k (r,x)ϕ(r,x)dx t s + Z (s,t)×R h −u k ϕ r + 1 2 (σ k ) 2 u k x ϕ x +u k x σ k σ k x +σ k σ k u u k x ϕ−u k x b k (x,u k )ϕ+h k ϕ i dxdr =0. (3.13) 30 Notethatu k → ˆ u,u k x → ˆ u x , andσ,σ x , σ u ,h areallbounded continuous functions, one could easily check that Z R u k (r,x)ϕ(r,x)dx t s → Z R ˆ u(r,x)ϕ(r,x)dx t s , Z (s,t)×R u k ϕ r dxdr→ Z (s,t)×R ˆ uϕ r dxdr, Z (s,t)×R 1 2 (σ k ) 2 u k x ϕ x dxdr→ Z (s,t)×R 1 2 σ 2 (ˆ u)ˆ u x ϕ x dxdr, Z (s,t)×R u k x σ k σ k x +σ k σ k u u k x ϕdxdr→ Z (s,t)×R ˆ u x σ(ˆ u)σ x (ˆ u)+σ(ˆ u)σ u (ˆ u)ˆ u x ϕdxdr, Z (s,t)×R h k ϕdxdr→ Z (s,t)×R h(ˆ u)ϕdxdr, as k goes to infinity. There is one troubled term left, R R u k x b k (x,u k )ϕdx . Define B k (x,z) = Z z 0 b k (x,y)dy, B(x,z) = Z z 0 b(x,y)dy. 31 Clearly, B k and B are continuous, and B k converges to B point-wise. Consider B k (x,u k ) and B(x,ˆ u). Since b k , u k are bounded by constants M b , M respectively, we have B k (x,u k )−B(x,ˆ u) = Z u k 0 b k (x,y)dy− Z ˆ u 0 b(x,y)dy ≤ Z u k 0 |b k (x,y)−b(x,y)|dy+ Z ˆ u u k b(x,y)dy ≤ Z M 0 |b k (x,y)−b(x,y)|dy+ Z ˆ u u k M b dy . Becauseb k converges tobalmosteverywhere, byDominatedConvergence theorem, we have Z M 0 |b k (x,y)−b(x,y)|dy→ 0. At the same time, since u k → ˆ u uniformly, Z ˆ u u k M b dy → 0 Therefore, B k (x,u k )−B(x,ˆ u) goesto0ask approaches infinity. Thus, B k (x,u k ) converges to B(x,ˆ u). Using a similar argument, we can show that Z R Z u k 0 b k x (x,y)ϕdydx→ Z R Z ˆ u 0 b x (x,y)ϕdydx. 32 Remember that b(t,x,y) is uniformly Lipschitz continuous in x, so b k x and b x are both bounded. Notice that ∂ ∂x B k (x,u k )= Z u k 0 b k x (x,y)dy+b k (x,u k )u k x , ∂ ∂x B(x,ˆ u)= Z ˆ u 0 b x (x,y)dy+b(x,ˆ u)ˆ u x . Applying integration by part, one will get Z R u k x b k (x,u k )ϕdx = Z R h ∂ ∂x B k (x,u k )− Z u k 0 b k x (x,y)dy i ϕdx =− Z R B k (x,u k )ϕ x dx− Z R Z u k 0 b k x (x,y)ϕdydx →− Z R B(x,ˆ u)ϕ x dx− Z R Z ˆ u 0 b x (x,y)ϕdydx = Z R h ∂ ∂x B(x,ˆ u)− Z ˆ u 0 b x (x,y)dy i ϕdx = Z R ˆ u x b(x,ˆ u)ϕdx. Finally, put all the terms together, and let k→∞ in (3.13), we have Z R ˆ u(r,x)ϕ(r,x)dx t s + Z (s,t)×R h − ˆ uϕ r + 1 2 σ 2 (ˆ u)ˆ u x ϕ x + ˆ u x σ(ˆ u)σ x (ˆ u)+σ(ˆ u)σ u (ˆ u)ˆ u x −b(ˆ u) ϕ+h(ˆ u)ϕ i dxdr = 0. The proof is complete. 33 In the above prove, we are not claiming the convergence of b ǫ t,x,u ǫ to b t,x,ˆ u , which is not true as we mentioned before b ǫ t,x,α 9 b t,x,α . Instead, thanks to the extra term u x that is multiplied to b(ˆ u), we showed the convergence of u ǫ x b ǫ t,x,u ǫ to ˆ u x b t,x,ˆ u . In fact, for points in the set {x : ˆ u(x) 6= α}, convergence of u ǫ x b ǫ t,x,u ǫ is without a doubt. Additionally, the set {x : ˆ u(x) = α and ˆ u x (x) 6= 0} has a measure 0 since all the points are isolated. All these lead to the convergence and consequently the result above. 3.3 Solving the FBSDEs: A Weak Solution As shown in the above discussion, the limit function ˆ u is a distributional solution to PDE(0). But this is far from enough. In the original Four-Step scheme, one would hope to find the solution to the PDE to be at least in C 1,2 , so that Ito’s formula would be applicable. As of now, we only have ˆ u(t,x) being continuous and once differentiable in x. We need ˆ u with higher regularity. In the following, we study the regularity of ˆ u with the help of some results from PDE literatures. 3.3.1 Regularity of ˆ u The following definitions and theorems are enlisted to “raise” the regularity of ˆ u, so to speak. 34 Definition3.5. Afunctionu∈W 1,2 p (Q T )iscalledasolutiontothenon-divergence type equation u t −a(t,x)u xx +b(t,x)u x +c(t,x)u =f (3.14) if it satisfies the equation on Q T as well as the boundary equations. Definition 3.6. A function u is in W 1,p ∗ (Q T ), 1 < p <∞, if u∈ W 1,0 p (Q T ) and there exist functions F ∈L p (Q T ,R n ), g∈L p (Q T ) such that u t =divF −g in Q T in the sense that for all ϕ∈C ∞ 0 (Q T ) with ϕ= 0 for t =T, Z Q T uϕ t dxdt= Z Q T (F∇ϕ+gϕ)dxdt. Furthermore, the norm is defined by kuk W 1,p ∗ (Q T ) =kuk W 1,0 p (Q T ) +inf ( Z Q T (|F| p +|g| p )dxdt 1/p ) . Detailed definition and discussion of W 1,p ∗ (Q T ) could be found in Byun and Wang [11]. 35 Definition 3.7. A function u in W 1,p ∗ (Q T ) is called a solution to the divergence type equation u t − ∂ ∂x (a(t,x)u x )+b(t,x)u x +c(t,x)u =f, if for any ϕ∈C ∞ ((0,T)×R) such that ϕ(t,·)∈C ∞ 0 (R) for all t∈ [0,T], Z Ω uϕdx t s + Z (s,t)×Ω −uϕ r +a(t,x)u x ϕ x +b(t,x)u x ϕ+c(t,x)uϕ dxdr. = Z (s,t)×Ω fϕdxdr. Theorem below from Kim and Krylov [27] is the cornerstone in our process of finding a solution to the forward-backward SDEs. Theorem 3.8 (Kim & Krylov). Suppose the coefficients a, b, and c of (3.14) are measurablefunctionsdefinedonΩ T =(0,T)×R, andthere existspositiveconstants δ∈ (0,1] and K such that δ≤|a|≤δ −1 , |b|≤K, |c|≤K. Then for any f ∈L 2 (Ω T ), there exists a unique u∈W 1,2 2 (Ω T ) satisfying (3.14) in (0,T)×R. In addition, there is a constant N =N(δ,K,T) such that kuk W 1,2 2 (Ω T ) ≤Nkfk L 2(Ω T ). 36 With the help of these results, we introduce the following lemma. Lemma3.9. If h is in L 2 ((0,T)×R), the limit function ˆ u(t,x) is in W 1,2 2 ((0,T)× R). Proof. Consider the following linear non-divergence type equation: w t (t,x)− 1 2 ˜ σ 2 (t,x)w xx (t,x)− ˜ b(t,x)w x (t,x) =− ˜ h(t,x), (3.15) where ˜ f(t,x) =f(T −t,x,ˆ u(t,x)), for f =σ,b,h. Noticethatin(3.15),coefficient ˜ b(t,x)wasusedinsteadofb(T−t,x,w(t,x)). With ˆ u already been found, we made the substitution for the purpose of converting the original quasi-linear PDE into a linear one. The fact is, linear PDE has been extensively studied and there are more results are available to use. Giving the assumptions (H1) - (H4), it’s clear that the coefficient ˜ σ, and ˜ b of (3.15) meet the requirements stated in Theorem 3.8. At the same time, it’s true that ˜ h∈L 2 ((0,T)×R). ByTheorem3.8,thereexistsauniquew∈W 1,2 2 ((0,T)×R) satisfying (3.15). Thus, for all 0≤s≤t≤T and ϕ∈ C ∞ ((0,T)×R), we have Z R wϕdx t s + Z (s,t)×R h w r ϕ− 1 2 ˜ σ 2 w xx ϕ− ˜ bw x ϕ dxdr =− Z (s,t)×R ˜ hϕdxdr. 37 Using integration by part, one will get Z R wϕdx t s + Z (s,t)×R h −wϕ r + 1 2 ˜ σ 2 w x ϕ x + ˜ σσ x (ˆ u)+˜ σσ u (ˆ u)ˆ u x − ˜ b w x ϕ dxdr =− Z (s,t)×R ˜ hϕdxdr. (3.16) Equation above implies that w is also a solution to the following divergence type linear equation on (0,T)×R: w t − ∂ ∂x n 1 2 ˜ σ 2 w x o + h ˜ σσ x (ˆ u)+˜ σσ u (ˆ u)ˆ u x − ˜ b i w x =− ˜ h. (3.17) On the other hand, ˆ u is a distributional solution. From the proof of Theorem 3.4, Z R ˆ uϕdx t s + Z (s,t)×R h − ˆ uϕ r + 1 2 ˜ σ 2 ˆ u x ϕ x + ˜ σσ x (ˆ u)+˜ σσ u (ˆ u)ˆ u x − ˜ b ˆ u x ϕ dxdr =− Z (s,t)×R ˜ hϕdxdr. ˆ u evidently solves the PDE in the weak sense. If one can show ˆ u is in W 1,p ∗ (Q T ), then ˆ u would be a solution to the divergence type equation. In fact, by choosing ϕ ∈ C ∞ 0 (Q T ) , the first term of the equation above goes away. Rearrange the equation and the rest becomes Z Q T ˆ uϕ r dxdr = Z Q T h 1 2 σ 2 ˆ u x ϕ x + ˆ u x σσ x +(ˆ u x ) 2 σσ u −bˆ u x +h ϕ i dxdr. 38 Denote F = 1 2 σ 2 ˆ u x , g = ˆ u x σσ x +(ˆ u x ) 2 σσ u −bˆ u x +h. Because of the boundedness of all the coefficients, one can easily check that F,g are indeed in L 2 . Hence, ˆ u is in W 1,2 ∗ (Q T ). And by definition, ˆ u(t,x) is a solution to the divergence type equation (3.11). Finally, we have w and ˆ u are both solutions to the same divergence type linear equation (3.17). However, solution to the divergence type equation is unique. We conclusion that ˆ u =w, which in turn implies ˆ u∈W 1,2 2 ((0,T)×R). Thekeyoftheproofisthelinearizationoftheinitialproblem,whichreducesthe difficulty by a great amount. After that, it’s just a matter of utilizing Theorem 3.8. And thanks to the uniqueness of solution to the divergence type equation, regularity of ˆ u is “raised”. 39 3.3.2 Solution to the FBSDEs Proceeding with the plan based on the Four-Step scheme, the next step is to use ˆ u to decouple the forward-backward SDEs, and solve them individually. Notice that, ˆ u is in W 1,2 2 , still not in C 1,2 . The traditional Ito’s formula is not applicable inthis case. Nonetheless, we were able tocarryonwith the assistance ofthebelow theorem from Krylov [28], the usually called Ito-Krylov formula. Theorem 3.10. Consider the process X t =x 0 + Z t 0 b r dr+ Z t 0 σ r dW r . Let Q be a bounded region in R T and x 0 ∈ Q be fixed. Let τ Q be the first exit time of the process (X) from the the region Q, let τ be some Markov time such that τ ≤τ Q . Suppose that there exist constants K,δ >0 such that |σ t |+|b t |≤K, |σ|≥δ for all t< τ. Then for any v∈W 1,2 (Q) and t≥ 0, v(τ,X τ )−v(t,X t ) = Z τ t v t (r,X r )+ 1 2 σ 2 r v xx (r,X r )+b r v x (r,X r ) dr+ Z τ t σ r v x (r,X r )dW r . Remark. The above theorem is essentially Theorem 2.10.1 in Krylov [28], we simplified the statement for our own use. 40 The Ito-Krylov formula is considered to be a generalized version of the Ito’s formula. It’s particularly useful when coefficients of the SDE are not so smooth. And because of it, we have the following theorem. Theorem 3.11. There exist a weak solution to the forward-backward SDEs (3.1). Proof. Step 1, we look into the the forward SDE:        dX t = ¯ b(t,X t )dt+¯ σ(t,X t )dW t , X 0 =x, (3.18) where ¯ b(t,x) =b(t,x,v(t,x)), ¯ σ(t,x) =σ(t,x,v(t,x)), v(t,x) = ˆ u(T −t,x). Rememberthat(3.1)istransformedintoaforwardequationbysettingu(t,x)= v(T −t,x). One should go back using v to decouple and solve the SDE. Because of the discontinuity in ¯ b, finding a strong solution might be unachievable at the moment. As a result, we turn to look for a weak solution to (3.18). Since thesource ofdifficulty lies inthecoefficient ofthedriftterm, ourstrategy is to use the Girsanov theorem to avoid dealing with ¯ b directly. 41 Consider first the following SDE:        dX t = ¯ σ(t,X t )dW t , X 0 =x. It’s a general fact that for SDEs with bounded and Lipschitz continuous coeffi- cients, a unique strong solution exists. In this case, |¯ σ(t,x)− ¯ σ(t,y)|=|σ(x,v(t,x))−σ(y,v(t,y))| ≤k[|x−y|+|v(t,x)−v(t,y)|], (3.19) where k is some generic constant. The above inequality is true due to the assump- tion that σ(x,y) are Lipschitz continuous in both x and y. Meanwhile, by the mean value theorem, there exists a ξ such that v(t,x)−v(t,y)=v x (t,ξ)(x−y). Noticethatv∈ W 1,2 2 ((0,T)×R),sincev(t,x) = ˆ u(T−t,x)and ˆ u∈W 1,2 2 ((0,T)×R) as shown in Lemma 3.9. There exists a constants k v ≥ 0 such that |v x | ≤ k v in ((0,T)×R). Thus we have |v(t,x)−v(t,y)|≤ k v |x−y|. 42 Plug in (3.19), we get |¯ σ(t,x)−¯ σ(t,y)|≤ k|x−y|, whichimplies ¯ σ(t,x)isindeedLipschitzcontinuous. Therefore, thereexitsastrong solution X t , such that X t =x+ Z t 0 ¯ σ(s,X s )dW s (3.20) holds almost surely, and this X t is unique. To proceed, define the function θ(t,x) as θ(t,x) = ¯ b(t,x) ¯ σ(t,x) . Since ¯ b is bounded and ¯ σ is bounded from below, it’s clear that θ is bounded as well. And so one could easily check that M t , exp n Z t 0 θ(s,X s )dW s − 1 2 Z t 0 |θ(s,X s )| 2 ds o is a martingale under the initial probability measure P. Define a new probability measure Q by dQ dP =M T . 43 By the Girsanov theorem, ¯ W t =W t − Z t 0 θ(s,X s )ds (3.21) is now a Brownian motion under the new probability space (Ω,F,Q). Putting (3.20) and (3.21) together, we have X t −x = Z t 0 ¯ σ(s,X s )dW s = Z t 0 ¯ b(s,X s )ds+ Z t 0 ¯ σ(s,X s )d ¯ W s Recall the definition of weak solution, one could easily see that (X, ¯ W) defined asin(3.20)and(3.21)alongwith(Ω,F,Q)isaweaksolutionto(3.18). Moreover, X t does not explode in finite time due to the boundedness of ¯ b and ¯ σ. With the solution of the forward equation found, we move on to the next step. Step 2, we solve the backward SDE. Let Y t ,v(t,X t ), Z t ,σ(t,X t ,v(t,X t ))v x (t,X t ). (3.22) We are going to show that (Y,Z) is a solution to the backward SDE. Remember that ˆ u(t,x) =w(t,x)∈W 1,2 2 ((0,T)×R), and w is a solution to the nondivergence type linear equation (3.15). So we have ˆ u t (t,x)− 1 2 ˜ σ 2 (t,x)ˆ u xx (t,x)− ˜ b(t,x)ˆ u x (t,x) =− ˜ h(t,x), 44 or equivalently v t (t,x)+ 1 2 ¯ σ 2 (t,x)v xx (t,x)+ ¯ b(t,x)v x (t,x) =h(x,v(t,x)). For any 0≤t≤T, we have v(t,X t )∈W 1,2 2 (0,T)×R . Let τ n =inf{t :|X t |≥n} be a stopping time. Clearly, v(t,X t )∈ W 1,2 2 (0,τ n ∧T)×(−n,n) . Since (X t , ¯ W t )isaweak solution to(3.18)in(Ω,F,Q), by the Ito-Krylovformula, v(τ n ∧T,X τn∧T )−v(t,X t ) = Z τn∧T t h v s + ¯ σ 2 2 v xx + ¯ bv x i ds+ Z τn∧T t ¯ σv x d ¯ W s = Z τn∧T t h(s,X s ,v(s,X s ))ds+ Z τn∧T t σ(s,X s ,v(s,X s ))v x (s,X s )d ¯ W s . (3.23) Because X t does not explode in finite time, lim n→∞ τ n ∧T =T. 45 Also notice that h, v x and σ are all bounded. Let n go to infinity in the above equation (3.23), by dominated convergence theorem, we have v(T,X T )−v(t,X t ) = Z T t h(s,X s ,v(s,X s ))ds+ Z T t σ(s,X s ,v(s,X s ))v x (s,X s )d ¯ W s . Substituting (Y t , Z t ) into the equation above, together with the boundary con- ditionthatv(T,X T ) =g(X T ), onecouldeasily see that(Y t ,Z t )asdefined in(3.22) is indeed the solution to the backward equation. At last, putting the two parts together, we have          X t = Z t 0 b(s,Xs,Y s )ds+ Z t 0 σ(s,X s ,Y s )d ¯ W s , Y t =g(X T )− Z T t h(s,X s ,Y s )ds+ Z T t Z s d ¯ W s . What’s more, one can easily check that the following holds E Q n |b t |+|σ t | 2 +|h t |+|Z t | 2 o =E P n M T |b t |+|σ t | 2 +|h t |+|Z t | 2 o <∞. Therefore, (X,Y,Z) along with (Ω,F,Q) is a weak solution to (3.1). In this model, we only allow the drift of the forward equation to have jumps; coefficient of the volatility term remains to be nice and smooth. Reasons are the following. As one could have seen through the analysis above, we rely heavily on 46 thesolutiontothePDE(3.2)tosolvetheforward-backwardSDEs. Andit’sclearly shown that, the coefficient of the volatility term ends up being the coefficient of the second order term in PDE (3.2). A PDE with discontinuous second order term coefficient is much more difficult to deal with. A solution to the PDE might not even exists. Also, existence of solution to the forward SDE depends on the Lipschitz continuity of the remaining coefficient after “removing” the drift term. ForaSDEwithboththedriftandvolatilitybeingdiscontinuous, eventheexistence of a weak solution is hard to guarantee. 47 Chapter 4 Regime Switching Term Structure Model Given the discussion in Chapter 3, we now return to our proposed term structure model and see how the previous discussion applies. 4.1 Model Formulation Consider the following model:          X t =x+ Z t 0 [b(Y s )−βX s ]ds+ Z t 0 σdW s , Y t =E n g(X T )+ Z T t 1−e ζ(Xs) Y s ds F t o . (4.1) Here X t =ln(r t ) represents the short rate, Y t is the long term treasury bond price. 48 For the short rate model X t , in order to capture the regime switch, we let b :R→{B 1 ,B 2 } be a piece-wise constant function given by b(y)=          B 1 , y≤α, B 2 , y > α, whereB 1 ,B 2 (B 1 6=B 2 )andαareknownconstants. β andσarealsokeptconstant inthemodel. Asamatteroffact,ourX t isaversionoftheBlack-Karasinskimodel, which originally could be described as: dlnr t =k(θ t −lnr t )dt+σdW t . The BK model is among one of the popular term structure models currently used in industrial practice. This model is chosen owing to its four important positive features: (1)incorporatingthemeanreversion behavior(thetendency torisewhen it’s below the θ t , and fall when it’s above), (2) guaranteeing non-negative rates (short rates generated are log-normally distributed), (3) the capability of fitting theinitialtermstructureoftheinterestrate,and(4)easyextensiontoothersource of risk as desired. In the BK model, k represents the mean reversion speed; the larger k is, the faster the short rate converges to θ; θ describes the long term equilibrium short rate level, or the average short rate in the long run to which the short rate tends to revert; and σ measures the volatility, we keep it constant for 49 the simplicity of the model, one could certainly replace it with a smooth function σ(t,x,y). Let us take a further look at θ. θ indicates the short rate equilibrium level in the long run, naturally one would expect that θ ought to be dependent on long termeconomicfactors, forinstancethelongtermbondprice(equivalently thelong rate). When there are substantial changes happened to the long rate, it suggests that the aggregate economy is experiencing some structural transformation, or what we call here entering into another regime. And as a result, the long run equilibrium θ changes accordingly. Furthermore, θ in the BK model is a time- dependent function. If we treat θ as a continuous function of the t, it makes θ computationally difficult to calibrate. On the other hand, if θ was treated as one simple constant, it loses the ability to reproduce the yield curve. In this study, we modified θ as a piece-wise constant function depending on long term bond price for the reason that: (1) the jump of θ captures the regime switch and (2) a piece-wise constant function reduces the calibration difficulty and at the same time maintains the yield-curve reproducing ability to some degree. Therefore we construct the short rate model as in (4.1). Once Y t crosses the predetermined threshold level α, the function b in the drift will make a jump, resulting a dynamic change for the short rate. 50 Asforthelongratemodel, remember thatthebondprice satisfies theexpected discounted value: Y t =E n Z T t exp − Z s t r u du ds F t o . By Ito’s formula, dY t =(r t Y t −1)dt+Z t dW t forsome processZ. Notethatr t =e Xt . Toensure the boundedness ofr t , we apply a cut-off function ζ to X t , where ζ(x)=|x|∧K for some predetermined constant K. And we also assume that the terminal condition is given by Y T = g(X T ). All these combined lead to the long-term bond price model in (4.1). Based on the study in the previous section, we will be looking for (X,Y) along with some probability space (Ω,F,Q) thatsolves (4.1). Note that(4.1) could also be setting up as:                  dX t = [b(Y t )−βX t ]dt+σdW t , dY t = e ζ(Xt) Y t −1 dt+Z t dW t , X 0 =x, Y T =g(X T ). (4.2) Compared to (3.1), one could see that (4.2) is not simply a special case of the generalmodel. Takethedriftoftheforwardequationasanexample, b(Y t )−βX t is not bounded. Some adjustments would need to be made in order to solve (4.2). 51 But essentially, we will be following the steps as in the discussion of the general model. 4.2 A weak solution to the FBSDE Consider the short rate model: dX t = [b(Y t )−βX t ]dt+σdW t . As we just mentioned, the drift term is not bounded. We resolve this obstacle by performing a variable change. Let S t =e βt X t , by the usual variation of constants formula, we have dS t =e βt b(Y t )dt+e βt σdW t . One can see thatthe coefficients in the above model are now bounded (T is given). Compared with X t , the S t model shares the same characteristics as the forward 52 equation in the general model. Also once S t is solved, we could easily back outX t . Therefore, we study the following model instead of (4.2):                  dS t =e βt b(Y t )dt+e βt σdW t , dY t =[h(S t )Y t −1]dt+Z t dW t , S 0 =s, Y T =g(e −βT S T ), (4.3) where h(t,x) =e ζ(e −βt x) . Clearly, h(t,x) is still bounded. Following the Four-Step scheme, together with the time change at the same time, the PDE associated with the FBSDEs (4.3) is given as followed:        u t − 1 2 e 2β(T−t) σ 2 u xx −e β(T−t) b(u)u x +h(T −t,x)u−1= 0, u(0,x)=g(e −βT x). (4.4) 4.2.1 Solving the PDE Similar approach as before, because of the discontinuity in the function b(y), we apply the standard mollification to b, and turn to study PDE(ǫ):        u t − 1 2 e 2β(T−t) σ 2 u xx −e β(T−t) b ǫ (u)u x +h(T −t,x)u−1 =0 u(0,x)=g(e −βT x). (4.5) Assume that g is bounded and smooth, one can check that coefficients in (4.5) satisfyalltheconditionsrequiredinTheorem3.1. Therefore,byTheorem3.1,there 53 exists a classical solution to (4.5) in R T . Further more, the solution is bounded in R T and its derivative in x is also bounded in any bounded Q T ⊂R T . For any ǫ, we could find a u ǫ which solves the according PDE(ǫ). Adopting the same technique stated in the previous chapter, we could obtain a uniform conver- gent subsequence {u n,n } ∞ 1 with their derivatives{ ∂ ∂x u n,n } ∞ 1 also being convergent. Again, let us denote ˆ u(t,x) = lim n→∞ u n,n (t,x) as the limit functions. And we also have ˆ u x (t,x) = lim n→∞ ∂ ∂x u n,n (t,x). Using the same argument as in the proof as of Theorem 3.4, we have the following results. Lemma 4.1. ˆ u is a distributional solution to PDE(0): u t − 1 2 e 2β(T−t) σ 2 u xx −e β(T−t) b(u)u x +h(T −t,x)u = 1. (4.6) Following the method used in the general model, the next step should be using the PDE argument to show that the ˆ u we obtained above is in W 1,2 2 ((0,T)×R). However in this case, the same argument is not entirely true any more. Let us take a look back at PDE(0). In the general model, right-hand side of PDE(0) is 54 a L 2 ((0,T)×R) function. And here, right-hand side of PDE(0) is the constant 1, which clearly does not belong to L 2 ((0,T)×R). This discrepancy weakened our result, and consequently we can now only claim the following. Lemma 4.2. The limit function ˆ u(t,x) is in W 1,2 2,loc ((0,T)×R). Proof. Because ofthetroublecaused by theright-handside, we can’tsimply apply theproofofLemma3.9toourcase. Oneremedytodealwiththetroubledconstant 1 is to multiply a cut-off function η to PDE(0), and localize the problem to a bounded domain. Then rest of the proof could follow the ideas of Lemma 3.9. Let R be a positive number. Consider first the following linear non-divergence type equation on (0,T)×B 2R :                  w t − 1 2 ˜ σ 2 (t)w xx − ˜ b(t,x)w x +h(T −t,x)w = ˜ f, w(t,x)| (0,T)×∂B 2R = 0, w(0,x)=g(e −βT x)η(x), (4.7) where ˜ b(x) =e β(T−t) b(ˆ u), ˜ σ(t) =e β(T−t) σ, ˜ f =η− ˜ σ 2 ˆ u x η x − ˜ σ 2 2 ˆ uη xx − ˜ bˆ uη x , and the cut-off function η(x) is taken as η(x)∈C ∞ 0 (B 2R ), with η(x) =1 on B R . 55 It’s easy to check that ˜ σ and ˜ b remain to be bounded. Meanwhile, since (1) both ˆ u and ˆ u x are bounded in [0,T]×B 2R , and (2) η(x) ∈ C ∞ 0 (B 2R ), one could easily find that ˜ f ∈ L 2 ((0,T)×B 2R ). Therefore, we have the coefficients of (4.7) are all bounded along with ˜ f ∈ L 2 . By Theorem 3.8, there exists a unique w ∈ W 1,2 2 ((0,T)×B 2R ) satisfying (4.7). Simply using integration by part, we have Z B 2R wϕdx t s + Z (s,t)×B 2R h −wϕ t + 1 2 ˜ σ 2 (t)w x ϕ x − ˜ b(t,x)w x ϕ+h(T −t,x)wϕ i dxdt = Z (s,t)×B 2R ˜ fϕdxdr, (4.8) for all 0 ≤ s ≤ t ≤ T and any ϕ ∈ C ∞ ((0,T)×R) such that ϕ(t,·) ∈ C ∞ 0 (R). (4.8) implies that the function w is also a solution to the following divergence type linear equation on (0,T)×B 2R :                  w t − ∂ ∂x 1 2 ˜ σ 2 (t)w x − ˜ b(t,x)w x +h(T −t,x)w = ˜ f, w(t,x)| (0,T)×∂B 2R = 0, w(0,x)=g(e −βT x)η(x). (4.9) Ontheotherhand,considerthefunction(ˆ uη). Weclaimthat,(ˆ uη)isasolution to the divergence type equation (4.9) as well. To show that, first we have (ηˆ u)ϕ t = ˆ u(ηϕ) t . 56 The equation above is true due to η is only a function in x. Second, using integra- tion by part along with the chain rule, one will get Z (s,t)×B 2R 1 2 ˜ σ 2 (ηˆ u) x ϕ x dxdr = Z (s,t)×B 2R 1 2 ˜ σ 2 ˆ u x (ηϕ) x − 1 2 ˜ σ 2 ˆ u x η x ϕ+ 1 2 ˜ σ 2 ˆ uη x ϕ x dxdr, and Z (s,t)×B 2R ˜ b(ηˆ u) x ϕdxdr = Z (s,t)×B 2R ˜ bˆ u x (ηϕ)+ ˜ bˆ uη x ϕ dxdr, for all 0≤ s≤ t≤ T. Once again, the independent variables are left out for the simplicity of the equations. Reordering the terms, we have Z (s,t)×B 2R −(ηˆ u)ϕ r + 1 2 ˜ σ 2 (ηˆ u) x ϕ x − ˜ b(ηˆ u) x ϕ+h(ηˆ u)ϕ dxdr = Z (s,t)×B 2R − ˆ u(ηϕ) r + 1 2 ˜ σ 2 ˆ u x (ηϕ) x − ˜ bˆ u x (ηϕ)+hˆ u(ηϕ) dxdr + Z (s,t)×B 2R − 1 2 ˜ σ 2 ˆ u x η x ϕ+ 1 2 ˜ σ 2 ˆ uη x ϕ x − ˜ bˆ uη x ϕ dxdr. Notethat(ηϕ)isinC ∞ ((0,T)×R). Andbecause ˆ uisadistributionalsolution to (4.6), by definition, Z (s,t)×B 2R − ˆ u(ηϕ) r + 1 2 ˜ σ 2 ˆ u x (ηϕ) x − ˜ bˆ u x (ηϕ)+hˆ u(ηϕ) dxdr =− Z B R ˆ u(ηϕ)dx t s + Z (s,t)×B R ηϕdxdr. 57 Meanwhile, using integration by part again, Z (s,t)×B R ˆ uη x ϕ x dxdr = Z (s,t)×B R − ˆ u x η x ϕ− ˆ uη xx ϕ dxdr. Thus we have Z (s,t)×B 2R − 1 2 ˜ σ 2 ˆ u x η x ϕ+ 1 2 ˜ σ 2 ˆ uη x ϕ x − ˜ bˆ uη x ϕ dxdr = Z (s,t)×B 2R − ˜ σ 2 ˆ u x η x ϕ− 1 2 ˜ σ 2 ˆ uη xx ϕ− ˜ bˆ uη x ϕ dxdr. Putting them all together, one would get Z B R ˆ u(ηϕ)dx t s + Z (s,t)×B 2R −(ηˆ u)ϕ r + 1 2 ˜ σ 2 (ηˆ u) x ϕ x − ˜ b(ηˆ u) x ϕ+h(ηˆ u)ϕ dxdr = Z (s,t)×B 2R ηϕ− ˜ σ 2 ˆ u x η x ϕ− 1 2 ˜ σ 2 ˆ uη xx ϕ− ˜ bˆ uη x ϕ dxdr = Z (s,t)×B 2R ˜ fϕdxdr. The formula above also suggests that (ηˆ u)∈ W 1,p ∗ ((s,t)×B 2R ). Therefore, (ηˆ u) indeed is a solution to the divergence type equation (4.9). As we shown above, both w and (ˆ uη) are solutions to the divergence type equation (4.9). Since solution to (4.9) is unique, so it’s only true that w = ˆ uη. 58 Note that η(x) =1 on B R and w∈W 1,2 2 ((0,T)×B 2R ), so we have ˆ u∈W 1,2 2 ((0,T)×B R ). And because R is arbitrary, we conclude that ˆ u∈W 1,2 2,loc ((0,T)×R). Other than the multiple uses of chain rule and integration by part, the proof above essentially adopted the exact same reasoning as in Lemma 3.4. Because of the issue caused by the constant 1, we had to manipulate the cut-off function η to reach the goal of “raising” the regularity. This eventually weakened our result. 4.2.2 Solving the FBSDEs Even though ˆ u in this case is not as nice as the previous results, we still managed to find a solution to the model (4.1). Lemma 4.3. There exist a weak solution to forward-backward SDEs(4.1). 59 Proof. Step one, consider the forward SDE:        dS t = ¯ b(t,S t )dt+e βt σdW t , S 0 =s, (4.10) where ¯ b(t,x)=e βt b(v(t,x)), v(t,x) = ˆ u(T −t,x). Same as the general case, a strong solution to (4.10) were not found because of the discontinuity of ¯ b(t,x). We search for the weak solution as followed. Let θ(t,x) = ¯ b(t,x) e βt σ , and M t = exp n Z t 0 θ(r,S r )dW r − 1 2 Z t 0 |θ(r,S r )| 2 dr o . One could check that θ(t,x) is a bounded function, so M t is a martingale under the probability measure P. Define a new probability measure Q by dQ dP =M T . 60 We claim (S t ,W t ) defined as S t ,s+ Z t 0 e βr σdW r , ¯ W t ,W t − Z t 0 θ(r,S r )dr, along with the probability space (Ω,F,Q) is a weak solution to (4.10). Indeed, the above claim is correct. By the Girsanov theorem, ¯ W is a Brownian motion in (Ω,F,Q). Furthermore, we have dS t =e βt σdW t =e βt σ θ(t,S t )dt+d ¯ W t = ¯ b(t,S t )dt+e βt σd ¯ W t . (4.11) It’s clear that the pair (S, ¯ W) solves (4.10) in the weak sense. What’s more, since |e βt σ|≤|e βT σ|<∞, S t does not explode in finite time. Step two, define Y t ,v(t,S t ), we are going to prove that Y t is the solution to the backward equation. As we had shown before, the non-divergence type equation (4.7) has a solution w on (0,T)× B 2R . At the same time, we have ˆ uη = w. Thus, for (t,x) ∈ (0,T)×B 2R , (ˆ uη) t − 1 2 ˜ σ 2 (ˆ uη) xx − ˜ b(x)(ˆ uη) x +h(ˆ uη) = ˜ f. 61 By the definition of η(x), η(x) =1 on B R . We find that on the region (0,T)×B R , ˜ f =η−˜ σ 2 ˆ u x η x − ˜ σ 2 2 ˆ uη xx − ˜ bˆ uη x =1. And the PDE above becomes ˆ u t − 1 2 ˜ σ 2 ˆ u xx − ˜ b(x)ˆ u x +hˆ u =1. Because v(t,x) = ˆ u(T −t,x), alternatively we have v t + 1 2 e βt σv xx + ¯ b(t,x)v x =h(t,x)v−1. (4.12) Next, we define a stopping time by, τ R = inf{t :|S t |≥R}. For any 0≤ t≤τ R ∧T, it’s clear that v(t,S t )∈ W 1,2 2 (0,τ R ∧T)×B R . 62 By the Ito-Krylov formula along with (4.11), we have v(τ R ∧T,S τ R ∧T )−v(t,S t ) = Z τ R ∧T t h v t + 1 2 e 2βr σ 2 v xx + ¯ b(r,S r )v x i dr+ Z τ R ∧T t e βr σv x d ¯ W r . (4.13) Since v x in bounded on the region (0,τ R ∧T)×B R , E Q t n Z τ R ∧T t e βr σv x d ¯ W r o = 0. Hence, by taking conditional expectation under the probability measure Q with respect toF t in (4.13), the equation is turned into E Q t n v(τ R ∧T,S τ R ∧T ) o −v(t,S t ) =E Q t Z τ R ∧T t h v t + 1 2 e 2βt σ 2 v xx + ¯ b(r,S r )v x i dr . Substituting (4.12) into the equation above, one gets E Q t n v(τ R ∧T,S τ R ∧T ) o −v(t,S t ) =E Q t Z τ R ∧T t h h(r,S r )v−1 i dr . (4.14) Next, we let R goto infinity in (4.14). As we stated above, S t does not explode in finite time, hence lim R→∞ τ R ∧T =T. 63 Also, v and h are both bounded, by dominated convergence theorem, the right handsideof (4.14)doesconvergeto,E Q t R T t h h(t,S t )v−1 i ds ,afiniteexpression. Finally we have, E Q t n v(T,S T ) o −v(t,S t ) =E Q t Z T t h h(t,S t )v−1 i ds . Or equivalently Y t =E Q t g(e −βT S T )+ Z T t h 1−h(r,S r )Y r i dr . (4.15) The last step, define X t ,e −βt S t . (4.16) Applying Ito’s formula, one will get dX t =−βe −βt S t dt+e −βt dS t = [b(Y t )−βX t ]dt+σd ¯ W t . Therefore, we have X t =x+ Z t 0 [b(Y s )−βX s ]ds+ Z t 0 σd ¯ W s . 64 Substituting X t for S t in (4.15), one could easily get Y t =E Q t g(X T )+ Z T t h 1−e ζ(Xt) Y t i dt . Clearly, (X, Y) along with the probability space (Ω,F,Q) is a weak solution to (4.1). 4.2.3 Extension to Other Model The way we built the model (4.1) is not particularly designed for the BK model. One could use any short rate model he or she prefers and builds a custom regime switching model. For instance, if one takes the Hull-White model instead, it becomes          r t =r 0 + Z t 0 [b(Y s )−βr s ]ds+ Z t 0 σdW s , Y t =E n g(X T )+ Z T t 1−e ζ(Xs) Y s ds F t o , (4.17) aregimeswitchingHull-Whitemodel. Intermsoffindingasolutiontotheforward- backwardSDEs,onecouldfollowtheexactsamesteps,andfind(r t ,Y t )asasolution to(4.17). The mathematic principalbehind isnodifferent. Ofcourse, (r t ,Y t )itself would not be the same. One should select a suitable short rate model based on their specific needs. Some are hard to implement, while some are computational expensive. 65 One could see that, the method is rather flexible. It’s ready to be extended to othershortratemodels,aslongasthecoefficients donotgetweaker thanwhatwe have in our assumptions. As a matter of fact, we are going to perform numerical analysis on both (4.1) and (4.17) in the next chapter to confirm the flexibility, as well as other purposes. 66 Chapter 5 Numerical Experiments With the existence of a weak solution proved, it’s a natural step to show the uniqueness of the solution next. Our first attempt was to show the uniqueness of solution to the PDE associated with the forward-backward SDEs. However, the discontinuity of the coefficient b brings tremendous difficulty. The usual way to prove uniqueness requires the coefficients to be Lipschitz continuous. In our model, continuity is not even provided, not to mention Lipschitz continuity. Many attempts were made; unfortunately, none of them was successful. As a result, we turn to numerical experiments on the model in hope of gaining empirical support. Weacknowledge thatnumerical resultsareapproximations, conclusions couldhave beendifferentintheory. Nevertheless, theseempiricalresultscouldprovideuswith guidance for future study. WeperformednumericalanalysisonboththeHull-WhiteandBlack-Karasinski models. Severalexperimentswereconductedoneachmodeltoexaminethevalidity of our theory. 67 5.1 Regime Switching Hull-White Model 5.1.1 The Model Consider the following regime switching model using the Hull-White model as the short rate model:                  dr t =[b(Y t )−βr t ]dt+σdW t , dY t = r t Y t −1 dt+Z t dW t , r 0 =r, Y T =g(r T ), (5.1) where we take β = 0.1, σ = 0.02, T = 10, r 0 = 0.4, g(x)= 100, and b(y) =          0.01, y≤ 80, 0.004, y >80. Note that parameters above are randomly selected based on intuition behind the model. One could use other values as long as they are reasonable. Rewriting the short rate model in another form, we have        dr t =β(10%−r t )+σdW t , when Y t ≤ 80, dr t =β(4%−r t )+σdW t , when Y t >80. (5.2) 68 Let us take a closer look at the model. β controls the mean reversion speed; σ is the volatility of the short rate. These two parameters stay the same after the switch. The only thing changes is the level of the long term mean. Whenever the long-termbondpricecross thelineof80,thelongtermmeanswitches from10%to 4% or the other way around. Since higher the short rate gets, the more the bond get discounted, leading to a lower bind price. We claim here, when bond price is high, short rate trends to a lower long term mean; and short rate follows a higher long term mean when the bond price is low. 5.1.2 The Crank-Nicolson Method Following the path of our method, (5.1) is equivalent to                  dX t =e βt b(Y t )dt+e βt σdW t , dY t = e −βt X t Y t −1 dt+Z t dW t , X 0 = 0.04, Y T = 100, (5.3) where X t =e βt r t . And using the 4-step scheme, (5.3) leads to following PDE:        u t − 1 2 e 2β(T−t) σ 2 u xx −e β(T−t) b ǫ (u)u x +e −β(T−t) xu−1= 0, u(0,x)= 100, (5.4) where we took ǫ =1 as a start. 69 We first tried solving (5.8) with the widely used finite difference method. The Crank-Nicolsonmethodischosenasourmaintool. Simplyput,theCrank-Nicolson method could be thought of as the combination of explicit scheme and implicit scheme. Letkbethetimestepandhbethespacestep,Crank-Nicolsonmethodemploys the following approximations: u t (t n ,x i )≃ u(t n+1 ,x i )−u(t n ,x i ) k , (5.5) u xx (t n ,x i )≃ 1 2 u(t n+1 ,x i+1 )−2u(t n+1 ,x i )+u(t n+1 ,x i−1 ) 2h + 1 2 u(t n ,x i+1 )−2u(t n ,x i )+u(t n ,x i−1 ) 2h , (5.6) u x (t n ,x i )≃ 1 2 u(t n+1 ,x i+1 )−u(t n+1 ,x i−1 ) 2h + 1 2 u(t n ,x i+1 )−u(t n ,x i−1 ) 2h . (5.7) Asonecouldseein(5.6)and(5.7),thefirsthalfisapproximationusingtheimplicit scheme, whereas the send half comes from the explicit scheme. By combining the two schemes, the Crank-Nicolson method was proved to be unconditionally stable (see Thomas [43]). Togetherwith theinitialandboundaryconditions, thesolution u would be foundby repeatedly solving alinear system corresponding to the PDE. To accommodate the use of boundary and boundary conditions, we took 0≤x≤e βT 70 as the boundary for the reasons that (1) it’s reasonable to assume that 0≤r t ≤ 1, and (2) X t =e βt r t . We also employed the boundary conditions:        u(t,0)=100, u(t,e βT ) = 0. (5.8) Consider x in extreme situations. When x goes to 0, r goes to 0 as well. Not much discounting would happen and so u(t,0)=100. On the other side, when x→ e βT , r becomes very large, thus the discounted bond price essentially is reduced to 0. The result using the Crank-Nicolson method is rather unsatisfactory in the sense that it does not agree with the economic intuition. To explain, we plot the curve of u(t,x) when t is fixed. Theoretically speaking, for a fixed t, u should be a decreasing function of x. Because when x increases, the short rate r increases, and this should result in a decreasing bond price . Apparently it is not the case in Figure (5.1). Afterfurtherexamination oftheproblem, werealized thatthatessence of (5.1) was changed as we enforced the boundary conditions. In other words, solving the PDE (5.4) along with the boundary conditions (5.8) is not equivalent to solv- ing (5.1). It’s known that the bond price is a path-dependent function of short rate. Enforcement of the boundary condition defies the known fact. For instance, when the short rate hits 0 at some time t, it does not imply the short rate always 71 0 0.5 1 1.5 2 0 10 20 30 40 50 60 70 80 90 100 For fixed t t=T/4 t=T/2 t = 3T/4 Figure 5.1: HW Model: Plot of u(·,x) (C-N Method) stay 0. Hence, it’s inappropriate to just claim that bond price does not get dis- counted as the enforcement of u(t,0) = 100. As a matter of fact, because of the path-dependency, it’s virtually impossible to rigorously define the boundary con- ditions. Any non-precise use of the boundary condition equates an invitation to false assumption and unsound reasoning. The unsuccessful experiment above requires the finding of a more suitable method in response. 72 5.1.3 The Layer Method The Layer methodoriginallydescribed inMilstein [37]andMilstein andTretyakov [38] solves the issue with boundary condition. Introduction to Layer Method Consider the following forward-backward SDEs:                  dX t =b(t,X t ,Y t )dt+σ(t,X t ,Y t )dW t , dY t =h(t,X t ,Y t )dt+Z t dW t , X 0 =x 0 , Y T =g(X T ). Let T =t N >t N−1 >···> t 0 , Δt := T −t 0 N beaequidistantdiscretizationofthetimeinterval[t 0 ,T]. Duetothecorresponding connectionbetweenforward-backwardSDEsandquasi-linearPDE,alocalsolution to the PDE        u t + 1 2 σ 2 (t,x,u)u xx +b(t,x,u)u x −h(t,x,u)= 0, u(T,x)=g(x) 73 has a probabilistic representation as: u(t k ,x) =E n u t k+1 ,X t k ,x (t k+1 ) − Z t k+1 t k h s,X t k ,x (s),u s,X t k ,x (s) ds o , (5.9) where X t k ,x (s) is the solution to the SDE:        dX t =b(t,X t ,Y t )dt+σ(t,X t ,Y t )dW t , X(t k )=x. (5.10) To find the solution u, one will need to approximate the right hand side of (5.9). First, applying the weak Euler scheme with the simplest simulation of noise to X t , we have X t k ,x (t k+1 )≃ ˜ X t k ,x (t k+1 ):=x+b(t,x,u(t k ,x))Δt+σ(t k ,x,u(t k ,x)) √ Δt·ξ, (5.11) where ξ∼N(0,1) is the noise. Similarly, Z t k+1 t k h s,X t k ,x (s),u s,X t k ,x (s) ds≃h t k ,x,u(t k ,x) Δt. (5.12) Since u are in fact being solved backward, u(t k+1 ,x) is used as an approximation to u(t k ,x). And the following Layer method is proposed: (1) For k =N, ˜ u(t N ,x) =g(x); 74 (2) For k =N−1,··· ,1,0 ˜ u(t k ,x) = 1 2 ˜ u t k+1 ,x+b t k ,x,˜ u(t k+1 ,x) Δt−σ t k ,x,˜ u(t k+1 ,x) √ Δt + 1 2 ˜ u t k+1 ,x+b t k ,x,˜ u(t k+1 ,x) Δt+σ t k ,x,˜ u(t k+1 ,x) √ Δt −h(t k ,x,˜ u(t k+1 ,x))Δt. To proceed, one would also need to discretize the space variable x in order to obtain a numerical algorithm. As a result, the following algorithm is constructed. Algorithm 5.1 (Milstein-Tretyakov). Let ΔX be the distant between each space step, (1) For t =t N , ¯ u(t N ,x) =g(x); (2) For k =N−1,··· ,1,0, j =0,±1,±2,···, compute ¯ u(t k ,x j )= 1 2 ¯ u t k+1 ,x j +b t k ,x j ,¯ u(t k+1 ,x j ) Δt− √ Δtσ t k ,x j ,¯ u(t k+1 ,x j ) + 1 2 ¯ u t k+1 ,x j +b t k ,x j ,¯ u(t k+1 ,x j ) Δt+ √ Δtσ t k ,x j ,¯ u(t k+1 ,x j ) −h(t k ,x j ,¯ u(t k+1 ,x j ))Δt, 75 where ¯ u(t k ,x) is defined as the linear interpolation between two neighboring points, ¯ u(t k ,x) = x j+1 −x ΔX ¯ u(t k ,x j+1 )+ x−x j ΔX ¯ u(t k ,x j ), for x j ≤x≤x j+1 . In [38], it’s shown that |¯ u(t k ,x)−u(t k ,x)|≤ KΔt, where K does not depend on x, k and Δt. Algorithm 5.1 is then of order one. Once ¯ u(t,x) is computed, numerical integration ofthe forward-backward SDEs could easily be done by using a simple Euler scheme: (1) For k = 0, X 0 =x,Y 0 =u(t 0 ,x); (2) For k = 1,··· ,N X k =X k−1 +b(t k−1 ,X k−1 ,¯ u(b(t k−1 ,X k−1 ))Δt +σ(t k−1 ,X k−1 ,¯ u(b(t k−1 ,X k−1 )) √ Δt·ξ, Y k =u(t k ,X k ). 76 It’s also shown that the Euler scheme above has the mean-square order of conver- gence 1/2. Existence of the Solution We apply Algorithm 5.1 to PDE (5.8). Note that, the corresponding forward- backward SDEs to (5.8) are given by:                  dX t =e βt b ǫ (Y t )dt+e βt σdW t , dY t = e −βt X t Y t −1 dt+Z t dW t , X 0 = 0.04, Y T = 100. (5.13) In practice, execution of the algorithm requires a bounded domain. Just as in the Crank-Nicolson case, we used [0,e βT ] as the primary domain. Figure (5.2) shows the 3-D plot of the function ¯ u ǫ (t,x). Looking at Figure (5.2), one can’t help but to ask the question, does Figure (5.2) truly represent what the real solution should look like? Due to there is no explicit form of the solution, we could only generally assess the question through some common knowledge behind the equations. At first sight, one could easily see that Figure (5.2) has a smoother surface compared to the previous result, there is no ups and downs in the middle of the graph. And to achieve a better understanding of the graph, we plot the curves ¯ u ǫ (t,·) for several fixed values of x. Figure (5.3) shows the result. 77 Figure 5.2: HW Model: 3-D Plot of ¯ u ǫ (t,x) - (1) We first observe from Figure (5.3) that all the curves are trending upward. That means, for a fixed x, ¯ u ǫ (t,·) is a increasing function in t. This is logical in terms of economic senses. The reason is twofold. On one hand, because of the relationship r t =e −βt X t , (5.14) in the case of a fixed x, the short rate decreases as t increases, resulting a higher bond price. On the other hand, as t increases, there will be less time to discount the bond, which leads to a even higher price. Therefore, in aggregate, ¯ u ǫ (t,·) as 78 0 2 4 6 8 10 0 20 40 60 80 100 120 Fixed x ( ε = 1) t u ε (t,x) x=0 Lower Quartile Midpoint Upper Quartile Figure 5.3: HW Model: Plot of ¯ u ǫ (t,·) the bond price should be a increasing function in t. Indeed, this is the case as it’s shown in (5.3). Secondly, all the curves come back to the point ¯ u ǫ = 100 when T = 10. This is no surprise since we set g(x)=100 as the terminal condition. In addition, if we also put x into consideration, one could see that the curves incline toalower positionasxincreases. This implies that ¯ u ǫ (·,x)isandecreasing function of x, which once again is in line with the economic rationality. When t is fixed, due to (5.14), the short rate increases as x increases. And as we men- tioned before, higher the short rate is, lower the discounted bond price would be. Therefore, ¯ u ǫ (·,x) as the bond price should be a decreasing function in x. 79 We also plot the curves for fixed t. In Figure (5.4), one could observe that 0 0.5 1 1.5 2 2.5 3 0 10 20 30 40 50 60 70 80 90 100 Fixed t ( ε = 1) x u ε (t,x) t=0 t=T/4 t=T/2 t = 3T/4 Figure 5.4: HW Model: Plot of ¯ u ǫ (t,·) ¯ u ǫ (·,x)isindeed adecreasing function ofx, supportingthereasoning statedabove. Based on the preliminary assessments above, we believed that ¯ u ǫ (t,x) obtained from the the Layer method is in fact appropriate and reasonable. Convergence and Uniqueness of the Solution Employing the same logic we used in the proof of the existence, we gradually let ǫ goes to zero, and study the convergence of the solution. 80 For our experiment, we used ǫ ={ 5, 1, 0.1, 0.01, 0.001, 0.0001, 0.00001 }. For each pair, max|Δu| is computed. Table (5.1) shows the results. ǫ ={5,1} ǫ ={1,0.1} ǫ ={0.1,0.01} max(|Δu|) 0.1752 0.0789 0.0502 ǫ ={0.01,0.001} ǫ ={0.001,0.0001} ǫ ={0.0001,0.00001} max(|Δu|) 0.0547 0.0129 1.1213×10 −4 Table 5.1: Convergence of the Solution with Layer Method One could easily see that, max|Δu| is getting smaller and smaller when the value of ǫ declines. As matter of fact, in the last case, most values of Δu are simply 0. Difference only happens in a small region of the graph, as shown in Figure (5.5). We also plot the graph of ¯ u ǫ (t,x) when ǫ = 0.00001 (Figure (5.6)) for compar- ison with the previous result. Figure (5.6) looks identical to Figure (5.2) which is no surprise giving the results in Table (5.1). Note that ǫ above are not purposely selected to achieve the convergence of ¯ u ǫ . other values of ǫ were used as well to test the convergence. Results are consistent with the ones shown above. In the previous chapter, we find the weak solution through a subsequence of {u ǫ (t,x)}. However, it appears that in our experiments, {u ǫ (t,x)} converges as 81 Figure 5.5: HW Model: 3-D Plot of Δu Figure 5.6: HW Model: 3-D Plot of ¯ u ǫ (t,x) - (2) 82 long as ǫ goes to 0. And they all converge to the same limit function. If{u ǫ (t,x)} does not have to a unique limit, the numerical results would probably be a lot different. We acknowledge the fact that we did not test the convergence for all possible ǫ, which at the mean time is impossible to execute. But with the solid empirical results we have so far, it’s our belief that u ǫ (t,x) will converge as ǫ goes to zero, and the limit function is unique. The Regime Switching Effect Given the already proved existence of the solution and strong evidence of unique- ness from the empirical results, we now turn to examine the term structure model itself. With ¯ u ǫ obtained from the Layer method, we used the aforementioned Euler scheme to generate the short rates and the bond price. Following figures show a particular path of r t and Y t respectively. In Figure (5.7), we also plot the trajectories of expected short rate in the two regimes when there are no switches between regimes. Because of the volatility impact, one could barely see r t following any of the trends, let alone observing the occurrence of regime change. With Figure (5.8), one could infer the time and frequency of the switch. However,even with these additional information, it’s still too much noise in Figure (5.7) to determine when or even whether the regime switch take place. 83 0 2 4 6 8 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 r t (ε = 0.001, σ = 0.02) t r t r t E(r t ) with b = 0.01 E(r t ) with b = 0.004 Figure 5.7: HW Model: The Short Rate(r t ) Trajectory - (1) 0 2 4 6 8 10 60 65 70 75 80 85 90 95 100 Y t ( ε = 0.001, σ = 0.02) t Y t Figure 5.8: HW Model: The Bond Price ( Y t ) Trajectory - (1) 84 In order to illustrate the regime switching effect, we gradually decrease the value of σ to tone down the influence of the volatility. Figure (5.9) and Figure (5.10) show the path of r t with σ = 0.01 and σ = 0.0004 respectively. 0 2 4 6 8 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 r t (ε = 0.001, σ = 0.01) t r t r t E(r t ) with b = 0.01 E(r t ) with b = 0.004 Figure 5.9: HW Model: The Short Rate(r t ) Trajectory - (2) Compared to Figure (5.7), in Figure (5.9) we cut σ by half, and r t starts to show some patterns. r t seems to follow the first regime at the beginning then shift totheotherregimeforthelatterpart. TheninFigure(5.10), duetotheverysmall value of σ, the volatility effects are very much muted off, and one could easily see that r t followed entirely the trends of the two regime. The switch happened only once, and it took place around t = 5.5, corresponding to the fact that Y t passed the trigger point only once at about t =5.5, see Figure (5.11). 85 0 2 4 6 8 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 r t (ε = 0.001, σ = 0.0004) t r t r t E(r t ) with b = 0.01 E(r t ) with b = 0.004 Regime Switch Figure 5.10: HW Model: The Short Rate(r t ) Trajectory - (3) 0 2 4 6 8 10 60 65 70 75 80 85 90 95 100 Y t ( ε = 0.001, σ = 0.0004) t Y t Regime Switch Figure 5.11: HW Model: The Bond Price(Y t ) Trajectory - (2) 86 5.2 Regime Switching Black-Karasinski Model One of the shortcomings of the Hull-White model is that short rate generated could be negative. To overcome that, we now turn to one of the alternatives: the Black-Karasinski Model. 5.2.1 The Model Consider the following model as we proposed in Chapter 3:                  dX t =[b(Y t )−βX t ]dt+σdW t , dY t = e Xt Y t −1 dt+Z t dW t , X 0 =x 0 , Y T =g(X T ), (5.15) where X t = ln(r t ), with the parameters β = 0.1, σ = 0.2, T = 10, x 0 = ln(0.4), g(x)= 100, b(y)=          β×ln(0.1), y≤ 80, β×ln(0.04), y > 80. 87 Parameter values above are selected similarly to the ones in the Hull-White model for the purpose of same level comparison. The short rate model could be rewritten as followed:        dX t =β(ln(10%)−X t )+σdW t , when Y t ≤ 80, dX t =β(ln(4%)−X t )+σdW t , when Y t > 80. (5.16) When the bondprice is above 80,X t hasa longrunaverage ofln(10%). Otherwise X t has a long run average of ln(4%). Because ofX t = ln(r t ), it is equivalent to say thatr t haslongrunaverage of10%ro4%depending onthebondprice. Therefore, (5.15) is essentially the same model as the previous model (5.1). 5.2.2 Numerical Results Same idea as before, applying mollification and variable transformation, we first look to find a solution to the PDE:        u t + 1 2 e 2βt σ 2 u xx +e βt b ǫ (u)u x +1−exp e −βt x u = 0, u(T,x)= 100. (5.17) Since 0 ≤ r t ≤ 1 and X t = ln(r t ), domain of the above PDE turns out to be (−∞,0], apparently unbounded. Numerical method requires discretization of the domain, which could not be performed on a unbounded domain. A simple remedy for the issue would be a change of variable. 88 Define m =e x , for x≤ 0. (5.18) Naturally we have m∈ [0,1]. Applying the chain rule that: ∂u ∂x = ∂u ∂m dm dx , (5.17) was transformed into        u t + 1 2 e 2βt σ 2 m 2 u mm + 1 2 e 2βt σ 2 +e βt b ǫ (u) mu m +1−m exp(−βt) u =0, u(T,m)= 100, (5.19) as we perform the variable change. Even though PDE (5.19) is a bit more compli- cated than the previous PDE (5.17), the above PDE is much easier to deal with thanaunbounded domain. Anotherway tohandeltheunbounded domainistrun- cation. We feel that using the variable change is more appropriate and would gain more precision. We are now ready to solve (5.19) using the same Layer method. Note that the forward-backward SDEs corresponding to (5.19) are now:        dM t = [ 1 2 e 2βt σ 2 +e βt b ǫ (Y t )]M t dt+σe βt M t dW t , dY t = M exp(−βt) t Y t −1 dt+Z t dW t . (5.20) 89 As a matter of fact, one could check that just by using Ito’s formula directly on M t = exp{e βt X t }, the same forward-backward SDEs as (5.20) would be obtained. This is to say that (5.15) and (5.20) are intrinsically the same model. We applied Algorithm 5.1 on (5.20). Figure (5.12) shows the solution ¯ u ǫ to PDE (5.19): Figure 5.12: BK Model: 3-D Plot of ¯ u ǫ (t,m) - (1) Just as we expected, Figure (5.12) resembles Figure (5.2): the two graphs bear similar shapes and patterns. After all, they both represent the bond price, it’s 90 only understandable when they are more or less alike. The difference stems from the independent variables. In Figure (5.2) we have X t =e βt r t . On the other side, the independent variableM t is related tothe shortrate through the connection M t =exp e βt ln(r t ) . (5.21) in Figure (5.12). 0 2 4 6 8 10 0 10 20 30 40 50 60 70 80 90 100 Fixed m ( ε = 0.001) t u(t,m) m=1/5 m =1/4 m =1/3 m=1/2 m=2/3 m=5/6 Figure 5.13: BK Model: Plot of ¯ u ǫ (t,·) 91 And because of (5.21), in the case of a fixed t, short rate raises as m increases. Meanwhile, when m is fixed, short rate will decline for an inclining t. By the same aforementionedeconomicrationality,weexpect ¯ u(t,m)remainstobeanincreasing functionintandadecreasingfunctioninm. Resultsmetouranticipation,asshown in the following plot of ¯ u(t,·). Convergence test were also ran for the model. One could see the results in Table (5.2) and Figure (5.14). Once again, ¯ u ǫ converges as ǫ goes to zero, and the limit function seems to be unique. ǫ ={5,1} ǫ ={1,0.1} ǫ ={0.1,0.01} max(|Δu|) 0.3891 0.1323 0.0762 ǫ ={0.01,0.001} ǫ ={0.001,0.0001} ǫ ={0.0001,0.00001} max(|Δu|) 0.0450 0.0034 1.0105×10 −5 Table 5.2: Convergence of ¯ u ǫ Regime switching effect of the model was examined as well.Results are affirma- tive as it is shown in the graph (5.15). To conclude, the Layer method nicely implemented both models. Convergence was confirmed, which is in line with the proof in the previous chapter. The regime switching characteristic in both cases were properly captured, showing legitimate- ness of the algorithm and flexibility ofour method. All the experiments justify the soundness of our model. Even though the uniqueness of the solution is still ques- tionable in theory, the strong empirical evidence give us the confidence to believe that uniqueness could actually be the case. 92 Figure 5.14: BK Model: 3-D Plot of ¯ u ǫ (t,m) - (2) 0 2 4 6 8 10 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 r t ( ε = 0.001, σ = 0.00002) t r t r t E(r t ) with b = 0.01 E(r t ) with b = 0.004 Regime Switch Figure 5.15: BK Model: Regime Switching of the Short Rate 93 Bibliography [1] Andersen, L., Andreasen, J.(2002), Volatile Volatilities, Risk, Volume 15, No. 12, (December), 163-168, 2002 [2] Ang, A., Bekaert, G. (2002), Regime Switches in Interest Rates, Journal of Business and Economic Statistics, Volume 20, No. 2 2002 [3] Antonelli, F., Ma, J. (2003), Weak Solutions of Forward-backward SDE’s, Stochastic Analysis and Applications, Volume 21, Issue 3, 2003 [4] Antonelli, F., Hamadene, S. (2006), Existence of the Solutions of Backward- forwardSDE’s with continuous monotonecoefficients,Statistics&Probability Letters, Volume 76, Issue 14, 1559-1569, 2006 [5] Bansal, R., Zhou, H. (2002), Term Structure of Interest Rates with Regime shifts, Journal of Finance, Volume 57, 2002 [6] Bali,T.,(2003)ModelingtheStochasticBehaviorofShort-termInterestRates: Pricing Implications for Discount bonds,JournalofBankingandFinanceVol- ume 27, Issue 2, 201-228, 2003 [7] Black, F., Derman, E., Toy, W. (1990), A One-factor Model of Interest Rates and Its Application to Treasury Bond Options, Financial Analyst Journal, January - February, 33-39, 1990 [8] Black, F., Karasinski, P. (1991), Bond and Option Pricing When Short Rates are Lognormal, Financial Analyst Journal, July- August, 1991 [9] Brace, A., Gatarek, D. et Musiela, M. (1997), The Market Model of Interest Rate Dynamics, Mathematical Finance, Vol. 7, No. 2, 127147 , 1997 94 [10] Brennan,M., Schwartz, E.(1982),An Equilibrium Model of Bond Pricing and a Test of Market Efficiency, Journal of Financial and Quantitative Analysis, Volume 17, No.3, 301-29, 1982 [11] Byun, S., Wang, L. (2005), Parabolic Equations in Reifenberg Domains, Archive for Rational Mechanics and Analysis, Volume 176, No. 2 , May, 2005 [12] Cox, J., Ingersol, J., Ross, S. (1985), A Theory of the Term Structure of Interest Rates,Econometrica, Volume 53, 385- 407, 1985 [13] Dai, Q., Singleton, K., Yang, W. (2007), Regime Shifts in a Dynamic Term Structure Model of U.S. Treasury Bond Yields, Review of Financial Studies, Volume 20, No. 5, 1669-1706, 2007 [14] Delarue,F.,Guatteri,G.(2006),WeakExistenceandUniquenessforForward- Backward SDEs, Stochastic Processes and their Applications, Volume 116, Issue 12, Pages 1712-1742, 2006 [15] Driffill, J., Kenc, T., Sola, M. (2003), An Empirical Examination of Term Structure Models with Regime Shifts EFMA 2003 Helsinki Meetings. [16] Duffie, D., Ma, J., Yong, J. (1995), Black’s Console Rate Conjecture, The Annals of Applied Probability, Volume 5, No. 2, 356-382, 1995 [17] Evans, L., Partial Differential Equations, Graduate Studies in Mathematics Vol 19, American Mathematical Society, 1998 [18] Evans, M. (2003), Real Risk, Inflation Risk, and the Term Structure, The Economic Journal, Volume 113, No. 487, pp. 345-389, 2003 [19] Folland, G. B., Real Analysis: Modern Techniques and Their Applications, Wiley; 2 edition (1999) [20] Hamilton, J.(1998),Rational Expectations Econometrics Analysis of Changes in Regimes: An Investigation of the Term Structure of Interest Rates, Journal of Economic Dynamics and Control, Volume 12, Issues 2-3, 385-423, 1988 [21] Heath, D., Jarrow, R., Morton, A. (1992), Bond Pricing and the Term Struc- ture of Interest Rates: A New Methodology for Continegnt Claim Valuation, Econometrica, Volume 60, 77-105, 1992 [22] Hu, Y., Peng, S. (1995), Solution of Forward-backward Stochastic Differential equations, Probability Theory and Related Fields, Volume 103, Number 2, 1995 95 [23] Hu, Y., Yong, J. (2000), Forward-backward Stochastic Differential Equations with Nonsmooth coefficients,StochasticProcesses andtheirApplications, Vol- ume 87, No. 1, pp. 93-106(14), May 2000 [24] Hull, J., White, A. (1990), Valuing Derivatives Securities Using the Explicit Finite Diffrences Method, Journal of Financial and Quantitative Analysis, Volume 25, 87-100, 1990 [25] Jia, G. (2008), A Class of Backward Stochastic Differential Equations with Discontinuous Coefficients, Statistics & Probability Letters, Volume 78, Issue 3, Pages 231-237, 2008 [26] Karatzas, I., Shreve, S., Brownian Motion and Stochastic Calculus, Springer, second edition, 1991 [27] Kim, D., Krylov, N.V. (2007), Parabolic Equations with Measurable Coeffi- cients, Potential Analysis, Volume 26, Number 4 / June, 2007. [28] Krylov, N.V., Controlled Diffusion Processes, Springer-valag, 1980 [29] Krylov, N.V., Lectures on Elliptic and Parabolic Equations in Sovolev Spaces, American Mathematical Society, 2008 [30] Ladyzenskaja, O., Solonnikov, V., Uralceva, N., Linear and Quasi-linear Equations of Parabolic Type, American Methematical Society, 1968 [31] Landen,C.(2000),BondPricing in a HiddenMarkov Model of the ShortRate, Finance and Stochastics, Volume 4, Issue 4, 371-389, 2000 [32] Lepeltier, J., San Martin, J. (1997) Backward Stochastic Differential Equa- tions with Continuous Coefficient, Statistics & Probability Letters, Volume 32, Issue 4, 425-430, 1997 [33] Ma, J., Protter, P., Yong, J. (1994) Solving Forward-backward Stochastic Dif- ferential Equations Explicitly - a Four Step Scheme, Probability Theory and Related Fields, Volume 98, No. 3, 1994 [34] Ma, J., Yong, J., Forward-backward Stochastic Differential Equations and Their Applications, Springer, 1999 [35] Ma, J., Zhang, J., Zheng, Z. (2008), Weak Solutions for Forward-backward SDEs - A Martingale Problem Apporach, The Annals of Probability, Volume 36, No. 6, 2092-2125, 2008 [36] Ma, J., Shen, J., Zhao, Y. (2008), On Numerical Approximations of Forward- Backward Stochastic Differential Equations, SIAM Journal on Numerical Analysis, Volume 46 Issue 5, 2008 96 [37] Milstein, G. N.(2002), The Probability Approach to Numerical Solution of Nonlinear Parabolic Equations, Numerical Methods for Partial Differential Equations Volume 18, Issue 4, pages 490C522, 2002 [38] Milstein, G. N., Tretyakov, M. V. (2006), Numerical Algoriths for Forward- backward Stochastic Differential Equations, SIAMJ.Sci.Comput. Volume28, Issue 2, pp. 561-582, 2006 [39] Morton, K. W., Mayers, D. F., Numerical Solution of Partial Differential Equations: An Introduction, Cambridge University Press; 2 edition (April 11, 2005) [40] Oksendal, B., Stochastic Differential Equations: An Introduction with Appli- cations, Springer; 6th edition, 2003 [41] Rebonato,R.(2004),InterestRate Term Structure Pricing Models: A Review, The Royal Society, Volume 460, 2004 [42] Schaefer, S., Schwartz E. (1984), A Two-Factor Model of the Term Structure: An Approximation Analytical Solution, Journal of Financial and Quantitative Analysis, Volume 19, No.4, 413-424, 1984 [43] Thomas, J. W.,Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics. 22. Berlin, New York: Springer- Verlag, 1995 [44] Vasicek, O. (1977), An Equilibrium Characterization of the Term Structure, Journal of Financial Economics, Volume 5, 177-188, 1977 [45] Wu, S., Zeng, Y. (2004), Affine Regime-Switching Models for Interest Rate Term Structure, in AMS (American Mathematical Society) Contemporary Mathematics, Vol. 351, ”Mathematics of Finance”, 375 - 386, 2004 97

Abstract (if available)

Abstract In this dissertation, we propose a regime switch term structure model built as forward-backward stochastic differential equations. We first generalize the model and study the forward-backward SDEs with discontinuous coefficients. Generally speaking, we adopt the tactics of the Four-Step scheme as our strategy of finding the solution. First, we find a solution v to the PDE associated with the FBSDEs. Then we use this v to decouple the FBSDEs, and solve each equation individually. A weak solution was successfully found. We then returned to the term structure model. We apply the framework of analysis from the general model to the regime switch term structure model, with some minor modifications. A solution, slightly weaker than the general case, was found for the term structure model, as well as the explicit relationship between the short rate and the long term bond price, all of which confirmed our model is fairly well-grounded. Numerical experiments were also conducted to show the validity of our theory. Results are rather satisfactory, providing us with strong empirical support.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Monte Carlo methods of forward backward stochastic differential equations in high dimensions

PDF

Topics on set-valued backward stochastic differential equations

PDF

Optimal investment and reinsurance problems and related non-Markovian FBSDES with constraints

PDF

Zero-sum stochastic differential games in weak formulation and related norms for semi-martingales

PDF

Pathwise stochastic analysis and related topics

PDF

On spectral approximations of stochastic partial differential equations driven by Poisson noise

PDF

Tamed and truncated numerical methods for stochastic differential equations

PDF

Conditional mean-fields stochastic differential equation and their application

$Stochastic differential equations driven by fractional Brownian motion and Poisson jumps$

PDF

Stochastic differential equations driven by fractional Brownian motion and Poisson jumps

PDF

Asymptotic problems in stochastic partial differential equations: a Wiener chaos approach

PDF

Dynamic approaches for some time inconsistent problems

PDF

Defaultable asset management with incomplete information

PDF

Numerical methods for high-dimensional path-dependent PDEs driven by stochastic Volterra integral equations

PDF

Dynamic network model for systemic risk

PDF

Optimal dividend and investment problems under Sparre Andersen model

PDF

Gaussian free fields and stochastic parabolic equations

PDF

Equilibrium model of limit order book and optimal execution problem

PDF

Global existence, regularity, and asymptotic behavior for nonlinear partial differential equations

PDF

Model, identification & analysis of complex stochastic systems: applications in stochastic partial differential equations and multiscale mechanics

PDF

Some topics on continuous time principal-agent problem

Asset Metadata

Creator Chen, Jianfu (author)

Core Title Forward-backward stochastic differential equations with discontinuous coefficient and regime switching term structure model

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 03/24/2011

Defense Date 01/10/2011

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag discontinuous coefficient,OAI-PMH Harvest,regime switching,stochastic differential equations

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ma, Jin (committee chair), Zapatero, Fernando (committee member), Zhang, Jianfeng (committee member)

Creator Email jianfuch@usc.edu,kennethcjf@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m3696

Unique identifier UC175658

Identifier etd-Chen-4264 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-444005 (legacy record id),usctheses-m3696 (legacy record id)

Legacy Identifier etd-Chen-4264.pdf

Dmrecord 444005

Document Type Dissertation

Rights Chen, Jianfu

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu