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Defaultable asset management with incomplete information

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Defaultable asset management with incomplete information

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Content DEFAULTABLE ASSET MANAGEMENT WITH INCOMPLETE INFORMATION by Huanhuan Wang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (APPLIED MATHEMATICS) May 2013 Copyright 2013 Huanhuan Wang Dedication To my family ii Acknowledgments I have written this thesis during the three years as a PhD student at department of Mathematics, University of Southern California. I would like to thank everyone who has contributed to the accomplishment of this thesis. I am grateful for the insight, motivation and support I have received. First of all, I would like to thank Professor Jin Ma for the great opportunity he oered me three years ago to work on my PhD dissertation under his supervision. I am wholeheartedly grateful for his valuable support during the past three years, for his insightful guidance and continuous condence and encouragement, especially in the key moments of my dissertation. I would also like to thank Professor Jianfeng Zhang for his generous advices, fruitful guidance, and critique during the years I was in USC. I would also like to express my gratitude to Professors Fernando Zapatero, Sergey Lototsky, and Remigijus Mikulevicius for serving on my dissertation committee and also for their valuable advices. iii I also would very much like to thank my academic siblings and cousins, class- mates, and friends Xinyang Wang, Jianfu Chen, Xin Wang, Tian Zhang, Jie Du, Jia Zhuo, Shanshan Xu, Wei Liu, Chunlei Xia, Liye Zhang, Zhicheng Luo, and Zemin Zheng for their unwavering support and understanding, and for sharing their constructive thoughts and precious experiences with me during this special period of my life. Their friendship and condence in me made my journey much more fun and rewarding. Last but not least, I wish to express my deepest gratitude to my family: my parents, Yuzhen Guo and Yihou Wang, and my wife Nan Xing, for their uncondi- tional love, support, understanding, and encouragement throughout my life, which made this dissertation possible. iv Table of Contents Dedication ii Acknowledgments iii Abstract vi Introduction 1 Chapter 1: BSDE 8 1.1 BSDE with a single jump . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 BSDE with multiple defaults . . . . . . . . . . . . . . . . . . . . . . 25 Chapter 2: Utility maximization problem with single default 33 2.1 Preliminary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.1 Literature review . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.2 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2 The problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3 The ltering approach . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.1 Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.2 Summary of intensity, density, and H hypothesis . . . . . . . 58 Chapter 3: The multiple defaults market with frailty 64 3.1 The market model . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 The ltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Utility maximization problem . . . . . . . . . . . . . . . . . . . . . 81 3.4 Conclution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Bibliography 90 v Abstract We consider a market where asset prices could be aected by multiple defaults along with possible factors including frailty. The aim of an investor is to maximize the expected utility of the terminal wealth, based on the observed data of the underlying asset(s) and the default history up to the current time. The main purpose is then to determine the conditional intensity of the future defaults, given the observed stock prices and the past defaults, without using the so-called \density hypothesis". The problem is naturally reduced to a nonlinear ltering problem, for which the so-called the H-hypothesis is known to fail. We show that the problem can be solved dynamically via a system of Zakai equations for the conditional densities between and at consecutive defaults. A related BSDE with jumps that has quadratic growth in both continuous and jump martingale integrands will also be studied, as a by product of the utility optimization problem. vi Introduction In this thesis we study a utility maximization problem in a market where the price of the underlying assets could be aected by a factor process which is possibly unobservable (e.g., frailty) and a nite sequence of events that may not be inde- pendent of the market itself (e.g,, natural disasters). The impact of the event not only induces an immediate jump of the price, but also causes a \regime change" to the model in the sense that the parameters of the asset price model will have a sudden change (or jump). The investor or decision maker can only design his/her strategy based on the history of the asset price and events, without knowing the actual statistical characteristics of the future events. Therefore the problem is essentially a utility maximization with partial information. The problem can be formulated mathematically as follows. Assume that the asset price follows a standard linear stochastic dierential equation: dS t =S t [a t dt + t dW t ]S t t dM t ; t 0; S 0 =s; (1) 1 whereW is a Brownian motion on a given probability space ( ;F;P),M is a pure jump process withm unit jumps, representing possible defaults, and is a piecewise constant process taking values in (0; 1], depending only on M, representing the impact of the defaults. We assume that the jump times of M are 0 <T 1 <T 2 < <T m , which could often be thought of as the ordered statistics of the default times of n names in a given market. The \regime change" eect of the model could be formulated by assuming that the coecients a and take the form: for =a;, t = 0 1 ft=0g + m X i=1 1 f(T i1 <tT i g i t ; t 0: (2) In other words, we can, and often will, write the SDE (1) as S i t =S i T i + Z t 0 S i s (a i s ds + i s dW s ); t2 [T i1 ;T i ]; S i T i =S i1 T i + T i ; i = 1; ;m: We note that the model of this kind has been studied by many authors in dierent context with dierent technical assumptions. For example, if m = 1 and = T 1 is a random time such that = 1, then S t =S t and hence S = S + S = 0, and thus S t 0 for t . Namely, the random time is in fact the bankruptcy time of the company. Such a case was considered by Lindtsay [?] in the context of option pricing. The case when m = 1, but 2 2 [0; 1] was studied by Carr-Madan [9] in the context of local volatility. The models involve martingales with one jump have also been a common subject in credit default literature (cf. e.g., the books [4], [5]) as well as the recent paper by Ankirchner et al. [2] involving a quadratic backward SDE with a single jump. The asset models with multiple default times have often been studied within the realm of multiple defaults (cf. [26], [27]). In particular, the utility maximization problem under such model was recently studied by Jiao-Kharroubi-Pham [26], in which the random times are assumed to be the ordered statistics of a marked point process representing the default scenarios of m names in a market. However, our premise towards the problem is quite dierent from theirs. Among other things, for example, we will not assume the so-called \density-hypothesis", which is the fundamental assumption in [26]. Instead, we shall provide a way to obtain the conditional density by observing only the asset price and the default history up to the current time. We now give a more specic description of our problem. Assume that the default free market is dened on a ltered probability space ( ;F;P;F) on which is dened a standard Brownian motion W . In what follows we shall denote the ltration generated by W as F W . Assume that market is impacted by m events, happened at random times 0 < T 1 < < T m , independent of W . For the sake of argument let us simply call T i 's the default times, and we denote H t = (H i t ; ;H m t ), whereH i t 4 = 1 fT i tg ,i = 1; ;m, and call it the \default indicator 3 process". The default history can then be characterized by the ltration H = fH t g t0 , whereH t 4 =fH s :stg. We also assume that there is a unobservable factor process X whose dynamics is given by an SDE X t = X 0 + Z t 0 b(X s ;H s )ds + Z t 0 (X s ;H s )dB s (3) + Z t 0 Z R K X (X s ;H s ;u)N (ds;du); whereB is a Brownian motion independent of the defaults andW ; andN (ds;du) is a standard Poisson random measure dened on R + R with L evy measure . Such a factor process is known as \frailty" (cf. e.g., [16]). We shall further assume that in the SDE (1) and (2) the coecients a i t =a i (X t ;H t ), wherea i 's are deterministic functions,T i 's are random times that are assumed to be the stopping times with respect to a ltration to which M is adapted, but not necessarily stopping times with respect to the ltration generated by S. The jumps in our asset price model and the dependence of the contingent claim on the default events lead our problem to a BSDE problem with jumps. Asset price models that assume a strictly positive price lose the sight of the jumps in the dynamic of the price. For the model we used in this part, we refer to Carr-Madan [9], where they emphasized on how to use credit default swap and equity option prices jointly to infer the deterministic function of calendar time in the hazard rate and the local 4 volatility surface. Our credit risk model is based on the technique of ltration enlargements. The ltrationF is generated by continuous processes, and its smallest extension G makes the default time aG-stopping time. This kind of ltration enlargement has been referred to as progressive enlargement of ltrations. For an introduction of progressive enlargement of ltrations we refer to [7]. In my thesis, I rst derive the default intensity process for each company that is still on the market with respect to on the observable information. I do not assume the existence of the conditional default density process introduced in [29]. At the same time, we allow the existence of an unobservable factor or frailty process X, which we will use the ltering techniques to derive the Zakai equation. Both the frailty process X and the default indicator processes H are in the drift of our observation S, which leads to a non uniformity problem between defaults, i.e. on each time interval [T i ;T i+1 ). In the results of defaultable models, the "H-hypothesis" is widely assumed. The hypothesis says the martingale of the smaller ltration is still a martingale under the enlarged ltration. In this thesis, I work on a model where the "H-hypothesis" does not hold any more, which is corresponding to the result in [32]. We will also show how to set up the BSDE 5 system with the absence of this hypothesis. The main technique used in this thesis to nd the conditional probability is ltering. To set up the preliminary work, we represent the default indicator process for each company, say, company i as H t;i = H 0;i + Z t 0 Z R (1H s;j )K i (X s ;H s ;u)N (ds;du); For this representation, we refer to [41]. In the second part of this thesis, a utility maximization problem is solved by a BSDE approach. Utility maximization problem is one of the most popular topics in Mathematical Finance, which has been worked on in many literatures. For economics agents with exponential utility preferences and holding non-defaultable contingent claims, we refer to [40] and [20]. We allow for defaultable contingent claims in the portfolio, and in this case the utility maximization coincides with the solution of a BSDE that is discontinuous at default time. For the motivation of this problem, we also refer to [4]. In [4], a single default problem with continuous asset price model was investigated. In our arguments, we just work with the assumption of intensity process instead of the Density Hypothesis and H-hypothesis. We use the ltering 6 techniques to lter out the conditional default probability, furthermore we can derive the dynamic of the intensity process, which will be updated after each default. Our BSDE approach of the utility maximization problem is based on solving the intensity process. 7 Chapter 1 BSDE In this chapter we consider a problem of a quadratic BSDE with jumps. The exis- tence and uniqueness results will be applied into the utility maximization problem in the following chapters. 1.1 BSDE with a single jump LetW be ad-dimensional Brownian motion on a probability space ( ;F;P), where d2N. We also dene H t = 1 ftg , where is a random time: !R + . Then we dene the following ltration enlargement G t =F W t _(H t ); respect to which is a stopping time. In this part , we assume the following hypothesis Hypothesis 1.1.1. Any square integrable (F W ;P)-martingale is a square inte- grable (G;P)-martingale. 8 Under this hypothesis, the Brownian motion W is still a Brownian motion in the enlarged ltrationG. In next part, we will have a situation where this hypothesis is not satised, and how we deal with the BSDE problem without it. In this part, we assume the hypothesis holds. Then we dene the compensated version of H t as M t :=H t Z ^t 0 s ds; (1.1) which is aG-martingale. Here we assume that =f t :t 0g is a non-negative, F W -adapted process, such that Ef R T 0 j t j 2 dtg <1, and call it the conditional default intensity process. Let us consider the following BSDE on t2 [0;T ]: Y t =A + Z T t f(s;Z s ;U s )ds Z T t Z s dW s Z T t U s dM s : (1.2) HereA =A 1 1 f>Tg +A 2 1 fTg is aG T -measurable random variable. A 1 andA 2 are boundedF W T -measurable random variables. For technical reasons in what follows we assume that the generator takes the following form: f(t;z;u) = [l(t;z) +p(t;u)](H t ) +q(t;z)H t ; (1.3) 9 wherel,p, andq are measurable functions. Furthermore, we shall make use of the following Standing Assumptions. Assumption 1.1.2. The functionsl,p, andq are continuous in all variables, and enjoy the following properties: (i) There are constants 0 > 0 and C > 0, such that l(t;z) +p(t;u) 0 +C[jzj 2 + 2 t juj 2 ]; (t;z;u)2 [0;T ]R d R; where =f t g is the conditional intensity process dened in (1.1). (ii) There exists constant L 0 > 0 such that for all z and z 0 2R jl(t;z)l(t;z 0 )j +jq(t;z)q(t;z 0 )jL 0 (1 +jzj +jz 0 j)jzz 0 j; (t;z;z 0 )2 [0;T ]R d R d : (iii) p(t;u) 0, for all (t;u)2 [0;T ]R, such that jp(t;u)p(t;u 0 )jL 0 p t (1 +juj +ju 0 j)juu 0 j; (t;u;u 0 )2 [0;T ]RR: (iv) There exists a constant M 0 0 such that jp(t;u)p(t;u 0 )jL 0 0 p t juu 0 j; (t;u;u 0 )2 [0;T ] (1;M 0 ] (1;M 0 ]: 10 We remark that under Assumption 1.1.2 the generatorf has a quadratic growth in both z and u, and the assumption (iii) implies (iv) for alljuj;ju 0 j M 0 (with L 0 0 =L 0 (1 + 2M 0 )). Thus the assumption (iv) amounts to simply saying that the generator is uniformly Lipschitz in the variable u even as u!1. Throughout this thesis we shall also use the following notations. Denition 1.1.3. We denote byH 2 T (G t ) the set of all (G t )-predictable processes X t satisfying E Z T 0 jX t j 2 dt1 and byH 1 T (G t ) the set of essentially bounded (G t )-predictable processes. We will show some auxiliary results, which will be used in the proof of the existence and uniqueness results for the BSDE. We will prove the process R t 0 Z s dW s is a BMO (Bounded Mean Oscillation) martingale. Let = ( t ;G t ) t2[0;T ] be a uniformly integrable martingale on [0,T] with 0 = 0, and for 1<1 we set kk BMOp = sup E[j T j p jG ] 1 p (1.4) where the supremum is taken over all stopping times with values in [0;T ]. The classf :kk BMOp <1g is denoted by BMO p , andkk BMOp is a norm on this 11 space. From the H older inequality it follows that forp<q,BMO p BMO q . And the reverse inclusion was proved in [30], so we write simply BMO for this space. Without loss of generality, we usekk BMO 2 as the BMO norm, where kk BMO 2 = sup E[j T j 2 jG ] 1 2 = sup E[h; i T h; i jG ] 1 2 ; (1.5) whereh; i is the quadratic variation of . In the following lemma we introduce some useful properties of BMO martin- gales. Lemma 1.1.4. Let be a continuous BMO martingale. Then we have: 1. The exponential of , E() t = expf t 1 2 h; i t g; (1.6) is a uniformly integrable martingale. 2. E() T satises E(E() T ) = 1, and thus the measure dQ =E() T dP is a probability measure. 3. The process ^ = h; i is a BMO martingale respected to the measure Q. 12 4. It is always possible to nd a p> 1 such thatE()2L P : And the p can be nd by the function (x) =f1 + 1 x 2 log 2x 1 2(x 1) g 1 2 1; (1.7) for all 1<x<1, which is a continuous decreasing function such that (1+) =1 and (1) = 0. And ifkk BMO < (p), thenE()2L p . Lemma 1.1.5. Assume that (Y;Z;U)2H 1 T (R)H 2 T (R d ))H 1 T (R) is solution of the BSDE (5.6), then R t 0 Z s dW s is a BMO-martingale. Proof. Without loss of generality, letL 1 be an upper bound of the processjYj and jUj. By Ito's formula, for a2R, e aYt =e aY 0 + Z t 0 ae aY s Z s dW s + Z t 0 ae aY s U s dM s Z t 0 ae aY s f(s;Z s ;U s )ds + 1 2 Z t 0 a 2 e aY s jZ s j 2 ds + X st e aY s (e aYs 1aY s ): BecausejUj is bounded byL 1 , there exist positive constantsC 1 andC 2 , such that f(s;Z s ;U s )C 1 (jZ s j 2 +jU s j 2 )C 2 . For a 0, e aYt e aY 0 + Z t 0 ae aY s Z s dW s + Z t 0 ae aY s U s dM s + ( 1 2 a 2 aC 1 ) Z t 0 e aY s jZ s j 2 ds Z t 0 aC 2 e aY s ds aC 1 Z t 0 e aY s jU s j 2 ds: 13 Taking conditional expectation, for arbitrary stopping times in [0,T], ( 1 2 a 2 aC 1 )E[ Z T e aY s jZ s j 2 dsjG ] E[e aY T e aY jG ] +E[ Z T aC 1 e aY s jU s j 2 dsjG ] + E[ Z T aC 2 e aY s dsjG ] Choose a = 3C 1 . Then 1 2 a 2 aC 1 = 3 2 C 2 1 , and 3 2 C 2 1 e 3L 1 C 1 E[ Z T jZ s j 2 dsjG ] (2 + 3C 2 1 L 2 1 T + 3C 1 C 2 T ))e 3L 1 C 1 ; which implies our result. We are now ready to prove the existence of the solution to the BSDE (1.2). Theorem 1.1.6. (Existence) The BSDE (1.2) admits a solution (Y;Z;U) 2 H 1 T (G t )H 2 T (G t )H 1 T (G t ). Proof. In the existence proof, we construct the solution of the BSDE with two continuous BSDE with terminal conditions A 1 , A 2 respectively. First, by the results in [31], there exists a solution ( ^ Y; ^ Z)2H 1 T (G t )H 2 T (G t ) for the jump-free BSDE ^ Y t =A 2 Z T t ^ Z s ds + Z T t q(s; ^ Z s )ds; (1.8) 14 and there exists constant K 0 > 0 such that K 0 ^ Y t K 0 ; 8t2 [0;T ]; a:s On the other hand, we consider the following BSDE Y r t =A 1 Z T t Z r s ds + Z T t r(s;Y r s ;Z r s )ds: where r(s;y;z) = l(s;z) + ( ^ Y s y) s . Again, by [31], we can nd a solution (Y r ;Z r )2H 1 T (G t )H 2 T (G t ), such that for some constant K 1 > 0, it holds that K 1 Y r t K 1 ; 8t2 [0;T ]; a:s; We will now prove the existence of the BSDE with generator k(s;y;z) 4 = h(s;y;z)1 fy2[K;1)g (1.9) +[r(s;y;z) +p(s; ^ Y s +K)[1 (y +K)]]1 fy2(1;K)g ; where K 4 =K 1 and h(s;y;z) 4 =l(s;z) + ( ^ Y s y) s +p(s; ^ Y s y): (1.10) 15 Clearly, the functionk is quadratic inz. We claim that it is Lipschitz continuous in y. Indeed,k is obviously Lipschitz fory2 (1;K). Now assume thatyK, and assume that the Assumption 1.1.2-(iv) holds for M 0 = K 0 +K 1 (i.e., the functionp is uniformly Lipschitz inu foruM 0 ). Note that for all yK 1 one has ^ Y r yY r +K 1 K 0 +K 1 <M 0 : Thus by Assumption (1.1.2), h is Lipschitz for y2 [K 1 ;1). Consequently, from the denition (1.9) we see that the functionk is Lipschitz continuous iny. It then follows from [31] that there exists a solution (Y k ;Z k )2H 1 T (G t )H 2 T (G t ) for the BSDE with generator k(s;y;z) and terminal condition A 1 . We will now prove that Y k Y r ;a:s. If this is true, then the solution of the BSDE with generator k will also be the solution for the BSDE with generator h. We rst dene s = l(s;Z k s )l(s;Z r s ) Z k s Z r s By the assumptions for f, we havej s j L 0 (1 +jZ r s j +jZ k s j);a:s. Such that t = R t 0 s dW s is a BMO martingale. Thanks to Lemma 1.1.4, ^ W t =W t R t 0 s ds 16 is a QBrownian motion, where Q is a new measure as dQ dP =E() T . Then we have Y k t Y r t = Z T t (Z k s Z r s )d ^ W s + Z T t (Y k s Y r s ) s ds + Z T t [p(s; ^ Y s Y k s )1 fY k s 2[K;1)g + p(s; ^ Y s +K)[1 (Y k s +K)]1 fY k s 2(1;K)g : The processes (Y k Y r ;Z k Z r ) solve the BSDE y t = Z T t z s d ^ W s + Z T t ( s +y s s )ds where s =p(s; ^ Y s Y k s )1 fY a s 2[K;1)g +p(s; ^ Y s +K)[1 (Y k s +K)]1 fY k s 2(1;K)g 0. The above BSDE has a unique closed form solution: Y k t Y r t = E Q [ R T t exp( R s t u du) s dsjG t ], which implies Y k Y r K;a:s. Now we can nish the construction of the solution for the BSDE (1.2). Set Y t = 8 > > < > > : ^ Y t t; Y k t >t; Z t = 8 > > < > > : ^ Z t t; Z k t >t; and U t = 8 > > < > > : 0 <t; ^ Y t Y k t t; 17 Notice that, on the setftg, we have Y t = Y t Y +Y = ( ^ Y t ^ Y ) + ( ^ Y Y k ) +Y k = Z t ^ Z s dW s Z t q(s; ^ Z s )ds + Z t 0 U s dH s + Z 0 Z k s dW s Z 0 h(s;Y k s ;Z k s )ds = Z t 0 Z s dW s + Z t 0 U s dM s Z t 0 f(s;Z s ;U s )ds The terminal condition Y T = ^ Y T =A 2 is satised. On the other setf >tg, M has no jump on [0;t] and we have Y t = Y k t = Z t 0 Z k s dW s Z t 0 h(s;Y k s ;Z k s )ds = Z t 0 Z s dW s + Z t 0 U s dM s Z t 0 f(s;Y s ;Z s )ds The terminal condition Y T =Y k T =A 1 1 fTg +A 2 1 f>Tg is satised. Therefore, (Y;Z;U) is a solution of the 1.2 with generator f, where Y and U are bounded. Then we can introduce our priori estimate which implies the uniqueness. Theorem 1.1.7. There exist constants l, and C > 0, which only depends on T;C 1 ;C 2 ;k sup t2[0;T ] jY 1 s jk 1 andk sup t2[0;T ] jY 2 s jk 1 , such that E[ sup t2[0;T ] jY t j 2 + Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds]C(E[jAj 2l ]) 1 l : 18 Proof. To prove this theorem, rst we have that for any 2R, by Ito's formula, e t Y 2 t = e T Y 2 T 2 Z T t e s Y s Z s dW s Z T t 2e s Y s U s dM s +2 Z T t e s Y s [f(s;Z 1 s ;U 1 s )f(s;Z 2 s ;U 2 s )]ds Z T t e s (Y 2 s +jZ s j 2 )ds X t<sT e s U 2 s (M s ) 2 : Using [M;M] t = R t 0 s ds +M t , it can be simplied to e t Y 2 t = e T Y 2 T 2 Z T t e s Y s Z s dW s Z T t e s (2Y s U s +U 2 s )dM s +2 Z T t e s Y s [f(s;Z 1 s ;U 1 s )f(s;Z 2 s ;U 2 s )]ds Z T t e s (Y 2 s +Z 2 s + s U 2 s )ds: Then we dene the following two processes I s = f(s;Z 1 s ;U 1 s )f(s;Z 2 s ;U 1 s ) Z s ; J s = f(s;Z 2 s ;U 1 s )f(s;Z 2 s ;U 2 s ) U s : And we have that f(s;Z 1 s ;U 1 s )f(s;Z 2 s ;U 2 s ) = I s Z s +J s U s . Because of the properties of f we have jI s jL 0 (1 +jZ 1 s j +jZ 2 s j); jJ s jL 0 (1 +jU 1 s j +jU 2 s j) p s L 0 (1 + 2L 1 ) p s : 19 Lemma 1.1.5 then ensures that R 0 I s dW s is a BMO martingale bounded by Ck R 0 (1 +jZ 1 s j +jZ 2 s j)dW s k BMO . We dene ^ W s =W s R t 0 I s ds, then e t Y 2 t = e T Y 2 T 2 Z T t e s Y s Z s d ^ W s Z T t e s (2Y s U s +U 2 s )dM s +2 Z T t e s Y s J s U s ds Z T t e s (Y 2 s +Z 2 s + s U 2 s )ds: Note that 2jY s J s U s j 2L 0 (1 + 2L 1 )j p s Y s U s j 2L 2 0 (1 + 2L 1 ) 2 jY s j 2 + s 2 jU s j 2 ; setting = 2L 2 0 (1 + 2L 1 ) 2 we obtain e t Y 2 t + Z T t e s (jZ s j 2 + s 2 jU s j 2 )ds e T A 2 2 Z T t e s Y s Z s d ^ W s Z T t e s (2Y s U s +U 2 s )dM s : Then we can dene a new measure Q by dQ dP =E(IW ) T , which implies ^ W is a Qmartingale. M is also a Qmartingale because of the independence of M and W . Dene R t =A 2 2 R T t Y s Z s d ^ W s R T t j2Y s U s +U 2 s jdM s , and we have 20 already proved Y 2 t CR t for some constant C. Apply Ito's formula, under the measure Q we have R p t =A 2p Z T t pR p1 s Y s Z s d ^ W s X t<sT ((R s + R s ) p R p s pR p1 s R s ): BecauseR s ; R s 0, and ((R s +j2Y s U s +U 2 s jM s ) p R p s pR p1 s M s ) 0, R p t A 2p Z T t pR p1 s Y s Z s d ^ W s Z T t pR p1 s j2Y s U s +U 2 s jdM s : Taking expectation on both sides for the above inequality, and applying Doob's L p -inequality, E Q [ sup t2[0;T ] jY t j 2p ]E Q [ sup t2[0;T ] jR t j p ]CE Q [jAj 2p ] for some C > 0;8p> 1: Also there are constants C 4 , C 5 , such that h Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds i p C 4 (jAj 2p + Z T 0 2Y s Z s d ^ W s + Z T 0 (2Y s U s +U 2 s )dM s p : 21 By the Burkholder-Davis-Gundy inequality E Q ( Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds) p C 5 E Q (jAj 2p +j Z T 0 (2Y s Z s ) 2 dsj p 2 + X 0<sT jU s j p jY s +U s j p M s ): We can choose C 6 > 1, such that E Q j Z T 0 (2Y s Z s ) 2 dsj p 2 2 p (2 p+1 C 6 E Q ( sup t2[0;T ] jY s j 2p ) + 1 2 p+1 C 6 E Q ( Z T 0 jZ s j 2 ) p ): Because of the bounded jump size of M andjU s jC 7 sup t2[0;T ] jY t j, we have E Q ( X 0<sT jU s j p jY s +U s j p M s ) = E Q n Z T 0 jU s j p jY s +U s j p dM s + Z T 0 jU s j p jY s +U s j p s ds o C 8 E Q n sup t2[0;T ] jY t j 2p o : Consequently E Q ( Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds) p C 9 E Q (jAj 2p + sup t2[0;T ] jY t j 2p : 22 By the property of BMO martingale of R 0 I s d ^ W s , there exists a constant p 0 > 1, such that E Q [E(I ^ W ) p 0 T ]<1. Using the Holder's inequality with p = p 0 p 0 1 , E n sup t2[0;T ] jY t j 2 o = E Q [E(I ^ W ) T ( sup t2[0;T ] jY t j 2 )] (E Q [E(I ^ W ) p 0 T ]) 1 p 0 (E Q [ sup t2[0;T ] jY t j 2p ]) 1 p C 10 (E Q [jAj 2p ]) 1 p : Since R 0 I s dW s is a BMO martingale with respect to P , 1.1.5 implies there exists a constant q 0 > 1, such that E[E(IW ) q 0 T ] <1. Apply the Holder's inequality again with q = q 0 q 0 1 E sup t2[0;T ] jY t j 2 C 10 (E[E(IW ) T jAj 2p ]) 1 p C 10 (E[E(IW ) q 0 T ]) 1 q 0 (E[jAj 2pq ]) 1 pq C 11 (E[jAj 2pq ]) 1 pq : Similarly we have E( Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds)C 12 (E[jAj 2pq ]) 1 pq : Setting l =pq, C = maxfC 10 ;C 11 g, the theorem is proved. Theorem 1.1.8. (Uniqueness) The BSDE problem 1.2 has a unique solution (Y;Z;U)2H 1 T (R)H 2 T (R d )H 1 T (R). 23 Proof. Let (Y i ;Z i ;U i )2H 1 T (R)H 2 T (R d )H 1 T (R) i = 1; 2 be two solutions for BSDE Problem 1.2. Since A = 0, using Theorem 1.2.3, E[ sup t2[0;T ] jY t j 2 + Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds]C(E[jAj 2l ]) 1 l = 0: Namely, Y 1 =Y 2 , Z 1 =Z 2 , and U 1 =U 2 a:s: 24 1.2 BSDE with multiple defaults In this section, we extend the results into the market with multiple default events. Namely, we consider the existence and uniqueness of the BSDE Y t =A + Z T t f(s;Y s ;Z s ;U s )ds Z T t Z s dW s Z T t m X i=1 U i s dM i;s (1.11) Assumption 1.2.1. The generator f satises: f(t;z;u) = [l(t;z) +p(t;u)](1H t ) +q(t;z)H t . There are constants 0 and C, such that l(t;z) +p(t;u) 0 +C[jzj 2 + 2 t juj 2 ]; wherejuj 2 4 = P m i=1 ju i j 2 . there exists constant L 0 > 0 such that for all z and z 0 2R jl(t;z)l(t;z 0 )j +jq(t;z)q(t;z 0 )jL 0 (1 +jzj +jz 0 j)jzz 0 j; p 0 and for all u, u 0 2R m jp(t;u)p(t;u 0 )jL 0 p t (1 +juj +ju 0 j)juu 0 j: 25 Theorem 1.2.2. (Existence)The BSDE Y t =A + Z T t f(s;Y s ;Z s ;U s )ds Z T t Z s dW s Z T t m X i=1 U i s dM i;s (1.12) has at least a solution (Y;Z;U)2H 1 T (G t )H 2 T (G t ) (H 1 T (G t )) m , where U = (U 1 ;U 2 ;:::;U m ) and A = P m1 n=1 A n 1 fTnT<T n+1 g +A 0 1 fT<T 1 g +A m 1 fTTmg . Proof. We have proved the existence for the specic casem = 1 in theorem (1.1.6). Then we can apply an induction process to prove the result in multiple-defaults scenario. Suppose we have a solution ( ~ Y; ~ Z; ~ U)2H 1 T (G t )H 2 T (G t )(H 1 T (G t )) m1 for the following BSDE Y t =A + Z T t f(s;Y s ;Z s ;U s )ds Z T t Z s dW s Z T t m1 X i=1 U i s dM i;s : (1.13) As we proved in theorem (1.1.6). There exists a solution ( ^ Y; ^ Z)2H 1 T (G t )H 2 T (G t ) for the jump-free BSDE ^ Y t =A m Z T t ^ Z s ds + Z T t q(s; ^ Z s )ds: (1.14) Then we can construct the multiple jumps BSDE's solution. Let Y t = 8 > > < > > : ^ Y t T m t; ~ Y t T m >t; 26 Z t = 8 > > < > > : ^ Z t T m t; ~ Z t T m >t; and U m t = 8 > > < > > : 0 T m <t; ^ Y t ~ Y t T m1 <tT m ; In addition, set ~ U 4 = (U 1 ;U 2 ;:::;U m ). The theorem is proved. Then we can introduce our priori estimate which implies the uniqueness. Theorem 1.2.3. There exist constants l, and C > 0, which only depends on T;C 1 ;C 2 ;k sup t2[0;T ] jY 1 s jk 1 andk sup t2[0;T ] jY 2 s jk 1 , such that E[ sup t2[0;T ] jY t j 2 + Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds]C(E[jAj 2l ]) 1 l : Proof. To prove this theorem, rst we have that for any 2R, by Ito's formula, e t Y 2 t = e T Y 2 T 2 Z T t e s Y s Z s dW s Z T t 2e s Y s U s dM s + 2 Z T t e s Y s [f(s;Z 1;s ;U 1;s )f(s;Z 2;s ;U 2;s )]ds Z T t e s (Y 2 s +jZ s j 2 )ds X t<sT e s U 2 s (M s ) 2 : 27 Using [M;M] t = R t 0 s ds +M t , it can be simplied to e t Y 2 t = e T Y 2 T 2 Z T t e s Y s Z s dW s Z T t e s (2Y s U s +U 2 s )dM s + 2 Z T t e s Y s [f(s;Z 1;s ;U 1;s )f(s;Z 2;s ;U 2;s )]ds Z T t e s (Y 2 s +Z 2 s + s U 2 s )ds: Then we dene the following three processes I s = f(s;Z 1;s ;U 1;s )f(s;Z 2;s ;U 1;s ) Z s ; J s = f(s;Z 2;s ;U 1;s )f(s;Z 2;s ;U 2;s ) U s : Because f(s;Z 1;s ;U 1;s )f(s;Z 2;s ;U 2;s ) = I s Z s +J s U s . As of the properties of f, we have jI s jL 0 (1 +jZ 1;s j +jZ 2;s j); jJ s jL 0 (1 +jU 1;s j +jU 2;s j) p s L 0 (1 + 2L 1 ) p s : The lemma 1.1.5 ensures that R 0 I s dW s is a BMO martingale bounded byCk R 0 (1+ jZ 1;s j +jZ 2;s j)dW s k BMO . We dene ^ W s =W s R t 0 I s ds, then e t Y 2 t = e T Y 2 T 2 Z T t e s Y s Z s d ^ W s Z T t e s (2Y s U s +U 2 s )dM s + 2 Z T t e s Y s J s U s ds Z T t e s (Y 2 s +Z 2 s + s U 2 s )ds: 28 Notice that 2jY s J s U s j 2L 0 (1 + 2L 1 )j p s Y s U s j 2L 2 0 (1 + 2L 1 ) 2 jY s j 2 + s 2 jU s j 2 . Let = 2L 2 0 (1 + 2L 1 ) 2 , we have e t Y 2 t + Z T t e s (jZ s j 2 + s 2 jU s j 2 )ds e T A 2 2 Z T t e s Y s Z s d ^ W s Z T t e s (2Y s U s +U 2 s )dM s : Then we can dene a new measure Q by dQ dP =E(IW ) T , which implies ^ W is a Qmartingale. M is also a Qmartingale because of the independence of M and W . Dene R t =A 2 2 R T t Y s Z s d ^ W s R T t j2Y s U s +U 2 s jdM s , and we have already proved Y 2 t CR t for some constant C. Apply Ito's formula, under the measure Q we have R p t =A 2p Z T t pR p1 s Y s Z s d ^ W s X t<sT ((R s + R s ) p R p s pR p1 s R s ): BecauseR s ; R s 0, and ((R s +j2Y s U s +U 2 s jM s ) p R p s pR p1 s M s ) 0, R p t A 2p Z T t pR p1 s Y s Z s d ^ W s Z T t pR p1 s j2Y s U s +U 2 s jdM s : Taking expectation on both sides for the above inequality, and applying Doob's L p -inequality, E Q [ sup t2[0;T ] jY t j 2p ]E Q [ sup t2[0;T ] jR t j p ]CE Q [jAj 2p ] for some C > 0;8p> 1: 29 Also there are constants C 4 , C 5 , such that ( Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds) p C 4 (jAj 2p + j Z T 0 2Y s Z s d ^ W s + Z T 0 (2Y s U s +U 2 s )dM s j p ): By the Burkholder-Davis-Gundy inequality E Q ( Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds) p C 5 E Q (jAj 2p +j Z T 0 (2Y s Z s ) 2 dsj p 2 + X 0<sT jU s j p jY s +U s j p M s ): We can choose C 6 > 1, such that E Q j Z T 0 (2Y s Z s ) 2 dsj p 2 2 p (2 p+1 C 6 E Q ( sup t2[0;T ] jY s j 2p ) + 1 2 p+1 C 6 E Q ( Z T 0 jZ s j 2 ) p ): On the other hand, because of the bounded jump size of M, and jU s j C 7 sup t2[0;T ] jY t j we have E Q ( X 0<sT jU s j p jY s +U s j p M s ) = E Q ( Z T 0 jU s j p jY s +U s j p dM s + Z T 0 jU s j p jY s +U s j p s ds) C 8 E Q ( sup t2[0;T ] jY t j 2p ): 30 Therefore E Q ( Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds) p C 9 E Q (jAj 2p + sup t2[0;T ] jY t j 2p ): By the property of BMO martingale of R 0 I s d ^ W s , there exists a constant p 0 > 1, such that E Q [E(I ^ W ) p 0 T ]<1. Using the Holder's inequality with p = p 0 p 0 1 , E sup t2[0;T ] jY t j 2 = E Q [E(I ^ W ) T ( sup t2[0;T ] jY t j 2 )] (E Q [E(I ^ W ) p 0 T ]) 1 p 0 (E Q [ sup t2[0;T ] jY t j 2p ]) 1 p C 10 (E Q [jAj 2p ]) 1 p : Since R 0 I s dW s is a BMO martingale with respect to P , 1.1.5 implies there exists a constant q 0 > 1, such that E[E(IW ) q 0 T ] <1. Apply the Holder's inequality again with q = q 0 q 0 1 E sup t2[0;T ] jY t j 2 C 10 (E[E(IW ) T jAj 2p ]) 1 p C 10 (E[E(IW ) q 0 T ]) 1 q 0 (E[jAj 2pq ]) 1 pq C 11 (E[jAj 2pq ]) 1 pq : Similarly we have E( Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds)C 12 (E[jAj 2pq ]) 1 pq : 31 Setting l =pq, C = maxfC 10 ;C 11 g, the theorem is proved. Theorem 1.2.4. (Uniqueness) The BSDE problem (1.2) has a unique solution (Y;Z;U)2H 1 T (R)H 2 T (R d )H 1 T (R). Proof. Let (Y i ;Z i ;U i )2H 1 T (R)H 2 T (R d )H 1 T (R),i = 1; 2 be two solutions for BSDE Problem 1.11. Since A = 0, using Theorem 1.2.3, E[ sup t2[0;T ] jY t j 2 + Z T 0 (jZ s j 2 + s 2 jU s j 2 )ds]C(E[jAj 2l ]) 1 l = 0: Namely, Y 1 =Y 2 , Z 1 =Z 2 , and U 1 =U 2 a:s: 32 Chapter 2 Utility maximization problem with single default 2.1 Preliminary 2.1.1 Literature review The modeling of credit risk can be divided into two classes: structural models and reduced form models. In the structural model, the total value of the rm's assets is directly used to determine the default event, which happens when the rm's value falls through some boundary. In this case the default time is a predictable stopping time with respect to the reference ltration. In the reduced form model, the rm's value process either is not modelled at all, or it is just an auxiliary variable. The default time is modelled as a stopping time that is not predictable. The random time of default event is a totally inaccessible stopping time. The main tool in the reduced form approach is the conditional probability of default given that default has not occurred. The hazard rate of default is also widely used in this case. In this thesis, we focus on the reduce form model. First, we x two 33 ltrations F I (= (F I t ) t0 ) G(= (G t ) t0 ), and the default time is a stopping time with respect toG which avoids anF I stopping time. As an introduction of the conditional default intensity process and the hazard rate model, we refer to [5]. Suppose we work under a probability measure P , the hazard process of the default time given by the information ltrationF I is always dened in the credit risk. Denition 2.1.1. The F I -hazard process of under P , denoted by , is dened by t := ln[P ( >tjF I t )]; 8t2R + : In most of the recent reduced-form models in credit risk, the hazard process is assumed to admit a integral representation for some non-negative,F I -progressively measurable stochastic process , with integrable sample paths. Hypothesis 2.1.2. The hazard process of admits the following integral rep- resentation t = Z t^ 0 s ds; 8t2R + : Although the assumption of the existence of such a conditional default intensity process is widely used, it is usually not easy to formulate the expression of that. 34 In the recent papers, the default density approach was investigated, where the conditional distribution of is assumed to admit a density with respect to some positive -nite measure on R + . With this hypothesis, the conditional default intensity process and conditional default distribution can be formulated. We rst introduce the following density hypothesis from [29]. Hypothesis 2.1.3. We assume that is a non-negative non-atomic measure on R + such that, for any tine t 0, there exists anF I t B(R + )-measurable function (!;)! t (!;) which satises P (2djF I t ) =: t ()(d); Pa:s: The family t () is called the conditional density of with respect to givenF I t . Then the distribution of is given byP ( >) = R 1 0 (u)(du). The conditional distribution of is also characterized by the survival probability function P t () :=P ( >jF I t ) = Z 1 t (u)(du) and P t :=P t (t) =P ( >tjF I t ) = Z 1 t t (u)(du) 35 Then we have the relation for any t, P t () =E(P jF I t ) a:s: namely, Z 1 t (u)(du) =E[ Z 1 (u)(du)jF I t ] a:s: There is a relation between the default intensity and density process. Proposition 2.1.4. The Gstopping time admits a Gintensity G t = 1 f>tg F I t = 1 f>tg t (t) S t ; Pa:s: The process M t := 1 ftg R t 0 G s (ds) is a G-martingale. Proof. We refer the proof to [29]. For t (), the parameter denotes the time of default. Then we consider the case where the information after the default time gives nothing new on the conditional distribution of the default. Under the following hypothesis, we have t () = (); 8t Pa:s:. Hypothesis 2.1.5. (Hypothesis H) Every square integrableF I -martingale is a G-martingale. 36 The hypothesis H also plays a very important role in the conditional default analysis, which is also called immersion property. For a detailed discussion of this hypothesis, we refer to [7], [15], [43], [32] and [42]. There are two useful properties covered in these literatures. Hypothesis H)8t 0;F I t =G t \F I 1 . For every t, the following assertions are equivalent: (1). Hypothesis H holds. (2). For every t,G t andF I 1 are conditionally independent with respect to F I t . Moreover (2) implies8t 0;F I t =G t \F I 1 . Under the density hypothesis (2.1.3), both the conditional intensity and the conditional distribution with respect to the reference ltration F I can be formu- lated using the density process . This was pointed out by [29]. Furthermore the default intensity with respect to the algebraG t :=F I t _(^t) before the default time (on the setft<g) also can be obtained. If the hypothesis (2.1.5) is simultaneously assumed, the conditional intensity and density are equivalent even after the default (on the setftg). In this chapter and next chapter, we will show that we can still formulate the conditional intensity process without the assumption of the existence of the conditional default density process. The method we will use is the ltering. It 37 works well both for before-default and after-default cases even with the absence of the H hypothesis. We will see in chapter 3 that the term structure and the formulation of the conditional default intensity after default are very important when there are multiple default opportunities on the market. 2.1.2 The model We x a probability space ( ;F;P), and all stochastic processes considered will be (F t )-adapted. And we consider a random time as the default time of a company in our portfolio, which is aFstopping time. Our nancial market consists a risky asset and a risk-free asset. Suppose the price of the risky asset has the following dynamic dS t =a(H t )S(t)dt +S(t)dW t S(t)dM t ; where W is anF t -Brownian motion and M is anF t -martingale, which is inde- pendent of B. The H t here is the default indicator process, i.e. H t = 1 ftg . We further suppose that a : [0;T ]!R is aG t measurable function having the form of a t (H) =a 1 (t)1 ftg +a 2 (t)1 ft<g ; where a 1 ;a 2 : [0;T ]!R are continuous. 38 > 0 is a constant. 1 represents the rate of the default. Furthermore, we will model the default time as the rst jump of a poisson point process on the probability space ( ;F;P). Let A =f0; 1g and (A;B A ) be a measurable space. A point process on A is a mappingp : D p !A, whereD p is a countable subset of (0;1). Andp denes a counting random measure N p (dt;dx) on (0;1)A by N p ((0;t]U) =]fs2D p ;st;p(s)2Ug; t> 0; U2B A And we dene the compensator of N p as ^ N p (dtdu) =q(t;du;!)dt. First we postulate U =f1g, hence = infft :p(t)2U; t2 [0;T ]g: Suppose there exists a -nite measure m on a standard measurable space (E;B E ) and a predictable A =A[fg-valued process (H t ;u;!) :f0; 1gE !A 39 such that m(fu :(H t ;u;!)2Ug) =q(H t ;U;!); where is an extra point attached to A. Then by the Theorem 7.4 of Chapter 2 in [21], there exists a (F t )-Poisson point process q on E with the characteristic measure m such that N p ((0;t]U) = Z t 0 Z E I U ((H s ;u;!))N q (dsdu) = ]fs2D q ;st;(H s ;q(s);!)2Ug: We dene K(H s ;u;!) :=(H s ;u;!): We can further assume H t = 1 (t) =H 0 + Z t 0 Z E (1H s )K(H s ;u;!)N q (ds;du) (2.1) For simplicity reason, we use K(H t ;u), N(dt;du) instead of K(H t ;u;!) and N q (dt;du). 40 2.2 The problem In this section we consider a utility maximization problem based on the defaultable asset price model. We dene the compensated default indicator process in the asset price model as M t := 1 ftg Z t^ 0 s ds; where t is a non negative process and continuous int called the conditional default intensity process. Now we can introduce the partial or incomplete information which is generated by the following process. dZ t =a(H t )dt +dW t We denote the ltrationsF I as (Z) andG asF I _(). Such that M t is a G martingale, and t isF I -measurable. Then we will move on to the utility maximization problem. Without loss of generality, we assume the interest rate r equals 0. For an admissible portfolio, let t represent the value invested in the risky asset S. Denition 2.2.1. The class (G), of G-admissible trading strategies is the set if all G-predictable processes such that R T 0 2 t dt<1, P-a.s. 41 Then by the argument of self-nancing portfolio, the wealth process of a strat- egy satises dV t () = t (a(H t )dt +v(t)dW t dM t ); V 0 =v (2.2) Suppose there is also a defaultable contingent claim or a liability X st time T in the portfolio. The payout at time T of it can be written as X T =X 1 1 fTg +X 2 1 f>Tg ; (2.3) whereX 1 andX 2 areF I T measurable. Then we will focus on the following problem: sup 2(G) E P fu(V v T ()X)g where u is the utility function, v is the initial wealth invested in the market. We consider the exponential utility function u(x) = 1e x , with > 0. For the details of utility functions, we refer to [20]. Then we are ready to introduce: Proposition 2.2.2. The optimal trading strategy for the utility maximization problem with self-nancing asset investing (3.1) and the contingent claim (3.2) is the solution of a(H t ) +( t Z t ) + [e (Ut+ t ) ](1H t ) t = 0: (2.4) 42 Here Z t is from the solution of the BSDE Y t =X T + Z T t f(s;Y s ;Z s ;U s )ds Z T t Z s dW s Z T t U s dM s ; (2.5) where f s = Z s a(H s ) (a(H s )) 2 2 2 [e (Us+) 1(U s +)](1H s ) s : (2.6) Proof. By the denition of the utility function, we have sup Efu(V v T ()X T )g = 1 + sup E(e V v T () e X T ): Suppose we can nd a processY withY T =X T , such that the processe V v t () e Y T is aG-supermartingale under P for any admissible strategy 2 (G) and is a martingale under P for some admissible strategy 2 (G). Then E(e V v T ( ) e X T ) = e V v t ( ) e Xt E(e V v T () e X T ) for86= ; namely, sup Efu(V v T ()X T )g =Efu(V v T ( )X T )g: 43 Suppose Y t is the solution for the BSDE Y t =X T + Z T t f s ds Z T t Z s dW s Z T t U s dM s ; and f is the generator such that the processe V v t ()+Yt is aG-supermartingale under P for any admissible strategy and is a martingale under P for some admissible strategy. Then we will nd the expression of f. Apply Ito's formula, d(V t Y t ) = t (a(H t )dt +dW t ) +f t dtZ t dW t (U t + t )dM t = [ t a(H t ) +f t ]dt + ( t Z t )dW t (U t + t )dM t ; and e (VtYt) =e (V T Y T ) + R T t e (V s Y s ) d(V s Y s ) R T t 1 2 2 e (V s Y s ) d[VY;VY ] c s P t<sT [e (VsYs) e (V s Y s ) e (V s Y s ) (V s Y s )] =e (V T Y T ) + R T t e (V s Y s ) [f s + s a(H s ) 1 2 ( s Z s ) 2 ]ds + R T t e (V s Y s ) [( s Z s )dW s (U s + s )dM s ] P t<sT e (V s Y s ) M s [e (Us+ s ) 1(U s + s )]: (2.7) 44 The last equality is because (V s Y s ) =(U s + s )M s and (M s ) n = M for any n 1. We also have P t<sT e (V s Y s ) M s [e (Us+ s ) 1(U s +)] = Z T t e (V s Y s ) [e (Us+ s ) 1(U s + s )]dM s + Z T t e (V s Y s ) [e (Us+ s ) 1(U s + s )](1H s ) s ds Let us choose t such that minimizing t a(H t ) + 1 2 ( t Z t ) 2 + [e (Ut+ t ) 1(U t + t )](1H t ) t : (2.8) Then we can set f t = t a(H t ) 1 2 ( t Z t ) 2 [e (Ut+ t ) 1(U t + t )](1H t ) t : (2.9) such that the martingale property is satised. Taking derivative of (2.8) respect to , s is the solution toa(H t )+( t Z t )+[e (Ut+ t ) ](1H t ) t = 0. Then f t = t a(H t ) 1 2 ( t Z t ) 2 + [1 +(U t + t )](1H t ) t + a(H t )( t Z t ) +(1H t ) t 45 Based on the results in chapter 1, we have the following theorems. Theorem 2.2.3. The BSDE problem (2.5) has a unique solution (Y;Z;U) 2 H 1 T (R)H 2 T (R d )H 1 T (R) if the generator f satises the assumption (1.1.2). Proof. Refer to theorem (1.1.6) and theorem (1.2.4) Besides, there are two issues should be noticed: (1) Usually it is not trivial to get the expression of the conditional default inten- sity. In [29], they provided a method to formulate based on the assump- tion of the existence of the consitional default density process we introduced in section 2. In that case, by Proposition 2.2, we have t = t (t) P t ; Pa:s: where P t can be calculated by P t = P ( > tjF I t ) = R 1 t t (u)du. In next section, we will introduce another approach for the default intensity pro- cess without the density assumption, where we use the ltering method to formulate it. (2) The Hypothesis H also plays an essential role in the BSDE arguments in the literature. For the case when Hypothesis holds, we refer to [2] and [37] for 46 the BSDE argument. In our model, the Hypothesis H is not satised, and we will see the diculty and solution for the BSDE problem in this case. Hence, our objectives consists formulating the conditional default intensity process and prove the existence and uniqueness results of the BSDE in Proposition 2.2.2. 47 2.3 The ltering approach 2.3.1 Filtering In this section, we will apply the ltering techniques to formulate the intensity process. Our signal process is the default indicator process H t = 1 ftg . Our observation process is the rate of the asset price change Z t , satisfying dZ t =a(H t )dt +dW t : Recall that W t is a standard Brownian Motion on ( ;F;P), independent of H. The ltration for the observable information isF I t =F Z t . The full information G t =F I t _F H t . In order to derive the conditional distribution of the default event as well as the expression of the conditional intensity process t , we will apply the ltering techniques in the rest of this section. First we are ready for the Girsanov measure transformation. Dene dP dQ jFt :=L t , where L t = exp( Z t 0 a(H s ) dZ s 1 2 Z t 0 [ a(H s ) ] 2 ds): Under the equivalent measure Q, Z is a standard Brownian motion. Lemma 2.3.1. L is a Q-martingale, and E Q [L t ] = 1. 48 Proof. This is a direct implication of the bounded property of a(H t ). Furthermore, underP ,Z t has the original dynamics, and sinceW is orthogonal to the martingale M, the law of H still follow the original SDE. Our objective is to determine the conditional expectation (t)(f)(:= E P (f(H t )jF I t )) under the original measure P . Here f(x)2C 2 b (R). Lemma 2.3.2. (Kallianpur-Striebel's formula) The optimal lter (t) can be represented as (t)(f) = p(t)(f) p(t)(1) ; (2.10) where p(t)(f) =E Q (f(H t )L t jF I t ); and E Q refer to the expectation with respect to the measure Q. Proof. For any A2F I t , we have Z A E Q (f(H t )L t jF I t ) E Q (L t jF I t ) dP = E Q [1 A E Q (f(H t )L t jF I t ) E Q (L t jF I t ) L t ] = E Q [1 A E Q (f(H t )L t jF I t ) E Q (L t jF I t ) E Q (L t jF I t )] = E Q (1 A f(H t )L t ): 49 On the other hand, Z A E P (f(H t )jF I t )dP =E P (1 A f(H t )) =E Q (1 A f(H t )L t ) The lemma is proved. Then we can derive the Zakai's equation for the unnormalized lter p(t)(f), and FKK equation for the optimal lter (t)(f). Before that, we rst introduce the following lemma Lemma 2.3.3. Suppose that f and g are predictable processes on the stochastic basis ( ;F;F t ;Q) satisfying E Q Z T 0 jf s jds +E Q Z T 0 jg s j 2 ds<1: Then,we have E Q ( Z t 0 f s dsjF I t ) = Z t 0 E Q (f s jF I s )ds; (2.11) E Q ( Z t 0 g s dZ s jF I t ) = Z t 0 E Q (g s jF I s )dZ s ; (2.12) 50 Proof. Suppose f is simple, i.e. f s = k X j=1 f j 1 (a j ;b j ] (s); where(a j ;b j ], j=1,2,...,k are disjoint subintervals of [0;t], andf j isF a j measurable, j=1,2,...,k. Let G s;t =(Z u Z s :sut): Then we have F I t =F I a j _G a j ;t Then E Q ( Z t 0 f s dsjF I t ) = k X j=1 E Q (f j (b j a j )jF I t ) = k X j=1 E Q (f j jF I a j _G a j ;t )(b j a j ) = k X j=1 E Q (f j jF I a j )(b j a j ) = Z t 0 E Q (f s jF I s )ds; 51 where the equality prior to the last one follows from the independence of incre- ments of the Brownian motion Z. Therefore equation (2.11) holds for simple pro- cesses. If f 0, we can take an increasing sequence of simple processes converging pointwise to f. Then the equation (2.11) follows from the monotone convergence theorem. For general f, we can take f = f + f ; where f + ;f 0, then we can apply the monotone convergence theorem. Now let us prove equation (2.12). Suppose g is simple, g s = k X j=1 g j 1 (a j ;b j ] (s); whereg j isF a j measurable, j=1,2,...,k. Again using the independence of increments for Z, we get E Q ( Z t 0 g s dZ s jF I t ) = k X j=1 E Q (g j (Z b j Z a j )jF I t ) = k X j=1 E Q (g j jF I t )(Z b j Z a j ) = k X j=1 E Q (g j jF I a j )(Z b j Z a j ) = Z t 0 E Q (g s jF I s )dZ s 52 For a general g, we can approximate it by a sequence of simple processes g n such thatjg n s jjg s j; a.s. for allst: Then E Q j Z t 0 g n s dZ s j 2 E Q Z t 0 jg s j 2 dZ s <1; and hence R t 0 g n s dZ s :n 1 is uniformly integrable. Thus E Q ( Z t 0 g s dZ s jF I t ) = lim n!1 E Q ( Z t 0 g n s dZ s jF I t ) = lim n!1 E Q ( Z t 0 g n s jF I s )dZ s = Z t 0 E Q (g s jF I s )dZ s : Now we are ready to introduce the theorem Theorem 2.3.4. (Zakai's equation) The unnormalized lter p(t)(f) satises the following stochastic dierential equation: p(t)(f) =p(0)(f) + Z t 0 p(s)(af)dZ s + Z t 0 p(s)(K[f](H s ))ds where K[f](x) = Z E [f(x + (1x)K(x;u))f(x)]F N (du) 53 Proof. First by Ito's formula, dL t =L t a t (H t )dZ t and, For any semi-martingale Y t , apply Ito's formula to f(Y t ) f(Y t ) =f(Y 0 ) + Z t 0 f 0 (Y s )dY s + 1 2 Z t 0 f 00 (Y s )d[Y c ;Y c ] s + X 0<st ff(Y s )f(Y s )f 0 (Y s )4Y s g; where Y c denotes the continuous part of Y . In our model, we have H t = 1 (t) =H 0 + Z t 0 Z E (1H s )K(H s ;u)N(ds;du) hence, df(H t ) = Z E [f(H t + (1H t )K(H t ;u))f(H t )]N(dt;du) By integration by parts, dL t f t =f t L t a t (H t )dZ t +L t Z E [f(H t + (1H t )K(H t ;u))f(H t )]N(dt;du): 54 namely, L t f(H t ) = f(0;H 0 ) + Z t 0 f(H s )L s a s (H s )dZ s + Z t 0 Z E L s [f(H s + (1H s )K(H s ;u))f(H s )]N(dt;du) Take conditional expectation on both sides and apply Lemma[2.3.3], p(t)(f) =p(0)(f) + Z t 0 p(s)(af)dZ s + Z t 0 p(s)(K[f](H s ))ds Then we can derive the stochastic dierential equation for the optimal lter. Theorem 2.3.5. (Kushner-FKK equation)The optimal lter(t)(f) satises the following stochastic dierential equation: For any bounded and C 2 function f, (t)(f) = (0)(f) + Z t 0 (s)(f)(s)(a) 2 (s)(a)(s)(af) +(s)(K[f](H s ))ds + Z t 0 (s)(af)(s)(f)(s)(a)dZ s : Proof. By the denition (t)(f) = p(t)(f) p(t)(1) : 55 By the Zakai's equation, p(t)(1) = 1 + Z t 0 p(s)(a)dZ s : Then we have p(t)(1) 1 = 1 + Z t 0 p(s)(a) 2 p(s)(1) 3 ds Z t 0 p(s)(a) p(s)(1) 2 dZ s : By integration by parts, we get d(t)(f) =p(t)(f)dp(t)(1) 1 +p(t)(1) 1 dp(t)(f) +d[p(t)(1) 1 ;p(t)(f)] t Therefore, (t)(f) = (0)(f) + Z t 0 (s)(f)(s)(a) 2 (s)(a)(s)(af) +(s)(K[f](H s ))ds + Z t 0 (s)(af)(s)(f)(s)(a)dZ s : From the following theorem, we can see the lter model we discussed does not satises the immersion property, where some results of the default density theory could not be obtained. 56 Theorem 2.3.6. The ltering model of Z and H does not satisfy the immersion property between the ltrationF I t andG t . Proof. To prove our result, it is sucient to give anF I t -martingale, which is not aG t -martingale. First we dene the innovation process v t by dv t =dZ t (t)(a)dt: (2.13) Under P , for t>s, E P (v t jF I s ) = E P (Z t Z t 0 (r)(a)drjF I s ) = E P (Z t Z s Z t s (r)(a)drjF I s ) +v s Since dW t =dZ t a t (H t )dt, we obtain E P (v t jF I s ) = E P (W t W s + Z t s a r (H r )(r)(a)drjF I s ) +v s = Z t s E P ((E P (a r (H r )jF I r )a r (H r ))jF I s )dr +v s = v s ; Therefore, v t is a F I t -martingale. Under P , we have dv t = dZ t (t)(a)dt = a t (H t )dt +dW t (t)(a)dt 57 namely, v t = Z t 0 a s (H s )ds +W t Z t 0 (s)(a)ds the quadratic variation of v t is t, i.e. [v;v] t = [W;W ] t =t. Hence, by Theorem 39 (Levy's Theorem) in [39], v t is anF I t Brownian motion under P . On the other hand, W t =Z t Z t 0 a s (H s )ds is aG t Brownian motion under P . We have v t W t = Z t 0 a s (H s )ds Z t 0 (s)(a)ds = Z t 0 a s (H s )(s)(a)ds: By the martingale representation theorem in [32], v t is not aG t -martingale under P . 2.3.2 Summary of intensity, density, and H hypothesis In this section, we apply the ltering results for deriving the conditional default intensity process and discuss the relation between intensity, density, and H hypoth- esis in a deeper insight. 58 First, we have the following equivalence result for density hypothesis and the hypothesis H: Theorem 2.3.7. Suppose Density Hypothesis holds, we have theL t -conditional density with respect to the measure . Then H Hypothesis holds i t () = (); 8t Pa:s:. Proof. For the \(" direction, we have P =P ( >jL ) = 1 Z 0 (u)(du) = 1 Z 0 u (u)(du): On the other hand, for any t>, P t () =P ( >jL t ) = 1 Z 0 t (u)(du) = 1 Z 0 u (u)(du) =P ; Pa:s: Then let t!1, we get P t () =E[P jL t ] when <t, which implies P ( >jL ) =P ( >jL 1 ): which is equivalent to the Hypothesis H. For the \)" direction, if the Hypothesis H or immersion property is valid, we haveL t =J t \L 1 , for every t> 0 (refer to [7]). Then, for8t t () =P (2djL t ) =E(1 f2dg jL t ) Pa:s: 59 and E(1 f2dg jL t ) = E(1 f2dg jJ t \L 1 ) = E(E(1 f2dg jJ )jJ t \L 1 ) = E(1 f2dg jJ \L 1 ) = E(1 f2dg jL ) dPa:s: namely, t () = (); 8t dPa:s: From our results of ltering, we can formulate the relation between density and the ltering. As the argument in the beginning of this section, we dened P t :=P ( >tjF I t ) = 1(t)(i) with i2C 1 b (R), and i(x) =x when x2 [0; 1]. By Theorem (3.5) and H 0 = 0, we have P t = 1 Z t 0 (s)(i)(s)(a) 2 (s)(a)(s)(ai) +(s)(K[i](H))ds Z t 0 (s)(ai)(s)(i)(s)(a)dZ s ; 60 We already proved v t =Z t R t 0 (s)(a)ds is anF I t Brownian motion, therefore P t = 1 Z t 0 (s)(ai)(s)(i)(s)(a)dv s Z t 0 (s)(K[i](H))ds: By the result in [19], the Doob-Meyer decomposition of the survival process P t is given by P t = 1 + ^ M t R t 0 u (u)(du) where ^ M t is the cadlag square-integrable F I t martingale. Then referring to the representation result of the Zakai's equation and the uniqueness of the Doob-Meyer decomposition we have d ^ M t =(t)(i)(t)(a)(t)(ia)dv t ; s (s) =(s)(K[i](H)) In [29], the intensity process was formulated under the density hypothesis by t := t(t) Pt , where P t = P ( > tjF I t ) = R 1 t t (u)du. But it is usually not trivial to get the expression of the density process t (u). The ltering arguments we provided gave another way to compute the intensity process, especially when the density process is not known. And in the case when the hypothesis H is not correct, the intensity also can be obtained by our ltering method. Theorem 2.3.8. TheF I t -intensity t of default follows the dynamics d t =m t dt +n t dv t 61 where m t = (t)(KK[i](H)) P t + (t)(K[i](H)) 2 P 2 t [1 + (t)(K[i](H)) P t ] + (t)(ia)(t)(i)(t)(a) P 2 t [(t)(K[i](H)a)(t)(K[i](H))(t)(a)] and n t = (t)(K[i](H)a) P t (t)(K[i](H))(t)(a) P t + [(t)(ia)(t)(i)(t)(a)] P 2 t Proof. Recall the dynamics for S t and t (t), dP t = [(t)(ia)(t)(i)(t)(a)]dv t (t)(K[i](H))dt: Apply Ito' formula dP 1 t =P 2 t dP t +P 3 t d[P;P ] t and d t = d(t)(K[i](H)) = [(t)(K[i](H)a)(t)(K[i](H))(t)(a)]dv t +(t)(KK[i](H))dt 62 Since t = t(t) Pt , and by the integration by parts, d t =m t dt +n t dv t m and n have the expression in the Theorem. At the end, we nish this subsection by the following example: Example 2.3.9. Suppose we have a defaultable claim X in the form of X =X 0 1 f>Tg +f()1 fTg where X 0 is a G T measurable random variable and f is bounded Borel func- tion.Then the price of the claim under partial information can be represented as E[XjF I t ] =X 0 Z 1 t E[(s)(K[i](H)jF I t ]ds + L t L t S t + R t 0 L s s (s)ds Z t 0 s (s)ds: 63 Chapter 3 The multiple defaults market with frailty In this chapter, we will extend our results into the market with multiple defaults and a frailty process. Again we x a probability space ( ;F;P). All the stochastic processes considered in this section areF-adapted. 3.1 The market model Let the processes H t = (H t;1 ;H t;2 ;:::;H t;m ) on the probability space ( ;F;P) be the default indicator processes of m companies. Again let A =f0; 1g and (A;B A ) be a measurable space. A point process onA is a mappingp : D p !A, where D p is a countable subset of (0;1). And p denes a counting random measure N p (dt;dx)on (0;1)A by N p ((0;t]U) =]fs2D p ;st;p(s)2Ug; t> 0; U2B A Then we dene the compensator of N p as ^ N p (dtdu) = ^ p(t;du;!)dt for some ^ p. 64 We suppose the default times of the companies are driven by a point process p, which means the ith default time is the ith time of p jumping to 1. Here we postulate U =f1g, so T i := infft :p(t)2U; t>T i1 g; T 0 := 0: Suppose there exists a -nite measure on a standard measurable space (E;B E ) and m A =A[fg-valued processes i (H t ;u;!) :f0; 1g m E !A ; i = 1; 2;:::m: such that, onft>T i1 g (fu : i (H t ;u;!)2Ug) = ^ p(t;U;!); where is an extra point attached to A. Then there exists a (F t )-Poisson point process q on E with the characteristic measure m such that N p ((0;t]U) = Z t 0 Z E I U ( i (H s ;u;!))N q (dsdu) = ]fs2D q ;st; i (H s ;q(s);!)2Ug: 65 We dene K i (H s ;u;!) = i (H s ;u;!): Then we suppose H i;t = 1 ( i t) =H i;0 + Z t 0 Z E (1H i;s )K i (H s ;u;!)N q (ds;du) (3.1) Then if we take the frailty process into account, we further assume the dynamic of each H i;t and the frailty process X satisfy: H i;t =H i;0 + Z t 0 Z R (1H i;s )K i (X s ;H s ;u)N (ds;du) 1im (3.2) X t =X 0 + Z t 0 b(X s ;H s )ds + Z t 0 (X s ;H s )dB s + Z t 0 Z R K X (X s ;H s ;u)N (ds;du) (3.3) Here H t = (H 1;t ;H 2;t ;:::;H m;t ) andN (ds;du) denotes a (P;F)-standard Poisson random measure on R + R, with compensator measureF N (du)ds. We assume thatK i (x;h;u)2f0; 1g for allx;h;u and all 1im. The processX t is referred to a frailty process with state space S X , which is related to the default condition of the m observable companies. The b and are bounded continuous R-valued functions on S X f0; 1g m . B is a (P;F)Brownian motion independent of H. The concept of frailty was rst introduced in statistical models in the literature 66 referring to a unobserved covariate. It is assumed to be static at the beginning, which is unreasonable in the nancial market. In the recent papers such as [16], the frailty was extended to a covariate to vary over time according to an autoregressive time-series specication. In our situation, we suppose the frailty process having the dynamic (3.3), which has an interaction with the default condition of the m companies in the market. We assume the existence and uniqueness to the system (3.2)-(3.3). This assumption implies the pair (X t ;H t ) is a (P;F t )-Markov process. Then we introduce some notations for future use. Let = ( 1 ; 2 ;:::; m ) be the unordered default times for the m companies and (T 1 ;T 2 ;:::;T m ) be the ordered version of default times i.e. we have 0 <T 1 <T 2 <:::<T m <1. For t 0, we denote i 4 =ft :T i t<T i+1 g which represents the scenario where i defaults have already happened before time t. The asset price process denoted as an Fadapted process S, which is related to the defaults of the m companies. S t =S 0 + Z t 0 S s a s (X s ;H s )ds + Z t 0 S s (H s )dW s m X i=1 Z t 0 S s i s d ~ M i;s (3.4) 67 wherea(X s ;H s ) = P m i=0 1 i(s)a i s (X s ),(H s ) = P m i=0 1 i(s) i and ~ M i;s = 1 f i <sg R s 0 (1H i;v ) i v dv, with i t being theFintensity process of companyi. a i t : [0;1) S X ! R are continuous and bounded. i 's are constants. i s (i = 1; 2;:::;m) are continuous deterministic functions of time and take values in (1; 1]. Denote the jump size of the asset price at the when company i defaults. W is anF Brownian motion independent of B and the defaults. Let D i (x;h) :=fu2 R : K i (x;h;u)6= 0g for any 1 i m and D X (x;h) :=fu2 R : K X (x;h;u)6= 0g, then by denition H t;i Z t 0 Z R (1H s;i )K i (X s ;H s ;u)F N (ds;du) =H t;i Z t^ i 0 F N (D i (X s ;H s ))ds is anFmartingale. Hence, i (X s ;H s ) :=F N (D i (X s ;H s )) is the uncon- ditional default intensity process of company i. Assumption 3.1.1. For8h2R n ; i (x;h)2C b 2 (R);i = 1; 2;:::;m: Assumption 3.1.2. For all 1 i m and all T > 0 we have E( R T 0 F N (D i (X s ;H s ))ds)<1. Assumption 3.1.3. For all 1 i 1 < i 2 m and all (x;h)2 S X f0; 1g m we haveF N (D i 1 (x;h)\D i 2 (x;h)) = 0. Then we consider the following process Z t =Z 0 + Z t 0 a s (X s ;H s )ds + Z t 0 (H s )dW s m X i=1 Z t 0 i s d ~ M i;s : (3.5) 68 Note that Z t is the rate of change of S t . 3.2 The ltering In the reduce form credit risk model, the investors only have partial observation in the market. The ltration of observationfF I t g t2[0;T ] contains all information on the rate of change of the asset price and all the default conditions of the m companies. Denition 3.2.1. We dene the observation or information ltration as the l- tration generated byZ and the full information as the ltration generated byZ and X. F I t 4 = F Z _F H G t 4 = F I t _F X t : According to this denition, we have the relation F I t G t F t : (3.6) Our purpose in this section is to describe the lter that is the conditional distribution of the signal X given the observationF I , which is the projection of X t onto the observable information ltration. 69 Using the fact M i;s = H i;s we can rewrite our observation processes Z t and H i;t , i = 1; 2:::m Z t =Z 0 + R t 0 [a s (X s ;H s ) + P m i=1 (1H i;s ) i (X s ;H s )]ds + R t 0 (H s )dW s P m i=1 R t 0 R R (1H i;s ) i s K i (X s ;H s ;u)N (ds;du) H i;t =H i;0 + R t 0 R R (1H i;s )K i (X s ;H s ;u)N (ds;du) 1im (3.7) Suppose we have a probability space ( ;F t ;Q) supporting a solution (H;X) of system (3.2) and (3.3). And is anF t Brownian motion under Q, which is independent of H and X. Dene ~ Z t 4 = t . Apply Girsanov change of measure with dR dQ jF T =L T , where L t 4 = exp( R t 0 as(Xs;Hs)+ P m i=1 (1H i;s ) i (Xs;Hs) ((Hs)) 2 d ~ Z s 1 2 R t 0 j as(Xs;Hs)+ P m i=1 (1H i;s ) i (Xs;Hs) (Hs) j 2 ds): Lemma 3.2.2. L t is a Q-martingale with respect toG t . Proof. By denition L t = exp( R t 0 as(Xs;Hs)+ P m i=1 (1H i;s ) i (Xs;Hs) ((Hs)) 2 d ~ Z s 1 2 R t 0 j as(Xs;Hs)+ P m i=1 (1H i;s ) i (Xs;Hs) (Hs) j 2 ds) = exp( R t 0 as(Xs;Hs)+ P m i=1 (1H i;s ) i (Xs;Hs) (Hs) d s 1 2 R t 0 j as(Xs;Hs)+ P m i=1 (1H i;s ) i (Xs;Hs) (Hs) j 2 ds) Applying Ito's formula dL t = L t [a t (X t ;H t ) + P m i=1 (1H i;t ) i (X t ;H t )] (H t ) d t (3.8) 70 Implied by the bounded condition of a and i (i = 1; 2;:::m), L t is a Q-martingale with respect toG t . Under R, the process ~ Z has the dynamics d ~ Z t = [a t (X t ;H t ) + m X i=1 (1H i;t ) i (X t ;H t )]dt +(H t )dW t : (3.9) Since is orthogonal to both B and the martingale from compensating the possion random measureN , the measure transformation has no changes in the law ofX andH. By theorem 6.3 in chapter 2 of [21], we deduceB is independent of the jumps. Hence underR, (X;H;Z) has the same joint law underP . Without loss of generality, we suppose P and R are identical. Remark 3.2.3. From the model setup, our partial information includes the l- tration generated by Z, which means we can observe the G-Brownian Motion W and jumps with noise, but not W and jumps themselves. In our model, W is notF I -adapted. The process Z can be taken to represent cumulative noisy price- observations for traded credit derivatives. For the interpretation of the noisy obser- vations of functions of the factor process, we refer to [41]. Under Q, we have Z t =Z 0 + ~ Z t m X i=1 Z t 0 Z R (1H i;s ) i s K i (X s ;H s ;u)N (ds;du): (3.10) 71 Dene (u;s) 4 = P m i=1 i s (1H i;s )K i (X s ;H s ;u). Again, for any f :R + S X !R by the Kallianpur-Striebel formula E P [f(t;X t )jF I t ] = E Q [f(t;X t )L t jF I t ] E Q [L t jF I t ] : (3.11) Before deriving the unnormalized lter for f2C 1;2 (R + S X ), we rst introduce the elementary lemma and corollary. Lemma 3.2.4. Let ( ;F;P 0 ) be a complete probability space. LetG 1 andG 2 be two sub--elds ofF, and Y be anF-measurable random variable. If (under P 0 ) G 1 W (Y )?G 2 , then E P 0 fYjG 1 _ G 2 g =E P 0 fYjG 1 g; P 0 -a.s.: Proof. For any F2G 1 and G2G 2 , we have E P 0 fE P 0 fYjG 1 _ G 2 gFGg = E P 0 fYFGg =E P 0 fYFgE P 0 fGg = E P 0 fE P 0 fYjG 1 gFgE P 0 fGg =E P 0 fE P 0 fYjG 1 gFGg: Note that since bothE P 0 fYjG 1 W G 2 g andE P 0 fYjG 1 g areG 1 W G 2 -measurable, the lemma follows from a simple Monotone-Class argument. 72 Corollary 3.2.5. LetY be anfG t g-adapted process, satisfyingE Q R t 0 jY s j 2 ds<1. Then for any 0ts, it holds that (1) E Q fY t L t jF I s gE Q fY t L t jF I t g = 0. (2) E Q f R s t Y v dB v jF I s g = 0. (3) E Q f R s t Y v d ~ Z v jF I s g = R s t E Q fY v jF I v gd ~ Z v . (4)IfF N (D i (x;h)\D X (x;h)) = 0 for i = 1; 2;:::;m, then E Q f Z s t Z R Y v N (du;dv)jF I s g = 0: (3.12) Proof. (1) We need only consider t < s. DenoteG 1 =F I t ,G 2 = (F I ) t s . Then F I s =G 1 W G 2 , andG 1 W (Y )?G 2 . Thus applying Lemma 3.2.4 with P 0 =Q we obtain (1). (2) By standard approximation, we need only show that (2) holds for any simple,fG t g-predictable processes Y t =Y 0 + P n i=1 Y i 1 (t i ;t i+1 ] (t), t 0, where Y i 2 G t i . We might assume without loss of generality that t = t 0 < t 1 < < t n = s. 73 Applying the similar arguments as those in part (1) and noting that B? H and B?Z we get E Q f Z s t Y v dB v jF I g = E Q f n X i=1 Y i (B t i +1^s B t i ^s )jF I s g = n X i=1 E Q fY i (B t i +1 B t i )jF I s g = n X i=1 E Q fY i (B t i +1 B t i )jF I t i g: Since B? H and B? Z, B is also anfF I t W F B t g-Brownian motion. Hence for each i, E Q fY i (B t i +1 B t i )jF I t i g =E Q fE Q fY i (B t i +1 B t i )jF I t i _ F B t i gjF I t i g = 0; proving (2). (3) and (4) can be proved using the same method. Suppose Y is simple, Y s = P k j=1 Y j 1 (a j ;b j ] (s), where Y j isG a j measurable, j = 1; 2;:::;k. Using the independence of increments for ~ Z, we get E Q ( Z s t Y v d ~ Z v jF I s ) = k X j=1 E Q (Y j ( ~ Z b j ~ Z a j )jF I s ) = k X j=1 E Q (Y j jF I s )( ~ Z b j ~ Z a j ) = k X j=1 E Q (Y j jF I a j )( ~ Z b j ~ Z a j ) = Z t 0 E Q (Y v jF I v )d ~ Z v 74 For a generalY , we can approximate it by a sequence of simple processesY n such thatjY n v jjY v j, a.s. for all tvs: Then E Q j Z s t Y n v d ~ Z v j 2 E Q Z s t jY v j 2 d ~ Z v <1; and hencef R s t Y n v d ~ Z v :n 1g is uniformly integrable. Thus E Q ( Z s t Y v d ~ Z v jF I s ) = lim n!1 E Q ( Z s t g n s d ~ Z v jF I s ) = lim n!1 E Q ( Z s t Y n v jF I v )d ~ Z v = Z t 0 E Q (Y v jF I v )d ~ Z v : (3) is proved. Theorem 3.2.6. (Zakai's equation) The unnormalized lter p(t)(f) 4 = E Q [f(t;X t )L t jF I t ] satises the following stochastic dierential equation: p(t)(f) =p(0)(f)+ Z t 0 p(s)(Af)ds+ Z t 0 p(s)( f 2 )d ~ Z s + Z t 0 Z R p(s)(K[f])N (du;ds): (3.13) where K[f](x) = Z R [f(x + (1x)K X (x;H;u))f(x)]F N (du) (3.14) A(s)f 4 = @f @s + @f @x b(X s ;H s ) + 1 2 @ 2 f @x 2 2 (X s ;H s ): (3.15) 75 (X s ;H s ) 4 =a s (X s ;H s ) + m X i=1 (1H i;s ) i (X s ;H s ): (3.16) Proof. For any xed 0t<sT consider the dierence p(s)(f)p(t)(f). p(s)(f) p(t)(f) =E Q [f(s;X s )L s jF I s ]E Q [f(t;X t )L t jF I t ] = E Q [f(s;X s )L s f(t;X t )L t jF I s ] +E Q [f(t;X t )L t jF I s ]E Q [f(t;X t )L t jF I t ] By Corollary (3.2.5-(1)) and applying It^ o's formula to f(t;X t ), we have f(t;X t ) = Z t 0 [ @f @v + @f @x b(X v ;H v ) + 1 2 @ 2 f @x 2 2 (X v ;H v )]dv + Z t 0 @f @x (X v ;H v )dB v + Z t 0 Z R f(v;X v +K X (X v ;H v ;u))f(v;X v )N (du;dv): Recall that L t = exp( Z t 0 (X v ;H v ) ((H v )) 2 d ~ Z v 1 2 Z t 0 j (X v ;H v ) (H v ) j 2 dv): (3.17) Further, applying It^ o's formula and integration by parts, we have, f(s;X s )L s f(t;X t )L t = R s t f(v;Xv )Lv (Xv;Hv ) (Hv ) 2 d ~ Z v + R s t L v [ @f @v + @f @x b(X v ;H v ) + 1 2 @ 2 f @x 2 2 (X v ;H v )]dv + R s t R R L v [f(v;X v +K X (X v ;H v ;u))f(v;X v )]N (du;dv) + R s t L v @f @x (X v ;H v )dB v (3.18) 76 Let us now take conditional expectation E Q fjF I s g on both sides above, and analyze the right side term by term. First note that by applying Lemma 3.2.4 we obtain that E Q f Z s t L v A(v)fdvjF I s g = Z s t E Q fL v A(v)fjF I v gdv: (3.19) Now, applying Corollary 3.2.5-2)-4), we see that E Q f Z s t f(v;X v )L v (X v ;H v ) (H v ) 2 d ~ Z v jF I s g = Z s t E Q f f(v;X v )L v (X v ;H v ) (H v ) 2 jF I v gd ~ Z v (3.20) E Q f Z s t L v @f @x (X v ;H v )dB v jF I s g = 0 (3.21) and E Q f R s t R R L v [f(v;X v +K X (X v ;H v ;u))f(v;X v )]N (du;dv)jF I s g = R s t R R E Q fL v [f(v;X v +K X (X v ;H v ;u))f(v;X v )]jF I v gN (du;dv): (3.22) We conclude p(t)(f) =p(0)(f)+ Z t 0 p(s)(Af)ds+ Z t 0 p(s)( f 2 )d ~ Z s + Z t 0 Z R p(s)(K[f])N (du;ds): (3.23) Then we can derive the stochastic dierential equation for the normalized lter. 77 Theorem 3.2.7. (Kushner-FKK equation)The optimal lter (t)(f) 4 = E P [f(t;X t )L t jF I t ] satises the following stochastic dierential equation (t)(f) = (0)(f) + R t 0 (s)(f)[(s)()] 2 + (s)(Af) (s)( 2 )(s)(f)ds + R t 0 (s)( f 2 ) (s)(f)(s)( 2 )d ~ Z s + R t 0 R R (s)(K[f])N(du;ds): (3.24) Proof. By (3.11), we have (t)(f) = p(t)(f) p(t)(1) . Applying Theorem 3.2.6 and Ito's formula p(t)(1) = 1 + Z t 0 p(s)( 2 )d ~ Z s (3.25) and [p(t)(1)] 1 = 1 + Z t 0 2 [p(s)( 2 )] 2 [p(s)(1)] 3 ds Z t 0 p(s)( 2 ) p(s)(1) 2 d ~ Z s (3.26) = 1 + Z t 0 [p(s)()] 2 [p(s)(1)] 3 ds Z t 0 p(s)( 2 ) p(s)(1) 2 d ~ Z s : By integration by parts, we get (t)(f) = (0)(f) + Z t 0 (s)(f)[(s)()] 2 + (s)(Af) (s)( 2 )(s)(f)ds + Z t 0 (s)( f 2 ) (s)(f)(s)( 2 )d ~ Z s (3.27) + Z t 0 Z R (s)(K[f])N(du;ds): 78 Compared with the unconditional default intensity process i (= 1; 2;:::;m) for company i, theF I -intensity is more important in the reduced-form credit risk models. Thanks to Theorem 3.2.7, we have ^ t;i 4 =E P ( i (X t ;H t )jF I t ) = (t)( i ): (3.28) Namely, ^ t;i = (0)( i ) + R t 0 (s)( i )[(s)()] 2 + (s)(A i ) (s)( 2 )(s)( i )ds + R t 0 (s)( i 2 ) (s)( i )(s)( 2 )d ~ Z s + R t 0 R R (s)(K[ i ])N(du;ds): (3.29) Next we can dene theF I t -innovation process in the market with multiple defaults. v t 4 = ~ Z t Z t 0 (s)( )ds: (3.30) Theorem 3.2.8. The innovation process v t is a F I t -Brownian motion under P . Proof. Under P , for t>s, E P (v t jF I s ) = E P ( ~ Z t (H t ) Z t 0 (u)( )dujF I s ) = E P ( ~ Z t (H t ) ~ Z s (H s ) Z t s (u)( )dujF I s ) +v s 79 Since W t = ~ Zt (Ht) R t 0 u (Hu) du is a PG Brownian motion E P (v t jF I s ) = E P (W t W s + Z t s u (H u ) (u)( )dujF I s ) +v s = E P (E P (W t W s jG s )) + Z t s E P (E P ( u (H u ) jF I u ) (u)( )jF I s )du +v s = v s ; Therefore, v t is a F I t -martingale. On the other hand v t =W t + Z t 0 s (H s ) ds Z t 0 (s)( )ds: (3.31) the quadratic variation of v t is t, i.e. [v;v] t = [W;W ] t =t. Hence, by Theorem 39 (Levy's Theorem) in [39], v t is anF I t Brownian motion under P . Because W t = ~ Z t (H t ) Z t 0 s (H s ) ds is aG t Brownian motion under P . And we have v t W t = Z t 0 s (H s ) ds Z t 0 (s)( )ds: By the Martingale representation theorem in [32], v t is not aG t -martingale under P . 80 3.3 Utility maximization problem In this section we consider the utility maximization problem under partial infor- mation in multiple defaults market by solving a backward stochastic dierential equation(BSDE) with jumps. We x a probability space ( ;G;P), whereG follows the denition in last section. Without loss of generality, we assume the interest rate r equals 0. From theorem 3.2.8, we have already known that v t is anF I t -Brownian motion under P . Then we dene the F I -measurable compensated jumps M i;t = 1 f i <tg R t 0 (1H i;s ) ^ i s ds(i = 1; 2;:::;m), with theF I -intensity process ^ i . The following proposition is natural. Proposition 3.3.1. M i;t is anF I t -martingale under P (i = 1; 2;:::;m). Proof. For 0<t<s<T , we have E P [M i;s jF I t ] =E P fE P [ ~ M i;s jF I s ]jF I t g =E P fE P [E P ( ~ M i;s jF I s )jF I t ]jG t g =E P fE P [E P ( ~ M i;s jG t )jF I t ]jF I s g =E P fE P [ ~ M i;t jF I t ]jF I s g =E P [ ~ M i;t jF I t ] =M i;t : (3.32) Before we introduce the utility maximization problem, we rst prove the fol- lowing martingale representation theorem for further use. 81 Theorem 3.3.2. Every (P;F I t )locally square integrable local-martingale R t has the decomposition R t =R 0 + Z t 0 f s dv s + m X i=1 Z t 0 g i;s dM i;s (3.33) wheref t is aF I t adapted process andg i;s 's areF I t predictable processes such that Z T 0 f 2 s ds< +1; Z T 0 g 2 i;s ^ i;s ds< +1 Pa:s:: Proof. Recall that L t = exp( Z t 0 (X s ;H s ) ((H s )) 2 d ~ Z s 1 2 Z t 0 j (X s ;H s ) (H s ) j 2 ds); (3.34) with (X s ;H s ) =a s (X s ;H s ) + P m i=1 (1H i;s ) i (X s ;H s ). And the measure Q is dened by dQ dPjGt = L 1 t . From Girsanov Theorem, under measure Q, ~ Zt = v t + R t 0 (s)( )ds is aG t -Brownian motion, which is independent ofH andX. Because the measure transformation doesn't aect the distribution of H and X, M i;t 's are F I martingales under Q. By applying Theorem 5.3.5 of [1], every (Q;F I t )local martingale ^ R t has the following representation property ^ R t = ^ R 0 + Z t 0 ' s d ~ Z s + m X i=1 Z t 0 i;s dM i;s ; (3.35) 82 for someF I t -adapted process ' t and mF I t predictable processes i;t Z T 0 ' 2 s ds< +1; Z T 0 2 i;s ^ s ds< +1 Qa:s:: LetR t be a (P;F I t )local martingale, by Kallianpur-Striebel formula and recalling dQ dPjF I t = (t)(L 1 ) = E P [L 1 t jF I t ] = expf R t 0 (s)( )dv s 1 2 R t 0 [(s)( )] 2 dsg, we have ^ R t = R t (E P [L 1 t jF I t ]) 1 is a (Q;F I t )local martingale. Denoting L 1 t as E P [L 1 t jF I t ] and applying product rule to R t = ^ R t L 1 t dR t = ^ R t dL 1 t +L 1 t d ^ R t +d[ ^ R;L 1 ] c t = ^ R t (t)( )dv t +L 1 t ' t dv t +L 1 t ' t (t)( )dt + m X i=1 L 1 t i;t dM i;t L 1 t (t)( )dt = [L 1 t ' t ^ R t (t)( )]dv t + m X i=1 L 1 t i;t dM i;t : We only need to dene f t 4 =L 1 t ' t ^ R t (t)( ) and g i;t 4 =L 1 t i;t . Suppose our nancial market consists a risky asset and a non-risky asset. Recall that the price of the risky asset (3.4) and (3.31) , and S t isF I t -adapted S t =S 0 + R t 0 S s a s (X s ;H s )ds + R t 0 S s (H s )dW s P m i=1 R t 0 S s i s d ^ M i;s =S 0 + R t 0 S s (s)(a)ds + R t 0 S s (H s )dv s P m i=1 R t 0 S s i s dM i;s (3.36) This asset price model allows jumps at random time or default time T j . The investor can always observe the default history of the companies in the portfolio, 83 which is like the default times in nance, death times in life insurance, or car accident times in auto insurance. Because the investors just have partial observation on the market (F I ), it is natural to consider theF I t adapted investing strategy t in the defaultable asset S t . Denition 3.3.3. The class (F I ), of F I -admissible trading strategies is the set if all F I -predictable processes such that Z T 0 j 0 t a t j 2 dt + Z T 0 j 0 t t j 2 dt + m X i=1 Z T 0 j t ^ i;t j 2 dt<1 Pa:s: (3.37) where T <1 is a xed expiration time. We specify here that at each timet, t represents the number of shares invested in the risky asset S. Then the wealth process V satises V t =V 0 + Z t 0 s (s)(a)ds + Z t 0 s (H s )dv s m X i=1 Z t 0 s i s dM i;s (3.38) Suppose the investor also has a liability A T depending on the defaults, which is F I T -measurable and bounded. A T = m X n=0 A n T 1 fT2ng ; (3.39) 84 where all theA n T 's areF I T measurable. Then we will focus on the following prob- lem: sup 2(F I ) E P fu(V v T ()A T )g where u is the utility function,v is the initial wealth invested in the market. Again we consider the exponential utility function u(x) = 1e %x , with %> 0. sup 2(F I ) E P fu(V v T ()A T )g = 1 + sup 2(F I ) E P (e %V v T () e %A T ): Suppose we can nd a processY withY T =A T , such that the processe %V v t () e %Y T is aF I -supermartingale under P for any admissible strategy 2 (F I ) and is a martingale under P for some admissible strategy 2 (F I ). Suppose Y t is the solution for the BSDE Y t =A T + Z T t f s ds Z T t Z s dv s Z T t m X i=1 U i s dM i;s ; such that the process t 4 =e V v t ()+Yt is aF I -supermartingale underP for any admissible strategy and is a martingale under P for some admissible strategy . To nd the expression of f, we rst note that d(V t Y t ) = ( t (t)(a) +f t )dt + ( t (H t )Z t )dv t m X i=1 ( t i t +U i t )dM i;t : 85 Then, applying Ito's formula we have e (VtYt) = e (V T Y T ) + Z T t e (VsYs) d(V s Y s ) Z T t 1 2 2 e (VsY s ) d[VY ] c s X t<sT [e (VsYs) e (V s Y s ) +e (V s Y s ) (V s Y s )] (3.40) = e (V T Y T ) + Z T t e (VsYs) [ s (s)(a) +f s 1 2 ( s Z s ) 2 ]ds + Z T t e (VsYs) ( s Z s )dv s m X i=1 Z T t e (VsYs) ( s i s +U i s )dM i;s X t<sT [e (VsYs) e (V s Y s ) +e (V s Y s ) (V s Y s )]: Using the facts (V t Y t ) = P m i=1 ( t i t +U i t )M i;t , M n i;t = M i;t for any n = 1; 2; 3;:::, and M i;t M j;t = 0 if i6=j, we have X t<sT [e (VsYs) e (V s Y s ) ] = X t<sT e (V s Y s ) (e (VsYs) 1) (3.41) = X t<sT e (V s Y s ) (e P m i=1 ( s i s +U i s )M i;s 1) = X t<sT e (V s Y s ) m X i=1 M i;s (e ( s i s +U i s ) 1); and e (V s Y s ) (V s Y s ) =e (V s Y s ) m X i=1 ( s i s +U i s )M i;s : (3.42) 86 Combing (3.41), (3.42), and (3.40), and noting thatdM i;t =dH i;t ^ i;t dt, we have t = e (VtYt) = e (V T Y T ) Z T t e (VsYs) [ s (s)(a) +f s 1 2 ( s Z s ) 2 ]ds Z T t e (VsYs) ( s Z s )dv s m X i=1 Z T t e (V s Y s ) ( s i s +U i s )dM i;s + X t<sT e (V s Y s ) m X i=1 [(e ( s i s +U i s ) 1( s i s +U i s )]M i;s (3.43) = e (V T Y T ) Z T t e (VsYs) [ s (s)(a) +f s 1 2 ( s Z s ) 2 ]ds Z T t e (VsYs) ( s Z s )dv s m X i=1 Z T t e (V s Y s ) (e ( s i s +U i s ) 1)dM i;s + Z T t e (VsYs) m X i=1 [(e ( s i s +U i s ) 1( s i s +U i s )](1H i;s ) ^ i;s ds: Clearly, in order to choosef such that is a supermartingale for all2 (F I ), and a martingale for some 2 (F I ), we rst choose s such that it maximizes Q(s;) 4 = s (s)(a) 1 2 ( s Z s ) 2 (3.44) m X i=1 [ 1 e ( s i s +U i s ) 1 ( s i s +U i s )](1H i;s ) ^ i;s : Next, we simply dene f s 4 =Q(s; ) = sup 2(F I ) Q(s;). Then, from (3.43) we see that for any 2 (F I ), t =e (V T Y T ) Z T t e (VsYs) [Q(s;) +f s ]ds + martingale 87 is a supermartingale, and it is aF I -martingale when f s =Q(s; ). We now show that does exist. First, we look at the rst order optimality condition. Taking derivative of (3.44) respect to , and letting s be the solution to (s)(a)( s Z s ) m X i=1 [ i s e ( s i s +U i s ) i s ](1H i;s ) ^ i;s = 0; (3.45) we have m X i=1 i s e ( s i s +U i s ) (1H i;s ) ^ i;s = (s)(a)( s Z s ) + m X i=1 i s (1H i;s ) ^ i;s : (3.46) Since lim !+1 @Q @ =1 and lim !1 @Q @ = +1; (3.47) there is a solution for (3.45). On the other hand, @ 2 Q @ 2 = 2 m X i=1 ( i s ) 2 e ( s i s +U i s ) (1H i;s ) ^ i;s < 0; (3.48) hence is the maximal critical point. 88 3.4 Conclution We summarized the following conclusions The ltering plays an important role when the density hypothesis and h- hypothesis do not hold. The utility maximization problem can be solved by considering a BSDE with jump. 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Abstract (if available)

Abstract We consider a market where asset prices could be affected by multiple defaults along with possible factors including frailty. The aim of an investor is to maximize the expected utility of the terminal wealth, based on the observed data of the underlying asset(s) and the default history up to the current time. The main purpose is then to determine the conditional intensity of the future defaults, given the observed stock prices and the past defaults, without using the so-called \density hypothesis"". The problem is naturally reduced to a nonlinear filtering problem, for which the so-called the H-hypothesis is known to fail. We show that the problem can be solved dynamically via a system of Zakai equations for the conditional densities between and at consecutive defaults. A related BSDE with jumps that has quadratic growth in both continuous and jump martingale integrands will also be studied, as a by product of the utility optimization problem.

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Asset Metadata

Creator Wang, Huanhuan (author)

Core Title Defaultable asset management with incomplete information

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Applied Mathematics

Publication Date 02/01/2013

Defense Date 05/14/2012

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

Tag BSDE,density,filtering,intensity,OAI-PMH Harvest,utility maximization,Zakai equation

Language English

Contributor Electronically uploaded by the author (provenance)

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

Creator Email howardw913@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c3-131096

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