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which is the optimal choice of seller i if the other sellers set prices with average pni = 1 n1 Pn j6=i p j . Now, let us de ne pn := 1 n Xn i=1 p i = an(1 xn) 2bn(1 xn) cn(1 xn) + bnyn : (3.1.15) Then, it is readily seen that pni = n n1 pn 1 n1p i , which means (plugging back into (3.1.14)) p ;n i = an 2bn + cn n1 1 n1 bnyn 1xn + 1 n1 n 2bn(1xn) cn(1xn)bnyn + 1 n pn: (3.1.16) For the sake of argument, let us assume that the coe cients (an; bn; cn; xn(Q); yn(Q)) converge to (a; b; c; x(Q); y(Q)) as n ! 1. Then, we see from (3.1.15) and (3.1.16) that 8>>>< >>>: lim n!1 pn = a(1 x) 2b(1 x) c(1 x) + by =: p; lim n!1 p ;n i = a 2b + c(1 x) by 2b(1 x) lim n!1 pn = a(1 x) (2b c)(1 x) + by =: p : (3.1.17) It is worth noting that if we assume that there is a \representing seller" who randomly sets prices p = pi with equal probability 1 n, then we can randomize the pro t function (3.1.13) as follows: n(p; p) = (an bnp + cn p) fp (xnp yn p)g; (3.1.18) where p is a random variable taking values fpig with equal probability, and p E[p], thanks to the Law of Large Numbers, when n is large enough. In particular, in the limiting case as n ! 1, we can replace the randomized pro t function n in (3.1.18) by: 1 = (p;E[p]) := (a bp + cE[p]) fp (xp yE[p])g: (3.1.19) 27
Object Description
Title  Equilibrium model of limit order book and optimal execution problem 
Author  Noh, Eunjung 
Author email  eunjungn@usc.edu;jennynoh86@gmail.com 
Degree  Doctor of Philosophy 
Document type  Dissertation 
Degree program  Applied Mathematics 
School  College of Letters, Arts and Sciences 
Date defended/completed  20180521 
Date submitted  20180810 
Date approved  20180810 
Restricted until  20180810 
Date published  20180810 
Advisor (committee chair)  Ma, Jin 
Advisor (committee member) 
Zhang, Jianfeng Lv, Jinchi Mikulevicius, Remigijus Lototsky, Sergey 
Abstract  In this dissertation, we study an equilibrium model of a limit order book (LOB) and an optimal execution problem. To generalize the previous results, we accommodate the idea of Bertrand price competition, as well as nonlocal meanfield stochastic differential equation (SDE) with evolving intensity and reflecting boundary conditions. To describe the equilibrium of LOB, we start with N sellers’ static Bertrand game, and extend the model to continuous time setting to formulate it as a meanfield type control problem of the representative seller, who wants to maximize the discounted lifelong expected utility. Using dynamic programming principle (DPP), we could form a HamiltonJacobiBellman (HJB) equation and prove the value function is the viscosity solution of it. And, we show the value function can be used to obtain the equilibrium density function of a LOB. ❧ Assuming the LOB has reached at the equilibrium, we solve the optimal execution problem of a buyer, who wants to purchase a certain number of shares during the finite time horizon with minimum cost. Again, we show that the Bellman principle of DPP holds in this case, and the value function is the viscosity solution of the HJB quasivariational inequality (QVI). Also, we investigate the optimal strategy when the QVI has a classical solution. 
Keyword  limit order book; optimal execution; equilibrium; Bertrand model; meanfield SDE with jump and reflecting boundary condition; discontinuous Skorohod problem; stochastic control 
Language  English 
Format (imt)  application/pdf 
Part of collection  University of Southern California dissertations and theses 
Publisher (of the original version)  University of Southern California 
Place of publication (of the original version)  Los Angeles, California 
Publisher (of the digital version)  University of Southern California. Libraries 
Provenance  Electronically uploaded by the author 
Type  texts 
Legacy record ID  uscthesesm 
Contributing entity  University of Southern California 
Rights  Noh, Eunjung 
Physical access  The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the author, as the original true and official version of the work, but does not grant the reader permission to use the work if the desired use is covered by copyright. It is the author, as rights holder, who must provide use permission if such use is covered by copyright. The original signature page accompanying the original submission of the work to the USC Libraries is retained by the USC Libraries and a copy of it may be obtained by authorized requesters contacting the repository email address given. 
Repository name  University of Southern California Digital Library 
Repository address  USC Digital Library, University of Southern California, University Park Campus MC 7002, 106 University Village, Los Angeles, California 900897002, USA 
Repository email  cisadmin@lib.usc.edu 
Filename  etdNohEunjung6728.pdf 
Archival file  Volume3/etdNohEunjung6728.pdf 
Description
Title  Page 32 
Full text  which is the optimal choice of seller i if the other sellers set prices with average pni = 1 n1 Pn j6=i p j . Now, let us de ne pn := 1 n Xn i=1 p i = an(1 xn) 2bn(1 xn) cn(1 xn) + bnyn : (3.1.15) Then, it is readily seen that pni = n n1 pn 1 n1p i , which means (plugging back into (3.1.14)) p ;n i = an 2bn + cn n1 1 n1 bnyn 1xn + 1 n1 n 2bn(1xn) cn(1xn)bnyn + 1 n pn: (3.1.16) For the sake of argument, let us assume that the coe cients (an; bn; cn; xn(Q); yn(Q)) converge to (a; b; c; x(Q); y(Q)) as n ! 1. Then, we see from (3.1.15) and (3.1.16) that 8>>>< >>>: lim n!1 pn = a(1 x) 2b(1 x) c(1 x) + by =: p; lim n!1 p ;n i = a 2b + c(1 x) by 2b(1 x) lim n!1 pn = a(1 x) (2b c)(1 x) + by =: p : (3.1.17) It is worth noting that if we assume that there is a \representing seller" who randomly sets prices p = pi with equal probability 1 n, then we can randomize the pro t function (3.1.13) as follows: n(p; p) = (an bnp + cn p) fp (xnp yn p)g; (3.1.18) where p is a random variable taking values fpig with equal probability, and p E[p], thanks to the Law of Large Numbers, when n is large enough. In particular, in the limiting case as n ! 1, we can replace the randomized pro t function n in (3.1.18) by: 1 = (p;E[p]) := (a bp + cE[p]) fp (xp yE[p])g: (3.1.19) 27 