A general comparison theorem for backward stochastic differential equations

A useful result when dealing with backward stochastic differential equations is the comparison theorem of Peng (1992). When the equations are not based on Brownian motion, the comparison theorem no longer holds in general. In this paper we present a condition for a comparison theorem to hold for backward stochastic differential equations based on arbitrary martingales. This theorem applies to both vector and scalar situations. Applications to the theory of nonlinear expectations are also explored.

[1]  Guangyan Jia.  Some uniqueness results for one-dimensional BSDEs with uniformly continuous coefficients , 2009 .

[2]  S. Peng.  A Generalized dynamic programming principle and hamilton-jacobi-bellman equation , 1992 .

[3]  Mark H. A. Davis.  Mathematics of Financial Markets , 2001 .

[4]  Samuel N. Cohen,et al.  Comparisons for backward stochastic differential equations on Markov chains and related no-arbitrage conditions , 2008, 0810.0055.

[5]  S. Peng,et al.  Filtration-consistent nonlinear expectations and related g-expectations , 2002 .

[6]  D. Sworder.  Stochastic calculus and applications , 1984, IEEE Transactions on Automatic Control.

[7]  S. Peng,et al.  BSDEs with random default time and their applications to default risk , 2009, 0910.2091.

[8]  H. Föllmer,et al.  Stochastic Finance: An Introduction in Discrete Time , 2002 .

[9]  Walter Schachermayer,et al.  The Mathematics of Arbitrage , 2006 .

[10]  Leonard Rogers,et al.  VALUATIONS AND DYNAMIC CONVEX RISK MEASURES , 2007, 0709.0232.

[11]  Qing Zhou.  On Comparison Theorem and Solutions of BSDEs for Lévy Processes , 2007 .

[12]  Robert J. Elliott,et al.  Backward Stochastic Difference Equations and Nearly Time-Consistent Nonlinear Expectations , 2011, SIAM J. Control. Optim..

[13]  Susanne Klöppel,et al.  DYNAMIC INDIFFERENCE VALUATION VIA CONVEX RISK MEASURES , 2007 .

[14]  S. Peng,et al.  Dynamically Consistent Nonlinear Evaluations and Expectations , 2005, math/0501415.

[15]  Samuel N. Cohen,et al.  A general theory of finite state Backward Stochastic Difference Equations , 2008, 0810.4957.

[16]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[17]  Mark H. A. Davis.  The Representation of Martingales of Jump Processes , 1976 .

[18]  David Heath,et al.  Coherent multiperiod risk adjusted values and Bellman’s principle , 2007, Ann. Oper. Res..

[19]  Samuel N. Cohen,et al.  Existence and Comparisons for BSDEs in general spaces , 2010 .

[20]  R. Situ.  Theory of Stochastic Differential Equations with Jumps and Applications: Mathematical and Analytical Techniques with Applications to Engineering , 2005 .

[21]  N. Karoui,et al.  Optimal derivatives design under dynamic risk measures , 2004 .

[22]  Ying Hu,et al.  On the comparison theorem for multidimensional BSDEs , 2006 .

[23]  M. Royer.  Backward stochastic differential equations with jumps and related non-linear expectations , 2006 .

[24]  M. Otmani.  Backward stochastic differential equations associated with Lévy processes and partial integro-differential equations , 2008 .

[25]  Robert J. Elliott,et al.  Solutions of Backward Stochastic Differential Equations on Markov Chains , 2008 .

[26]  Robert J. Elliott,et al.  Stochastic calculus and applications , 1984, IEEE Transactions on Automatic Control.

[27]  Emanuela Rosazza Gianin,et al.  Risk measures via g-expectations , 2006 .

[28]  G. Barles,et al.  Backward stochastic differential equations and integral-partial differential equations , 1997 .