Stochastic Differential Games and Viscosity Solutions of Hamilton–Jacobi–Bellman–Isaacs Equations

Rainer Buckdahn

200 citations0/0 files storedSIAM Journal on Control and Optimization

No readable copy stored yet — this record is metadata only.

No open-access location is known for this record yet.

Literature note Cite in a project Citation map

Stochastic processes and financial applications Nonlinear Partial Differential Equations Game Theory and Applications

Abstract

In this paper we study zero-sum two-player stochastic differential games with the help of the theory of backward stochastic differential equations (BSDEs). More precisely, we generalize the results of the pioneering work of Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp. 293–314] by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game. This extension of the class of admissible control processes has the consequence that the cost functionals become random variables. However, by making use of a Girsanov transformation argument, which is new in this context, we prove that the upper and the lower value functions of the game remain deterministic. Apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information, its combination with BSDE methods, in particular that of the notion of stochastic “backward semigroups" introduced by Peng [BSDE and stochastic optimizations, in Topics in Stochastic Analysis, Science Press, Beijing, 1997], allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way. The upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower Hamilton–Jacobi–Bellman–Isaacs equations, respectively. For this Peng's BSDE method is extended from the framework of stochastic control theory into that of stochastic differential games.

Paper brief

Loading…

Cite this work

Citation unavailable for this work.

BibTeX RIS CSL-JSON

No public copy is stored for this record, so only the metadata is cited.

Citations in this library

Citation map →

Stored only when both papers are in the corpus — 200 citations exist upstream.

References

1990Adapted solution of a backward stochastic differential equation 1997Backward Stochastic Differential Equations in Finance 1997Backward stochastic differential equations and integral-partial differential equations 1992A Generalized dynamic programming principle and hamilton-jacobi-bellman equation 2006Second‐order backward stochastic differential equations and fully nonlinear parabolic PDEs 1995Zero-sum stochastic differential games and backward equations

Cited by

2015Leader–follower stochastic differential game with asymmetric information and applications 2014Two Person Zero-Sum Game in Weak Formulation and Path Dependent Bellman--Isaacs Equation 2016Portfolio Optimization with Ambiguous Correlation and Stochastic Volatilities 2010A Pontryagin's Maximum Principle for Non-Zero Sum Differential Games of BSDEs with Applications 2010Some Recent Aspects of Differential Game Theory 2013On the Multidimensional Controller-and-Stopper Games 2008Dynamic Programming Principle for One Kind of Stochastic Recursive Optimal Control Problem and Hamilton–Jacobi–Bellman Equation

Explore the map →

Papers that cite the same references — the relational fallback while embeddings are still backfilling.

Imported from OpenAlex (CC0) — nothing here is generated.