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Stochastic Differential Games and Viscosity Solutions of Hamilton-Jacobi-Bellman-Isaacs Equations
Rainer Buckdahn, Juan Li
TL;DR
The paper addresses zero-sum stochastic differential games with BSDE-defined costs and controls that may use information from before the game. It translates Peng’s BSDE methods to games, proves a dynamic programming principle, and establishes that both value functions are unique viscosity solutions of their respective Hamilton-Jacobi-Bellman-Isaacs equations.
Problem
The paper studies stochastic differential games beyond the earlier setting by incorporating random cost functionals from pre-game information and more general BSDE-based running costs.
Method
The paper formulates the game with non-anticipating strategies and controlled BSDE costs, then applies Peng’s BSDE approach to derive dynamic programming and viscosity results.
Results
W and U are deterministic continuous viscosity solutions and unique in the stated growth class for the lower and upper Bellman-Isaacs equations, respectively.
Takeaways & Limitations
The paper extends BSDE methods from stochastic control to stochastic differential games and provides a direct route from dynamic programming to Bellman-Isaacs viscosity solutions.
Takeaways & Limitations
The game is posed over a fixed finite horizon with controls taking values in compact metric spaces, and coincidence of value functions is tied to Isaacs’ condition.
Abstract
from arXiv · showhide
In this paper we study zero-sum two-player stochastic differential games with the help of theory of Backward Stochastic Differential Equations (BSDEs). At the one hand we generalize the results of the pioneer work of Fleming and Souganidis 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 (which implies that the cost functionals become random variables), on the other hand the application of BSDE methods, in particular that of the notion of stochastic "backward semigroups" introduced by Peng allows to prove a dynamic programming principle for the upper and the lower value functions of the game in a straight-forward way, without passing by additional approximations. The upper and the lower value functions are proved 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 translated from the framework of stochastic control theory into that of stochastic differential games.
1 Introduction
The paper extends zero-sum stochastic differential games by allowing history-dependent controls and BSDE-defined costs, then characterizes the resulting value functions through dynamic programming and viscosity equations.
- Contributions: The framework allows controls to depend on the Brownian path before the game, making cost functionals random variables.This extends the setting of Fleming and Souganidis and harmonizes stochastic games with stochastic control theory.
- Contributions: Costs are defined by controlled backward stochastic differential equations with running driver f and terminal cost Φ.The BSDE has a unique solution under the paper’s assumptions.
- Game formulation: Admissible strategies are non-anticipating mappings from the opponent’s controls to one player’s controls.Player I uses α: Vt,T → Ut,T, while Player II analogously uses β: Ut,T → Vt,T.
- Main results: W and U are deterministic continuous viscosity solutions of the lower and upper Bellman-Isaacs equations, respectively.The equations use Hamiltonians formed by sup-inf and inf-sup optimization over the two players’ controls.
- Main results: The paper proves uniqueness for these viscosity solutions in a continuous-function class with a growth condition weaker than polynomial growth.This result is stated as an extension beyond the polynomial-growth setting.
- Proof strategy: The value functions are shown to satisfy a dynamic programming principle, enabling the derivation of their viscosity-solution properties.The paper applies Peng’s BSDE approach to stochastic differential games rather than stochastic control alone.
2 Preliminaries
This section establishes the Wiener-space and BSDE preliminaries used later, including existence, comparison, strict monotonicity, and stability estimates for BSDE solutions.
- Probabilistic setting: The stochastic basis is the classical Wiener space with canonical Brownian motion and its augmented natural filtration.The horizon is a fixed finite time T, and processes are adapted to this filtration.
- BSDE assumptions: The paper assumes BSDE drivers are progressively measurable, Lipschitz in their variables, and square-integrable at the origin.These are assumptions (A1) and (A2).
- Existence and uniqueness: Under assumptions (A1) and (A2), every square-integrable terminal variable yields a BSDE with a unique adapted solution.The paper adopts the Lipschitz framework for simpler calculations, although weaker assumptions can suffice.
- Comparison: The comparison theorem orders BSDE solutions according to their terminal values and drivers, with strict ordering under a positive-probability terminal gap.Strict monotonicity gives y1_0 > y2_0 when P(ξ1 > ξ2) > 0, under the stated additional conditions.
- Stability: A stability estimate controls the difference between two BSDE solutions when their terminal values and drivers differ.The estimate uses drivers differing through functions ϕi in H2(0,T;R), with β = 16(1 + C2).
3 Forward- Backward SDES (FBSDEs)
This section develops forward-backward SDE results needed to represent random initial conditions through a random field and establish regularity estimates for the associated solutions.
- Forward SDEs: The forward SDE coefficients satisfy linear-growth and Lipschitz conditions in the state variable.These assumptions provide well-posedness and estimates for the forward dynamics.
- Forward SDEs: Under the stated assumptions, the forward SDE has a unique strong solution with moment estimates for initial conditions in Lp.The constants depend on the Lipschitz and growth constants of b and σ.
- Backward SDEs: The backward component uses a progressively measurable driver f and terminal function Φ satisfying Lipschitz and linear-growth conditions.These conditions ensure the composed BSDE has a unique solution.
- Random fields: The random field u(t,x) is defined through the BSDE solution associated with the forward state initialized at x.The field is Lipschitz in x and has at most linear growth.
- Regularity: The field u is generally adapted and random, but additional assumptions can make it deterministic; under differentiability conditions it also has a continuous version.The paper additionally records 1/2-Hölder continuity in time under its stated hypotheses.
- Random initial conditions: For a general initial random variable ζ, the BSDE solution equals u(t,ζ), extending the representation from simple to L2 initial data.The proof approximates ζ by simple random variables and uses the established estimates.
4 Stochastic Differential Games and Associated Dynamic Programming Prin-
The paper formulates a two-player zero-sum stochastic differential game with Brownian-filtration controls, BSDE-defined costs, and nonanticipative strategies. It establishes deterministic value functions, a generalized dynamic programming principle, and regularity results under stated coefficient assumptions.
- Game setting: The game uses progressively measurable controls valued in compact metric spaces U and V, with state dynamics specified by controlled coefficients b and σ.The coefficients are assumed continuous in (t,u,v), Lipschitz in the state, and subject to global linear growth conditions.
- Game setting: Admissible controls may depend on the full Brownian-motion history, including information before the game begins, making cost functionals random variables.Controls on subintervals are progressively measurable, while strategies map the opponent’s controls nonanticipatively, including at stopping times.
- Value functions: The lower and upper value functions are formed from essential infima and suprema of BSDE-based cost functionals over controls and nonanticipative strategies.The cost functional is defined through the BSDE associated with the controlled state trajectory.
- Value functions: Despite initially being bounded measurable random variables, both value functions are deterministic under the paper’s assumptions.For the lower value function, Proposition 4.1 identifies W(t,x) almost surely with its expectation; the upper value function is treated analogously.
- Regularity: The lower value function is Lipschitz continuous in x and has linear growth, with |W(t,x)| ≤ C(1 + |x|).The stated spatial estimate is |W(t,x) − W(t,x′)| ≤ C|x − x′|.
- Dynamic programming: The paper adapts Peng’s stochastic backward semigroups to stochastic differential games to establish a generalized dynamic programming principle for the value functions.The lower value function satisfies the DPP for every intermediate interval [t,t+δ], with the proof completed by matching the two inequalities Wδ≤W and W≤Wδ.
5 Viscosity Solution of Isaacs’ Equation: Existence Theorem
The paper translates Peng’s BSDE approach to stochastic differential games and uses it to establish viscosity-solution results for the lower and upper value functions.
- 5 Viscosity Solution of Isaacs’ Equation: Existence Theorem: The upper value function U is likewise established as a viscosity solution of its associated Isaacs equation.The paper states that the argument is analogous to the proof for W.
- 5 Viscosity Solution of Isaacs’ Equation: Existence Theorem: The section applies Peng’s BSDE approach to prove that the lower value function W solves the lower Isaacs equation and the upper value function U solves the upper Isaacs equation.The lower equation is given as ∂tW(t, x) + H−(t, x, W, DW, D2W) = 0, while the upper equation is stated analogously for U.
- 5 Viscosity Solution of Isaacs’ Equation: Existence Theorem: The supersolution argument derives F0(t, x, 0, 0) ≤ 0 from the dynamic programming principle and the auxiliary ordinary differential equation.Here F0 is defined through the sup-inf operation over the two players’ controls.
- 5 Viscosity Solution of Isaacs’ Equation: Existence Theorem: The subsolution argument proves F0(t, x, 0, 0) ≥ 0 by contradiction, using measurable responses, the dynamic programming principle, and short-interval BSDE estimates.The contradiction begins by assuming F0(t, x, 0, 0) ≤ −θ < 0.
6 Viscosity Solution of Isaacs’ Equation: Uniqueness Theorem
The uniqueness section establishes comparison for viscosity subsolutions and supersolutions in a growth-controlled function space, yielding uniqueness for both Isaacs equations.
- 6 Viscosity Solution of Isaacs’ Equation: Uniqueness Theorem: Uniqueness is proved in Θ, the space of continuous functions satisfying a growth condition weaker than polynomial growth but more restrictive than exponential growth.The paper adapts ideas introduced for an integro-partial differential equation associated with a decoupled FBSDE with jumps.
- 6 Viscosity Solution of Isaacs’ Equation: Uniqueness Theorem: The comparison theorem states that every viscosity subsolution u1 and supersolution u2 of equation (6.1) satisfy u1(t, x) ≤ u2(t, x).The inequality holds for all (t, x) ∈ [0, T] × Rn under assumptions (H4.1) and (H4.2).
- 6 Viscosity Solution of Isaacs’ Equation: Uniqueness Theorem: The comparison proof uses a penalized difference of subsolution and supersolution together with a growth-control function and a contradiction argument.The proof analyzes maxima of ω − αχ and applies a differential inequality for χ.
- 6 Viscosity Solution of Isaacs’ Equation: Uniqueness Theorem: The lower value function W is the unique viscosity solution in Θ of equation (6.1), and the upper value function U is uniquely characterized by equation (5.2).The paper notes that W has at most linear growth and therefore belongs to Θ.
- 6 Viscosity Solution of Isaacs’ Equation: Uniqueness Theorem: Under Isaacs’ condition, the two Isaacs equations coincide, so uniqueness implies W(t, x) = U(t, x) and the stochastic differential game has a value.In the control-independent BSDE case, the paper also identifies these value functions with the Fleming–Souganidis value functions.