Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation

Pham, Triet; Zhang, Jianfeng

doi:10.48550/arxiv.1209.6605

Cited by 9 publications

(8 citation statements)

References 14 publications

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“…In many applications, see e.g. [21], we may have a representation formula for the process Y , but in general it is difficult to obtain representation formulae for M and A. So conditions imposed on Y are more tractable than those on M and A.…”

Section: Definition 24mentioning

confidence: 99%

“…This value process is a supermartingale (or, in general case, a g-supermartingale as introduced in Peng [17]), under the associated non-dominated class of mutually singular probaility measures. In Pham and Zhang [21], we studied a zero sum stochastic differential game and characterized the game value process as the unique viscosity solution of a path dependent Bellman-Isaacs equation. It is natural to conjecture that, under certain technical conditions, this value process should be a semimartingale under the underlying class of probability measures, which will enable us to characterize the value process as the solution to an extended second order BSDE with a non-convex generator.…”

Section: Introductionmentioning

confidence: 99%

“…This is important in applications because the semimartingale under consideration is typically a value process, , e.g. in [21], and thus has a representation. We prove that a progressively measurable process is a square integrable semimartingale if and only if it has finite norm in our sense.…”

Section: Introductionmentioning

confidence: 99%

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Some norm estimates for semimartingales

Pham

Zhang

2013

Electron. J. Probab.

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In this paper we introduce a new type of norms for semimartingales, under both linear and nonlinear expectations. Our norm is defined in the spirit of quasimartingales, and it characterizes square integrable semimartingales. This work is motivated by our study of zero-sum stochastic differential games [21], whose value process is conjectured to be a semimartingale under a class of probability measures. As a by product, we establish some a priori estimates for doubly reflected BSDEs without imposing the Mokobodski's condition directly.

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Section: Definition 24mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Some norm estimates for semimartingales

Pham

Zhang

2013

Electron. J. Probab.

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“…[12] and [8], there are by now several workarounds for this problem; they rely on approximations and exploit the continuity properties of the cost functions. We also refer to [23] and [24] for recent developments in different directions, in a setting where both players use controls. In [6], stochastic target games were analyzed when the target is of controlled-loss type; that is, of the form E ℓ(X u,α t,x,y (T ), Y u,α t,x,y (T )) ≥ m, where m is given and the loss function ℓ has certain continuity properties.…”

Section: Introductionmentioning

confidence: 99%

Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

Bouchard

Nutz

2016

Mathematics of OR

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We study a class of stochastic target games where one player tries to find a strategy such that the state process almost-surely reaches a given target, no matter which action is chosen by the opponent. Our main result is a geometric dynamic programming principle which allows us to characterize the value function as the viscosity solution of a non-linear partial differential equation. Because abstract measurable selection arguments cannot be used in this context, the main obstacle is the construction of measurable almost-optimal strategies. We propose a novel approach where smooth supersolutions are used to define almost-optimal strategies of Markovian type, similarly as in verification arguments for classical solutions of Hamilton-Jacobi-Bellman equations. The smooth supersolutions are constructed by an extension of Krylov's method of shaken coefficients. We apply our results to a problem of option pricing under model uncertainty with different interest rates for borrowing and lending.

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“…However those papers do not reach the same objective. Note that for the control against control zero-sum game, Pham and Zhang [26] and M.Sirbu [29] have overcome this restriction related to the independence of σ on the controls.…”

Section: Introductionmentioning

confidence: 99%

Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients

Hamadène

2014

Stochastics

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This paper is related to nonzero-sum stochastic differential games in the Markovian framework. We show existence of a Nash equilibrium point for the game when the drift is no longer bounded and only satisfies a linear growth condition. The main tool is the notion of backward stochastic differential equations which, in our case, are multidimensional with continuous coefficient and stochastic linear growth.

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Two Person Zero-sum Game in Weak Formulation and Path Dependent Bellman-Isaacs Equation

Cited by 9 publications

References 14 publications

Some norm estimates for semimartingales

Some norm estimates for semimartingales

Stochastic Target Games and Dynamic Programming via Regularized Viscosity Solutions

Existence of Nash equilibrium points for Markovian non-zero-sum stochastic differential games with unbounded coefficients

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