Two Person Zero-Sum Game in Weak Formulation and Path Dependent Bellman--Isaacs Equation
Triet Pham,Jianfeng Zhang +1 more
TLDR
The value process is characterized as the unique viscosity solution of the corresponding path dependent Bellman-Isaacs equation, a notion recently introduced by Ekren et al.Abstract:
In this paper we study a two person zero sum stochastic differential game in weak formulation. Unlike the standard literature, which uses strategy type controls, the weak formulation allows us to consider the game with control against control. We shall prove the existence of game value under natural conditions. Another main feature of the paper is that we allow for non-Markovian structure, and thus the game value is a random process. We characterize the value process as the unique viscosity solution of the corresponding path dependent Bellman-Isaacs equation, a notion recently introduced by Ekren et al. (Ann. Probab., 42 (2014), pp. 204-236) and Ekren, Touzi, and Zhang (Stochastic Process., to appear; preprint, arXiv:1210.0006v2; preprint, arXiv:1210.0007v2).read more
Citations
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Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II
TL;DR: In this article, the authors provide a notion of viscosity solutions for fully nonlinear parabolic path-dependent PDEs, and prove that their notion is consistent with the corresponding notion of classical solutions, and satisfies a stability property and a partial comparison result.
Journal ArticleDOI
Viscosity solutions of fully nonlinear parabolic path dependent PDEs: Part II
TL;DR: In this paper, the authors provided a full, well-posedness result under an additional assumption, formulated on some partial differential equation, defined locally by freezing the path, assuming further that such path-frozen standard PDEs satisfy the comparison principle and the Perron approach for existence.
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Measurability of semimartingale characteristics with respect to the probability law
Ariel Neufeld,Marcel Nutz +1 more
TL;DR: In this article, the authors give a general and unifying answer to measurability questions that arise in the context of quasi-sure analysis and stochastic control under the weak formulation.
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A Survey on the Combined Use of Optimization Methods and Game Theory
TL;DR: A combined use of game theory and optimization algorithms has been reviewed and a new categorization is presented for researches which have been conducted in this area.
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Robust Markowitz mean-variance portfolio selection under ambiguous covariance matrix
TL;DR: In this article, a robust continuous-time Markowitz portfolio selection problem is formulated into a min-max mean-variance problem over a set of non-dominated probability measures that is solved by a McKean-Vlasov dynamic programming approach, which allows the solution in terms of a Bellman-Isaacs equation in the Wasserstein space of probability measures.
References
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Book
Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
Book
Linear and Quasilinear Equations of Parabolic Type
TL;DR: In this article, the authors considered a hyperbolic parabolic singular perturbation problem for a quasilinear equation of kirchhoff type and obtained parameter dependent time decay estimates of the difference between the solutions of the solution of a quasi-linear parabolic equation and the corresponding linear parabolic equations.
Journal ArticleDOI
User’s guide to viscosity solutions of second order partial differential equations
TL;DR: The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorem, and continuous dependence may now be proved by very efficient and striking arguments as discussed by the authors.
Book
Multidimensional Diffusion Processes
TL;DR: In this paper, the authors propose extension theorems, Martingales, and Compactness, as well as the non-unique case of the Martingale problem, and some estimates on the transition probability functions.
Book
Second order parabolic differential equations
TL;DR: In this paper, the oblique derivative problem for quasilinear parabolic equations was studied and the theory of weak solutions was introduced. And the boundary gradient was used to estimate global and local gradient bounds.
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