Probabilistic interpretation of a system of semi-linear parabolic partial differential equations

doi:10.1016/S0246-0203(97)80101-X

Annales de l'Institut Henri Poincare (B) Probability and Statistics

Volume 33, Issue 4, 1997, Pages 467-490

https://doi.org/10.1016/S0246-0203(97)80101-X Get rights and content

Abstract

We obtain a probabilistic interpretation for systems of second order nonlinear parabolic partial differential equations by using a backward stochastic differential equation associated to a diffusion-transmutation process.

Résumé

Nous obtenons une formule probabiliste pour la solution de viscosité d'un système d'équations aux dérivées partielles paraboliques semilinéaires, à l'aide d'une équation différentielle stochastique rétrograde couplée à un processus de diffusion-transmutation. La nouveauté de ce travail tient à ce que l'opérateur linéaire du second ordre est différent d'une ligne à l'autre du système d'équations aux dérivées partielles.

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Cited by (51)

A representation theorem approach to probabilistic interpretation for viscosity solutions of Isaacs equations
2023, Journal of Mathematical Analysis and Applications
Citation Excerpt :
For non-Lipschitz framework, Pu and Zhang [22] and Zhuo, Dong and Pu [30] provided an approximation method of viscosity solutions under a monotonicity condition. To prove the probabilistic interpretation for semilinear (quasilinear) PDEs, different methods are employed under different conditions, such as strict comparison theorem for solutions of BSDEs under monotonicity conditions, see Pardoux, Pradeilles and Rao [18], Pardoux and Zhang [17], Pardoux and Tang [16], Shi and Zhao [24], Zheng, Zhang and Feng [29], etc. Usually, methods adopted for semilinear PDEs are not applicable to HJB and Isaacs equations.
The main objective of this paper is to show a certain representation theorem for generators of generalized backward stochastic differential equations (GBSDEs) and its application to probabilistic interpretation for viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations with nonlinear Neumann boundary value problems. In the representation theorem a time change method is adopted to treat the difficulty brought from the random measure contained in GBSDEs. By means of the representation theorem the viscosity solution of Isaacs equations with nonlinear Neumann problems is proved to be the value function of a two-player zero-sum stochastic differential game problem with state constraints and recursive cost functionals.
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The enhanced group classification of a semi linear generalization of a general bond-pricing equation is carried out by harnessing the underlying equivalence and additional equivalence transformations. We employ that classification to unearth the particular cases with a larger Lie algebra than the general case and use them to find non trivial invariant solutions under the terminal and the barrier option condition.
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This article has been retracted: please see Elsevier Policy on Article Withdrawal (https://www.elsevier.com/locate/withdrawalpolicy).
This article was retracted at the request of the Editors, in agreement with the authors. This paper has been retracted as the main results have a high level of similarity with the work published by E. Pardoux et al., Probabilistic interpretation of a system of semi-linear parabolic partial differential equations, Annales de l'Institut Henri Poincare (B) Probability and Statistics 33 (4) (1997) 467–490, http://www.sciencedirect.com/science/article/pii/S024602039780101X?via%3Dihub.
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Probabilistic interpretation of a system of semi-linear parabolic partial differential equations

Abstract

Résumé

System and Control Letters

BSDE's and integro-partial differential equations

Stochastics and Stochastics Reports

Malliavin Calculus for Processes with Jumps

Stoch. Monographs

User's guide to viscosity solutions of second order partial differential equations

Bull AMS

Superprocesses and partial differential equations

Ann. Probab.

Controlled Markov processes and viscosity solutions

Applications of Mathematics

Viscosity solutions for monotone systems of second-order elliptic PDEs

Commun. in Partial Differential equations

A General Theorem of Representation for Martingales

Proceed. of Symposia in Pure Math.

On some connections between probability theory and differential and integral equations