Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations

Shanjian Tang

doi:10.1016/j.crma.2006.03.025

Probability Theory/Optimal Control

Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations

Presented by: Jean-Michel Bony

Shanjian Tang ^1,²

¹ Department of Financial Mathematics and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
² Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, China

Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 773-778.

Abstract
Résumé

In this Note, assuming that the generator is uniform Lipschitz in the unknown variables, we relate the solution of a one dimensional backward stochastic differential equation with the value process of a stochastic differential game. Under a domination condition, an $F$ -consistent evaluation is also related to a stochastic differential game. This relation comes out of a min–max representation for uniform Lipschitz functions in terms of affine functions. The extension to reflected backward stochastic differential equations is also included.

Dans cette Note, supposant que le générateur soit une fonction uniformément lipschitzienne, nous présentons un lien entre les équations différentielles stochastiques rétrogrades et les jeux différentiels stochastiques. Sous une hypothèse de domination, une évaluation $F_{t}$ -consistante est associée avec un jeu différentiel stochastique. Ce lien est une conséquence d'une représentation du min–max type pour les fonctions lipschitzienne en termes de fonctions affines. Une formule duale est aussi donnée pour les équations différentielles stochastiques rétrogrades refléchies.

Received: 2005-08-18
Accepted: 2006-03-06
Published online: 2006-04-18

DOI: 10.1016/j.crma.2006.03.025

Author's affiliations:

Shanjian Tang ^{1, 2}

¹ Department of Financial Mathematics and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433, China
² Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, China

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     author = {Shanjian Tang},
     title = {Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {773--778},
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     number = {10},
     year = {2006},
     doi = {10.1016/j.crma.2006.03.025},
     language = {en},
}

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Shanjian Tang. Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations. Comptes Rendus. Mathématique, Volume 342 (2006) no. 10, pp. 773-778. doi : 10.1016/j.crma.2006.03.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.03.025/

References
Cited by

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