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
[Duale représentation comme les jeux différentielles stochastiques pour les équations différentielles stochastiques rétrogrades, et les évalutions dynamiques]

Présenté par : 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.

Résumé
Abstract

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.

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.

Reçu le : 2005-08-18
Accepté le : 2006-03-06
Publié le : 2006-04-18

DOI : 10.1016/j.crma.2006.03.025

Affiliations des auteurs :

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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     title = {Dual representation as stochastic differential games of backward stochastic differential equations and dynamic evaluations},
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     pages = {773--778},
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     volume = {342},
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     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/

Bibliographie
Cité par

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Shengjun Fan; Ying Hu; Shanjian Tang L1 solution to scalar BSDEs with logarithmic sub-linear growth generators, Systems Control Letters, Volume 177 (2023), p. 105553 | DOI:10.1016/j.sysconle.2023.105553
Shengjun Fan; Ying Hu Existence and uniqueness of solution to scalar BSDEs with $L \exp (μ \sqrt{2 \log (1 + L)})$ -integrable terminal values: the critical case, Electronic Communications in Probability, Volume 24 (2019) no. none | DOI:10.1214/19-ecp254
Ying Hu; Shanjian Tang Existence of solution to scalar BSDEs with $L \exp (\sqrt{\frac{2}{λ} \log (1 + L)})$ -integrable terminal values, Electronic Communications in Probability, Volume 23 (2018) no. none | DOI:10.1214/18-ecp127
Shanjian Tang; Fu Zhang Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations, Discrete Continuous Dynamical Systems - A, Volume 35 (2015) no. 11, p. 5521 | DOI:10.3934/dcds.2015.35.5521

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