# Stochastic Differential Games for Fully Coupled FBSDEs with Jumps

Stochastic Differential Games for Fully Coupled FBSDEs with Jumps This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. The upper and the lower value functions are defined by the doubly controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in Buchdahn (Stocha Process Appl 121:2715–2750, 2011 ), we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower second order integral-partial differential equations of Isaacs’ type combined with an algebraic equation, respectively. Furthermore, for a special case (when $$\sigma$$ σ and $$h$$ h do not depend on $$(y, z, k)$$ ( y , z , k ) ), under the Isaacs’ condition, we get the existence of the value of the game. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Applied Mathematics and Optimization Springer Journals

# Stochastic Differential Games for Fully Coupled FBSDEs with Jumps

, Volume 71 (3) – Jun 1, 2015
38 pages

/lp/springer_journal/stochastic-differential-games-for-fully-coupled-fbsdes-with-jumps-ryrDA64WS6
Publisher
Springer US
Subject
Mathematics; Calculus of Variations and Optimal Control; Optimization; Systems Theory, Control; Theoretical, Mathematical and Computational Physics; Mathematical Methods in Physics; Numerical and Computational Physics
ISSN
0095-4616
eISSN
1432-0606
D.O.I.
10.1007/s00245-014-9264-2
Publisher site
See Article on Publisher Site

### Abstract

This paper is concerned with stochastic differential games (SDGs) defined through fully coupled forward-backward stochastic differential equations (FBSDEs) which are governed by Brownian motion and Poisson random measure. The upper and the lower value functions are defined by the doubly controlled fully coupled FBSDEs with jumps. Using a new transformation introduced in Buchdahn (Stocha Process Appl 121:2715–2750, 2011 ), we prove that the upper and the lower value functions are deterministic. Then, after establishing the dynamic programming principle for the upper and the lower value functions of this SDGs, we prove that the upper and the lower value functions are the viscosity solutions to the associated upper and the lower second order integral-partial differential equations of Isaacs’ type combined with an algebraic equation, respectively. Furthermore, for a special case (when $$\sigma$$ σ and $$h$$ h do not depend on $$(y, z, k)$$ ( y , z , k ) ), under the Isaacs’ condition, we get the existence of the value of the game.

### Journal

Applied Mathematics and OptimizationSpringer Journals

Published: Jun 1, 2015

### References

• Backward stochastic differential equations in finance
Karoui, N; Peng, SG; Quenez, MC

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