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Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs

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  • Li, Juan

Abstract

In this paper we consider a mean-field backward stochastic differential equation (BSDE) driven by a Brownian motion and an independent Poisson random measure. Translating the splitting method introduced by Buckdahn et al. (2014) to BSDEs, the existence and the uniqueness of the solution (Yt,ξ,Zt,ξ,Ht,ξ), (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) of the split equations are proved. The first and the second order derivatives of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to x, the derivative of the process (Yt,x,Pξ,Zt,x,Pξ,Ht,x,Pξ) with respect to the measure Pξ, and the derivative of the process (∂μYt,x,Pξ(y),∂μZt,x,Pξ(y),∂μHt,x,Pξ(y)) with respect to y are studied under appropriate regularity assumptions on the coefficients, respectively. These derivatives turn out to be bounded and continuous in L2. The proof of the continuity of the second order derivatives is particularly involved and requires subtle estimates. This regularity ensures that the value function V(t,x,Pξ)≔Ytt,x,Pξ is regular and allows to show with the help of a new Itô formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.

Suggested Citation

  • Li, Juan, 2018. "Mean-field forward and backward SDEs with jumps and associated nonlocal quasi-linear integral-PDEs," Stochastic Processes and their Applications, Elsevier, vol. 128(9), pages 3118-3180.
    • Handle: RePEc:eee:spapps:v:128:y:2018:i:9:p:3118-3180
    DOI: 10.1016/j.spa.2017.10.011
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    References listed on IDEAS

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    1. Buckdahn, Rainer & Hu, Ying & Li, Juan, 2011. "Stochastic representation for solutions of Isaacs’ type integral–partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 121(12), pages 2715-2750.
    2. Li, Juan & Wei, Qingmeng, 2014. "Lp estimates for fully coupled FBSDEs with jumps," Stochastic Processes and their Applications, Elsevier, vol. 124(4), pages 1582-1611.
    3. Tao Hao & Juan Li, 2014. "Backward Stochastic Differential Equations Coupled with Value Function and Related Optimal Control Problems," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-17, March.
    4. Hui Min & Ying Peng & Yongli Qin, 2014. "Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-15, April.
    5. Buckdahn, Rainer & Li, Juan & Peng, Shige, 2009. "Mean-field backward stochastic differential equations and related partial differential equations," Stochastic Processes and their Applications, Elsevier, vol. 119(10), pages 3133-3154, October.
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    Cited by:

    1. Guo, Xin & Pham, Huyên & Wei, Xiaoli, 2023. "Itô’s formula for flows of measures on semimartingales," Stochastic Processes and their Applications, Elsevier, vol. 159(C), pages 350-390.
    2. Sin, Myong-Guk & Ri, Kyong-Il & Kim, Kyong-Hui, 2022. "Existence and uniqueness of solution for coupled fractional mean-field forward–backward stochastic differential equations," Statistics & Probability Letters, Elsevier, vol. 190(C).
    3. Buckdahn, Rainer & Chen, Yajie & Li, Juan, 2021. "Partial derivative with respect to the measure and its application to general controlled mean-field systems," Stochastic Processes and their Applications, Elsevier, vol. 134(C), pages 265-307.
    4. Fan, Xiliang & Huang, Xing & Suo, Yongqiang & Yuan, Chenggui, 2022. "Distribution dependent SDEs driven by fractional Brownian motions," Stochastic Processes and their Applications, Elsevier, vol. 151(C), pages 23-67.

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