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

By Juan Li. Stochastic Processes and their Applications, 2017

75 citations recorded by the source corpus.

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 Ito formula that it is the unique classical solution of the related nonlocal quasi-linear integral-partial differential equation (PDE) of mean-field type.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3a1c4a73d55832b2847eae5f4024d0c018982d08

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In Applied Mathematics

  1. Chenggui Yuan
  2. Yongqiang Suo
  3. Xing Huang
  4. Xiliang Fan

In Finance

  1. Chenggui Yuan
  2. Yongqiang Suo
  3. Xing Huang
  4. Xiliang Fan
  5. Xuerong Mao
  6. Xiaoyue Li
  7. George Yin

In Mathematical Physics

  1. Xing Huang

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