English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In this paper we consider a mean-field backward stochastic differentialequation (BSDE) driven by a Brownian motion and an independent Poisson randommeasure. Translating the splitting method introduced by Buckdahn, Li, Peng andRainer [6] to BSDEs, the existence and the uniqueness of the solution$(Y^{t,\xi}, Z^{t,\xi}, H^{t,\xi})$, $(Y^{t,x,P_\xi}, Z^{t,x,P_\xi},H^{t,x,P_\xi})$ of the split equations are proved. The first and the secondorder derivatives of the process $(Y^{t,x,P_\xi}, Z^{t,x,P_\xi},H^{t,x,P_\xi})$ with respect to $x$, the derivative of the process$(Y^{t,x,P_\xi}, Z^{t,x,P_\xi}, H^{t,x,P_\xi})$ with respect to the measure$P_\xi$, and the derivative of the process $(\partial_\mu Y^{t,x,P_\xi}(y),\partial_\mu Z^{t,x,P_\xi}(y), \partial_\mu H^{t,x,P_\xi}(y))$ with respect to$y$ are studied under appropriate regularity assumptions on the coefficients,respectively. These derivatives turn out to be bounded and continuous in $L^2$.The proof of the continuity of the second order derivatives is particularlyinvolved and requires subtle estimates. This regularity ensures that the valuefunction $V(t,x,P_\xi):=Y_t^{t,x,P_\xi}$ is regular and allows to show with thehelp of a new It\^{o} formula that it is the unique classical solution of therelated nonlocal quasi-linear integral-partial differential equation (PDE) ofmean-field type.

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