Controlled reflected mean-field backward stochastic differential equations coupled with value function and relatedPDEs

Juan  Li; Wenqiang  Li

doi:10.3934/mcrf.2015.5.501

Article Contents

2015, Volume 5, Issue 3: 501-516. Doi: 10.3934/mcrf.2015.5.501

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Controlled reflected mean-field backward stochastic differential equations coupled with value function and related PDEs

Juan Li^1, and
Wenqiang Li^1,

1.
School of Mathematics and Statistics, Shandong University, Weihai, Weihai, 264209

Received: February 2014

Revised: February 2015

Published: September 2015

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Abstract

In this paper, we consider a new type of reflected mean-field backward stochastic differential equations (reflected MFBSDEs, for short), namely, controlled reflected MFBSDEs involving their value function. The existence and the uniqueness of the solution of such equation are proved by using an approximation method. We also adapt this method to give a comparison theorem for our reflected MFBSDEs. The related dynamic programming principle is obtained by extending the approach of stochastic backward semigroups introduced by Peng [11] in 1997. Finally, we show that the value function which our reflected MFBSDE is coupled with is the unique viscosity solution of the related nonlocal parabolic partial differential equation with obstacle.

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Mathematics Subject Classification: Primary: 60H10, 60H30; Secondary: 90C39, 93E20.

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