Properties of gΓ-solution and risk measure induced by gΓ-solution

In this paper, some main properties of the g_Γ-solution of a constrained backward stochastic differential equation (CBSDE) are presented and a kind of risk measure induced by the g_Γ-solution is proposed. These fine properties make the risk measure become a satisfactory tool for solving the hedging problem in an incomplete market. More importantly, the infconvolution of the convex risk measures can be adopted to deal with some optimization problems involving a transformation of the initial risk measures. Some results about the dynamic version of the inf-convolution of the g_Γ-solutions will also be given, just like the usual case without constraints, the inf-convolution of two g_Γ-solutions of CBSDEs with different coefficients is equivalent to the g_Γ-solution of CBSDE with the inf-convolution of the two coefficients. In this case, it is possible to characterize the optimal risk transfer.