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A risk-neutral method is always used to price and hedge contingent claims incomplete market, but another method based on utility maximization or riskminimization is wildly used in more general case. One can find all kinds ofspecial risk measure in literature. In this paper, instead of using marketmodified risk measure, we use a kind of risk measure induced byg_\Gamma-solution or the minimal solution of a Constrained Backward StochasticDifferential Equation (CBSDE) directly when constraints on wealth and portfolioprocess comes to our consideration. Such g_\Gamma-solution and the risk measuregenerated by it is well defined on appropriate space under suitable conditions.We adopt the inf-convolution of convex risk measures to solve some optimizationproblem. A dynamic version risk measures defined through g_\Gamma-solution andsome similar results about optimal problem can be got in our new framework andby our new approach.

Inf-convolution of g_\Gamma-solution and its applications