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

TL;DR

This paper introduces a novel approach to risk measurement and optimization using inf-convolution of g_\Gamma-solution derived from constrained backward stochastic differential equations, expanding utility-based risk management methods.

Contribution

It develops a new framework for dynamic risk measures based on g_\Gamma-solution and applies inf-convolution to solve related optimization problems.

Findings

Defined a risk measure via g_\Gamma-solution under constraints.

Applied inf-convolution to convex risk measures for optimization.

Established dynamic risk measures and optimal solutions in the new framework.

Abstract

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