Constrained mean-variance investment-reinsurance under the Cram\'er-Lundberg model with random coefficients

TL;DR

This paper develops a mathematical framework for optimal investment and reinsurance strategies for insurers under a stochastic model with random coefficients, providing explicit solutions and conditions for existence and uniqueness.

Contribution

It introduces a novel approach to solve a constrained mean-variance problem under the Cramér-Lundberg model with jumps, using stochastic Riccati equations and BSDE techniques.

Findings

Explicit solutions for optimal strategies are derived.

Existence and uniqueness of solutions to the SREs are established.

The optimal strategies are expressed in linear feedback form.

Abstract

In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under a Cram\'er-Lundberg model with random coefficients. At any time, the insurer can purchase reinsurance or acquire new business and invest her surplus in a security market consisting of a risk-free asset and multiple risky assets, subject to a general convex cone investment constraint. We reduce the problem to a constrained stochastic linear-quadratic control problem with jumps whose solution is related to a system of partially coupled stochastic Riccati equations (SREs). Then we devote ourselves to establishing the existence and uniqueness of solutions to the SREs by pure backward stochastic differential equation (BSDE) techniques. We achieve this with the help of approximation procedure, comparison theorems for BSDEs with jumps, log transformation and BMO martingales. The efficient investment-reinsurance strategy and efficient mean-variance frontier are explicitly given through the solutions of the SREs, which are shown to be a linear feedback form of the wealth process and a half-line, respectively.