Constrained mean-variance investment-reinsurance under the Cramér-Lundberg model with random coefficients

ArXiv ID: 2406.10465 “View on arXiv”

Authors: Unknown

Abstract

In this paper, we study an optimal mean-variance investment-reinsurance problem for an insurer (she) under a Cramér-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.

Keywords: Mean-Variance Optimization, Reinsurance, Stochastic Control, Stochastic Riccati Equations, Backward Stochastic Differential Equations

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 1.5/10
  • Quadrant: Lab Rats
  • Why: The paper employs advanced stochastic control, BSDEs with jumps, and stochastic Riccati equations with no empirical implementation or backtesting. Its focus is entirely on theoretical existence, uniqueness, and analytical solutions.
  flowchart TD
    Start["Research Goal: Optimal Mean-Variance<br>Investment-Reinsurance"] --> Setup["Problem Setup<br>Constrained Stochastic LQ Control with Jumps"]
    Setup --> Solve["Solve via Stochastic<br>Riccati Equations SREs"]
    Solve --> Tech["Existence & Uniqueness<br>via BSDE Techniques<br>Log Transform & BMO Martingales"]
    Tech --> Results["Efficient Strategy: Linear Feedback<br>Efficient Frontier: Half-Line"]
    Results --> Outcomes["Outcomes:<br>Explicit optimal strategy &<br>Mean-Variance Frontier"]