Optimal reinsurance and investment via stochastic projected gradient method based on Malliavin calculus

Abstract

This paper proposes a new approach using the stochastic projected gradient method and Malliavin calculus for optimal reinsurance and investment strategies. Unlike traditional methodologies, we aim to optimize static investment and reinsurance strategies by directly minimizing the ruin probability. Furthermore, we provide a convergence analysis of the stochastic projected gradient method for general constrained optimization problems whose objective function has Hölder continuous gradient. Numerical experiments show the effectiveness of our proposed method.

Figures (6)

• Figure 1: Convergence behaviors of the minimum ruin probability with $n= 11$.
• Figure 2: Convergence behaviors of the minimum ruin probability with $n= 101$.
• Figure 3: Convergence behaviors of the minimum ruin probability with $n=1001$.
• Figure 4: Ruin probabilities of the proposed method and the adjustment coefficient approach with $\tilde{\gamma} = 0.01$.
• Figure 5: Ruin probabilities of the proposed method and the adjustment coefficient approach with $\tilde{\gamma} = 0.1$.

Theorems & Definitions (27)

• Lemma 1.1
• proof
• Lemma 1.2
• Lemma 1.3
• Lemma 3.1
• proof
• Theorem 3.1
• proof
• Proposition 3.1
• proof