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

Paper: arXiv 2405.17841

Abstract

This paper studies an optimal investment-reinsurance problem for an insurer (she) under the Cramér–Lundberg model with monotone mean–variance (MMV) criterion. At any time, the insurer can purchase reinsurance (or acquire new business) and invest in a security market consisting of a risk-free asset and multiple risky assets whose excess return rate and volatility rate are allowed to be random. The trading strategy is subject to a general convex cone constraint, encompassing no-shorting constraint as a special case. The optimal investment-reinsurance strategy and optimal value for the MMV problem are deduced by solving certain backward stochastic differential equations with jumps. In the literature, it is known that models with MMV criterion and mean–variance criterion lead to the same optimal strategy and optimal value when the wealth process is continuous. Our result shows that the conclusion remains true even if the wealth process has compensated Poisson jumps and the market coefficients are random.

Complexity vs Empirical Score

• Math Complexity: 9.0/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper heavily relies on advanced stochastic calculus, BSDEs with jumps, and random coefficient models, indicating very high mathematical density. It lacks any empirical validation, backtesting, or data implementation, focusing purely on theoretical derivation and solution existence in a controlled mathematical environment.

Research Flowchart

  flowchart TD
  A["Research Goal:<br/>Optimal Investment-Reinsurance<br/>under MMV Criterion"] --> B

  subgraph B ["Data & Inputs"]
      B1["Cramér--Lundberg Model<br/>(Claims as Jumps)"]
      B2["Market: Risk-free &<br/>Multiple Risky Assets"]
      B3["Constraints:<br/>General Convex Cone"]
  end

  B --> C["Methodology:<br/>Backward Stochastic Differential<br/>Equations BSDEs with Jumps"]

  C --> D["Computational Process:<br/>Solving BSDEs to derive<br/>Optimal Strategy & Value"]

  D --> E{"Key Findings"}

  subgraph E ["Outcomes"]
      E1["Optimal Strategy<br/>(Derived explicitly)"]
      E2["Optimal Value<br/>(Derived explicitly)"]
      E3["Equivalence Proof:<br/>MMV & Mean-Variance yield<br/>identical results despite<br/>jumps & randomness"]
  end

Cited References (16)

1. Introduction To Nonlinear Optimization: Theory, Algorithms, And Applications With Matlab (2017)
2. Optimization by Vector Space Methods (1970) — link
3. A sufficient condition for the uniform integrability of exponential martingales (1979)
4. Semimartingale Theory and Stochastic Calculus (2019) — link
5. Constrained Stochastic LQ Control with Random Coefficients, and Application to Portfolio Selection (2005) — link
6. Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework (2000) — link
7. Optimal investment–reinsurance strategy for mean–variance insurers with square-root factor process (2015) — link
8. Filtering a Markov Modulated Random Measure (2009) — link

1 of 16 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.