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Bayesian optimal investment and reinsurance with dependent financial and insurance risks

Nicole Bäuerle, Gregor Leimcke

Abstract

Major events like natural catastrophes or the COVID-19 crisis have impact both on the financial market and on claim arrival intensities and claim sizes of insurers. Thus, when optimal investment and reinsurance strategies have to be determined it is important to consider models which reflect this dependence. In this paper we make a proposal how to generate dependence between the financial market and claim sizes in times of crisis and determine via a stochastic control approach an optimal investment and reinsurance strategy which maximizes the expected exponential utility of terminal wealth. Moreover, we also allow that the claim size distribution may be learned in the model. We give comparisons and bounds on the optimal strategy using simple models. What turns out to be very surprising is that numerical results indicate that even a minimal dependence which is created in this model has a huge impact on the control in the sense that the insurer is much more prudent then.

Bayesian optimal investment and reinsurance with dependent financial and insurance risks

Abstract

Major events like natural catastrophes or the COVID-19 crisis have impact both on the financial market and on claim arrival intensities and claim sizes of insurers. Thus, when optimal investment and reinsurance strategies have to be determined it is important to consider models which reflect this dependence. In this paper we make a proposal how to generate dependence between the financial market and claim sizes in times of crisis and determine via a stochastic control approach an optimal investment and reinsurance strategy which maximizes the expected exponential utility of terminal wealth. Moreover, we also allow that the claim size distribution may be learned in the model. We give comparisons and bounds on the optimal strategy using simple models. What turns out to be very surprising is that numerical results indicate that even a minimal dependence which is created in this model has a huge impact on the control in the sense that the insurer is much more prudent then.

Paper Structure

This paper contains 22 sections, 16 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Optimal Investment and Reinsurance Model
  3. The aggregated claim amount process
  4. The financial market
  5. The strategies
  6. The optimization problem
  7. A Model with Learning
  8. Filtering
  9. The Reduced Control Problem
  10. The Solution
  11. The HJB equation
  12. Candidate for an optimal strategy
  13. Verification
  14. Existence result for the value function
  15. Comparison results
  16. ...and 7 more sections

Key Result

Proposition 3.1

The ${\mathfrak G}$-intensity kernel of $\Psi=(T_n,(Y_n,Z_n))$, denoted by $\hat{\nu}(t,d(y,z))$, is given by

Figures (1)

  • Figure 1: Optimal strategy in the case of complete observation as a function of $L$ with logaritmically scaled $x$-axis.

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Proposition 3.1
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • Lemma 4.4
  • ...and 22 more