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Optimal control under uncertainty: Application to the issue of CAT bonds

Nicolas Baradel

Abstract

We propose a general framework for studying optimal issue of CAT bonds in the presence of uncertainty on the parameters. In particular, the intensity of arrival of natural disasters is inhomogeneous and may depend on unknown parameters. Given a prior on the distribution of the unknown parameters, we explain how it should evolve according to the classical Bayes rule. Taking these progressive prior-adjustments into account, we characterize the optimal policy through a quasi-variational parabolic equation, which can be solved numerically. We provide examples of application in the context of hurricanes in Florida.

Optimal control under uncertainty: Application to the issue of CAT bonds

Abstract

We propose a general framework for studying optimal issue of CAT bonds in the presence of uncertainty on the parameters. In particular, the intensity of arrival of natural disasters is inhomogeneous and may depend on unknown parameters. Given a prior on the distribution of the unknown parameters, we explain how it should evolve according to the classical Bayes rule. Taking these progressive prior-adjustments into account, we characterize the optimal policy through a quasi-variational parabolic equation, which can be solved numerically. We provide examples of application in the context of hurricanes in Florida.

Paper Structure

This paper contains 31 sections, 13 theorems, 137 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The framework
  3. General framework
  4. CAT bond framework
  5. The control
  6. The CAT bonds process
  7. The output process
  8. Bayesian updates
  9. Evolution of the intensity
  10. Evolution of the parameters $\gamma_{0}$ and $\upsilon_{0}$
  11. Parametrization of the set $\mathbf{M}^{\lambda}$
  12. Gain function
  13. Value function characterization
  14. Viscosity solution properties
  15. Sub-solution property
  16. ...and 16 more sections

Key Result

Proposition 2.1

Under Assumption h_lambda, the process $M_{s}^{t,m^{\lambda}}$ is

Figures (9)

  • Figure 7.1: Representation of $h_{0}$ over one year with $d_{0} = 1^{\text{st}}$ July, $d_{1} = 15^{\text{th}}$ November, $\hat{\alpha} = 8$ and $\hat{\beta} = 6$.
  • Figure 7.2: Representation of an OEP curve, with the parameter $(\mu, \sigma, \xi)$ defined in the text and with the prior $(p^{\alpha}, p^{\beta}) := (25, 50)$.
  • Figure 7.3: Simulated path of the optimal strategy of the insurer (true value is $\lambda_0 = 0.6$).
  • Figure 7.4: Cash distribution (with 200 000 simulations) for $\lambda_{0} = 0.6$ (left) and $\lambda_{0} = 0.5$ (right) with the optimal control (solid dark blue) and without any CAT bond (dashed black).
  • Figure 7.5: Simulated path of the optimal strategy of the insurer.
  • ...and 4 more figures

Theorems & Definitions (42)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.1
  • Remark 2.4
  • Proposition 2.1
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.1
  • ...and 32 more