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Robust Optimal Investment and Reinsurance Problems with Learning

Nicole Bäuerle, Gregor Leimcke

TL;DR

The paper addresses robust optimal investment and reinsurance under partial information with learning of uncertain parameters. It blends Bayesian learning (Dirichlet priors for thinning and a finite-support prior for intensity) with a generalized HJB framework, solved via Clarke gradients to handle nondifferentiability. The authors derive an explicit investment strategy $\xi^*(t)=\frac{\mu-r}{\sigma^2}\frac{1}{\alpha}e^{-r(T-t)}$ and an implicitly defined reinsurance rule obtained from a root of $(1+\theta)\kappa=h(t,p,q,a)$, together with computable bounds and a verification theorem, and they establish existence of the value function with an augmented state $(X^{\xi,b},p,q)$. Numerical results illustrate learning dynamics and show how partial information compares to the complete-information benchmark across a multi-line setting.

Abstract

In this paper we consider an optimal investment and reinsurance problem with partially unknown model parameters which are allowed to be learned. The model includes multiple business lines and dependence between them. The aim is to maximize the expected exponential utility of terminal wealth which is shown to imply a robust approach. We can solve this problem using a generalized HJB equation where derivatives are replaced by generalized Clarke gradients. The optimal investment strategy can be determined explicitly and the optimal reinsurance strategy is given in terms of the solution of an equation. Since this equation is hard to solve, we derive bounds for the optimal reinsurance strategy via comparison arguments.

Robust Optimal Investment and Reinsurance Problems with Learning

TL;DR

The paper addresses robust optimal investment and reinsurance under partial information with learning of uncertain parameters. It blends Bayesian learning (Dirichlet priors for thinning and a finite-support prior for intensity) with a generalized HJB framework, solved via Clarke gradients to handle nondifferentiability. The authors derive an explicit investment strategy and an implicitly defined reinsurance rule obtained from a root of , together with computable bounds and a verification theorem, and they establish existence of the value function with an augmented state . Numerical results illustrate learning dynamics and show how partial information compares to the complete-information benchmark across a multi-line setting.

Abstract

In this paper we consider an optimal investment and reinsurance problem with partially unknown model parameters which are allowed to be learned. The model includes multiple business lines and dependence between them. The aim is to maximize the expected exponential utility of terminal wealth which is shown to imply a robust approach. We can solve this problem using a generalized HJB equation where derivatives are replaced by generalized Clarke gradients. The optimal investment strategy can be determined explicitly and the optimal reinsurance strategy is given in terms of the solution of an equation. Since this equation is hard to solve, we derive bounds for the optimal reinsurance strategy via comparison arguments.

Paper Structure

This paper contains 19 sections, 19 theorems, 135 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Optimal Investment and Reinsurance Model
  3. The claim arrival 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. Candidate for an Optimal Strategy
  12. Verification
  13. Existence result for the value function
  14. Comparison Results with the complete Information Case
  15. Numerical Results
  16. ...and 4 more sections

Key Result

Theorem 3.2

The posterior distribution of ${\bar{\alpha}}$ given $q_t=c$ with $c=(c_D)_{D\subset\mathbb{D}}\in\mathbb{N}_0^\ell$ is a Dirichlet distribution with parameter vector $\bar{\beta}+c=(\beta_D+c_D)_{D\subset\mathbb{D}}$.

Figures (3)

  • Figure 1: A trajectory of the filter process $(p_t)_{t\ge0}$ under the assumptions that $\bar{\pi}_\Lambda=(2/5,2/5,1/5)$ and $\Lambda=4$, where $p_t=(p_1(t),p_2(t),p_3(t))$ with $p_1(t)=\mathbb{P}(\Lambda=2|{\mathcal{G}}_t)$, $p_2(t)=\mathbb{P}(\Lambda=4|{\mathcal{G}}_t)$ and $p_3(t)=\mathbb{P}(\Lambda=5|{\mathcal{G}}_t)$.
  • Figure 2: A priori upper (red) and lower bound (orange) for the optimal reinsurance strategy and two paths of the reinsurance strategy $(b^\star_{u(p_{t-}),w(q_{t-})}(t))_{t\in[0,T]}$ with $u(p)\mathrel{\mathop{\raisebox{0.55pt}{\small$:$}}}= \sum_{k=1}^m\lambda_k p_k$ and $w(q)\mathrel{\mathop{\raisebox{0.55pt}{\small$:$}}}=((\beta_D+q_D)/\|\bar{\beta}+q\|)_{D\subset\mathbb{D}}$.
  • Figure 3: Paths of the surplus process in case of full reinsurance (red), constant retention level of 0.5 (blue) and the reinsurance strategy $(b^\star_{u(p_{t-}),w(q_{t-})}(t))_{t\in[0,T]}$ with $u(p)\mathrel{\mathop{\raisebox{0.55pt}{\small$:$}}}= \sum_{k=1}^m\lambda_k p_k$ and $w(q)\mathrel{\mathop{\raisebox{0.55pt}{\small$:$}}}=((\beta_D+q_D)/\|\bar{\beta}+q\|)_{D\subset\mathbb{D}}$ (black).

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1: Dirichlet distribution; DeGroot1970, p. 49
  • Theorem 3.2: DeGroot1970, Thm 9.8.1
  • Corollary 3.3
  • Theorem 3.4
  • Proposition 3.5
  • proof
  • Lemma 4.1
  • proof
  • ...and 25 more