Optimization Problems in Finance and Insurance

Abstract

We will now apply the theory of infinite horizon Markov Decision Models to solve some optimization problems in finance. In Section 9.1 we consider a consumption and investment problem with random horizon which leads to a contracting Markov Decision Model with infinite horizon as explained in Section 7.6.1. Explicit solutions in the case of a power utility are given. In Section 9.2 a classical dividend pay-out problem for an insurance company is investigated. In this example the state and action space are both discrete which implies that all functions on E×A are continuous and we can work with Theorem 7.2.1. Here the Markov Decision Model is not contracting. The main part of this section is to show that there exists an optimal stationary policy which is a so-called band-policy. In special cases this band-policy reduces to a barrier-policy, i.e. it is optimal to pay out all the money which is above a certain threshold. In Section 9.3 we consider a utility maximization problem in a financial market where the stock prices are Piecewise Deterministic Markov Processes. This optimization problem is contracting and our results from Chapters 7 and 8 allow a characterization of the value function and some computational approaches which complement the classical stochastic control approach via the Hamilton-Jacobi-Bellman equation. Some numerical results are also given. In Section 9.4 we study the liquidation of a large amount of shares in so-called dark pools. This is a continuous-time Markov Decision Chain with finite time horizon (see Section 8.3). Using the discretetime solution approach we are able to derive some interesting properties of the optimal liquidation policy.