# COLLOQUIUM

## Stochastic Optimal Control for Dynamic Reinsurance Policies

### Lin Zhuo M.S. Defense

When applying a reinsurance policy the surplus of the insurance company $X(t)$ is governed by a stochastic differential equation $dX(t) = b(X(t),u(t)) dt + \sigma(X(t),u(t))dw(t)$, where $w(\cdot)$ is a standard Brownian motion, and the stochastic control process $u(t)$ satisfying $0\leq u(t)\leq 1$ represents one of the possible reinsurance policies available at time $t$. The aim of this thesis is to find a reinsurance policy that maximizes the expected total return function $J(x, u(\cdot))=\mathbf E_{x} \int_0^\tau e^{-\delta t} l(X(t),u(t))dt$, where $l(X(t),u(t))$ is a return function, $\delta > 0$ is the discount rate, $\tau$ is the time of ruin and $x$ refers to the initial surplus. Since optimal control problems generally do not have analytic solutions, we introduce and employ a numerical method using Markov chain approximation on the basis of dynamic programming to solve these problems.