# 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 astandard Brownian motion, and the stochastic control process $u(t)$satisfying $0\leq u(t)\leq 1$ represents one of the possible reinsurancepolicies available at time $t$. The aim of this thesis is to find areinsurance 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 discountrate, $\tau$ 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 chainapproximation on the basis of dynamic programming to solve these problems.