Stochastic Control in Finance

Lecturer: H. Pham
Period: Term  2
Schedule: 3h per week

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Stochastic control is a classical topic in applied mathematics and occurs in many practical situations when we have to take decisions under uncertainty. It has known important developments over the last years inspired especially by problems in mathematical finance. To mention some applications: - hedging and pricing of options, - portfolio selection, - risk management, - real options and investment on energy markets, -  high frequency trading.  The aim of these lectures is to give an overview  of the main methods and results in this area. 

We first present the standard approach by dynamic programming equation and verification, and point out the limits of this method. We then move on to the viscosity solutions approach: it requires more theory and technique, but  provides the general  mathematical tool for dealing with stochastic control in a Markovian  context. Next, we focus on a special and important class of stochastic control, namely singular stochastic control, which arises in many applications including portfolio selection with transaction costs and (ir)reversible investment in corporate finance. The last part will be devoted to the martingale approach for portfolio optimization. The various methods presented in these lectures will be illustrated by several applications arising in economics and finance.