Financial Mathematics, Stochastic Portfolio

Stochastic Portfolio Theory is a branch of Mathematical Finance. Robert Merton studied the portfolio allocation problem as a Stochastic Control Problem. In his model, an investor allocates his/her wealth between a risky asset and a riskless asset, then chooses the consumption rate to maximize the total expected utility. Dr. Hussain is interested in studying portfolio allocation models, on finite and infinite time horizons, that incorporate the past (recent/complete) performance of the portfolio. In such a setting the risky asset is modeled using a non- Markovian process and therefore the portfolio allocation problem becomes a stochastic optimal control problem with delays. The goal is to find the optimal control strategies and the value function i.e. maximum total expected utility. Newly developed functional Ito's calculus initiated by Dupire can be used to obtain the associated Hamilton-Jacobi-Bellman Equation (HJB) for such models. Obtaining explicit solution of HJB equation might not be possible.  Method of sub/super solutions and viscosity solutions can be used to prove the existence of solution of the HJB equation. Moreover, he  is currently working on several portfolio allocation models with delays where state processes have stochastic volatility.  Dr. Hussain is also interested in financial risk management side of the finance in addition to several other areas directly or indirectly related to Operations Research and Optimization with applications in finance, health care and energy.