Optimal Investment Problems and Volatility Homogenization Approximations

We describe some stochastic control problems in financial engineering arising from the need to find investment strategies to optimize some goal. Typically, these problems are characterized by nonlinear Hamilton-Jacobi-Bellman partial differential equations, and often they can be reduced to linear PDEs with the Legendre transform of convex duality. One situation where this cannot be achieved is in a market with stochastic volatility. In this case, we discuss an approximation using asymptotic analysis in the limit of fast mean-reversion of the process driving volatility. Simulations illustrate that marginal improvement can be achieved with this approach even when volatility is not fluctuating that rapidly.