Nonlinear problems modeling stochastic volatility and transaction costs

The option pricing problem when the asset is driven by a stochastic volatility process and in the presence of transaction costs leads to solving a nonlinear partial differential equation (PDE). The nonlinear term in the PDE reflects the presence of transaction costs. Under a particular market completion assumption we derive the nonlinear PDE whose solution may be used to find the price of options. Under suitable conditions, we give an algorithmic scheme to obtain the solution of the problem by an iterative method. We prove theoretically the existence of strong solutions to the problem.