Small noise expansion and importance sampling

Consider the second-order differential operators ℒ₀ = -∂_t - a₁(y)∂_x - a₂(y)∂_xx and ℒ̃ = -b₁(y)∂_y - b₂(y)∂_yy - c(y)∂_xy and let u_ε(t, x, y) be the solution of the parabolic problem ℒ₀ + εℒ̃u = 0 on [0, T) × ℝ² with terminal condition u_ε(T, x, y) = φ(x), for given ε ∈ ℝ. We provide an explicit asymptotic expansion of the solution u_ε around the value ε = 0. The expansion coefficients of any order are determined by an explicit induction scheme involving the derivatives of u₀ with respect to x. The results are applied for the computation of European contingent claim prices by Monte Carlo simulations in stochastic volatility models, which are popular in the financial literature. The asymptotic expansion is used as accelerator in an importance sampling variance reduction procedure.