Integration by parts formula for locally smooth laws and applications to sensitivity computations

Abstract

We consider random variables of the form F=f(V₁, …, V_n), where f is a smooth function and V_i, i∈ℕ, are random variables with absolutely continuous law p_i(y) dy. We assume that p_i, i=1, …, n, are piecewise differentiable and we develop a differential calculus of Malliavin type based on ∂lnp_i. This allows us to establish an integration by parts formula E(∂_iϕ(F)G)=E(ϕ(F)H_i(F, G)), where H_i(F, G) is a random variable constructed using the differential operators acting on F and G. We use this formula in order to give numerical algorithms for sensitivity computations in a model driven by a Lévy process.