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Elementary introduction to Malliavin calculus and advanced Monte- Carlo methods II

The aim of these lectures is to provide an elementary introduction to Hörmanders famous results on linear diffusion equations from a probabilistic point of view, and to show how these results can be applied to computational finance in the context of advanced Monte-Carlo methods of diffusion models with stochastic volatility. The lectures are aimed at an audience with some modest knowledge of stochastics and with interest in numerical applications. The focus is on basic concepts of Malliavin calculus and their use in probabilistic schemes such that course participants should be able to read numerical research papers where Malliavin calculus is applied. The course contains the following topics: i) A general overview concerning Hörmanders regularity results for linear parabolic Cauchy problems with L-invertible diffusion matrices, motivation of Malliavin calculus, and why it is essential for studying stochastic volatility models and for computing Greeks in finance. ii) Integration by parts formula, Differential operators of Malliavin calculus operators on Wiener space. iii) L-invertibility of the diffusion matrix as a consequence of the Hörmander condition, and introduction to density estimates of Kusuoka-Stroock type. iv) Weighted Monte-Carlo Methods, optimal weights, density estimates and proxy schemes for semi-elliptic equations. In this first lecture we provide a general introduction to Hörmanders theorem and how it motivated Malliavin’s original work. The original work is Hörmanders paper [Hörmander, L. ]. In [Malliavin, P. a)], and [Malliavin, P. b)] provides a direct proof that smooth densities of hypoelliptic operators. Most stochastic volatility models satisfy the Hörmander condition, and are hypoelliptic only (not strictly elliptic). However in the most simple cases a more direct approach is possible (cf. [ Belomestny, D., Kampen, J., Schoenmakers, J.G.M.]).