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In a market driven by a Lévy martingale, we consider a claim ξ . We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for ξ : one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark-HaussmannOcone theorem.

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We introduce the forward integral with respect to a pure jump Lévy process and prove an Itô formula for this integral. Then we use Mallivin calculus to establish a relationship between the forward integral and the Skorohod integral and apply this to obtain an Itô formula for the Skorohod integral. 235 Infin. Dimens. Anal. Quantum. Probab. Relat. Top. 2005.08:235-258. Downloaded from www.worldscientific.com by UNIVERSITY OF MICHIGAN on 03/05/15. For personal use only. 236 G. Di Nunno et al.Brownian motion. This discovery led to an enormous increase in the interest in the Malliavin calculus both among mathematicians and finance researchers and since then the theory has been generalized and new applications have been found. In particular, Malliavin calculus for Brownian motion has been applied to compute the greeks in finance, see e.g. Refs. 2, 16 and 17. Moreover, anticipative stochastic caluclus for Brownian motion involving the forward integral (beyond the semimartingale context) has been applied to give a general approach to optimal portfolio and consumption problems for insiders in finance, see e.g. Refs. 9, 10 and 31. An extension of the Malliavin method to processes with discontinuous trajectories was carried out in 1987 by Bichteler, Gravereaux and Jacod. 11 However, their work is focused on the original problem of the smoothness of the densities of the solutions of stochastic differential equations, a question that does not deal with the other more recent aspects of the Malliavin calculus. For related works on stochastic calculus for stochastic measures generated by a Poisson process on the real line see Refs. 13,28,29,38, 44 and 45, for example. Recently two types of Malliavin derivative operators D (m) t groups,