Martingale representation of functionals of Lévy processes. (English) Zbl 1060.60058
The paper presents a version of a Clark-Ocone-Hausman formula for functionals on Lévy processes. It concentrates on the case of pure jump processes. The general case can be obtained by combination of the results in this paper and the classical Clark-Ocone-Hausman formula. To obtain the formula, a Gross (or Malliavin) type derivative \(D\) corresponding to some fixed square integrable Lévy process \(L\) with no Brownian motion part is introduced. This is done by using a specific chaos decomposition for functionals \(F \in L^2(\mathcal{F}_T,\mathbb{P})\), where \(\mathcal{F}_T = \overline{\sigma(L_t \mid 0 \leq t \leq T)}_+\), in a similar way as the classical Malliavin derivative operator can be defined on the Wiener chaos decomposition [see D. Nualart, “The Malliavin calculus and related topics” (1995; Zbl 0837.60050)]. Using this derivative operator, a formula
\[
F = \mathbb{E}(F)+\int_0^T \int_{\mathbb{R} \setminus 0} \mathbb{E}[D_{t,z}F\mid \mathcal{F}_{t-}](\mu-\pi)(dz,dt)
\]
where \(\mathbb{E}[D_{t,z}F\mid \mathcal{F}_{t-}]\) denotes the predictable projection and \(\mu-\pi\) is the compensated Poisson random measure, is derived for functionals \(F\) in \(L^2(\mathcal{F}_T,\mathbb{P}) \cap \mathbb{D}_{1,2}\). It is then proven, that the operator \(D\) as defined in this paper coincides on the class \(\mathbb{D}_{1,2}\) with an operator earlier defined by J. Picard [Probab. Theory Relat. Fields 105, 481–511 (1996; Zbl 0853.60064)]. This result is a generalization of a corresponding result which was proven by D. Nualart and V. Vives [in: Séminaire de Probabilités XXIV 1988/89. Lect. Notes Math. 1426, 154–165 (1990; Zbl 0701.60048)] in the case of a Poisson process. In the final section of the article, the obtained Clark-Ocone-Hausman formula is applied to compute the risk minimizing hedging strategy for a contingent claim in a financial market consisting of one bond \(B_t\) and one stock \(S_t\) given by \(B_t = 1\); \(S_t = S_0\cdot\exp(L_t-ct)\). To obtain the risk minimizing hedging strategy, the newly obtained Clark-Ocone-Hausman formula is used to identify the terms in the Kunita-Watanabe decomposition of the contingent claim.
Reviewer: Christian-Oliver Ewald (Bristol)
MSC:
| 60H07 | Stochastic calculus of variations and the Malliavin calculus |
| 91B28 | Finance etc. (MSC2000) |
| 60G51 | Processes with independent increments; Lévy processes |
Keywords:
Lévy processes; Poisson random measure; martingale representation; hedging; Malliavin calculusReferences:
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