On some expectation and derivative operators related to integral representations of random variables with respect to a PII process

Stéphane Goutte (1) , Nadia Oudjane (2, 3) , Francesco Russo (4, 5)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 EDF - EDF
3 LAGA - Laboratoire Analyse, Géométrie et Applications
4 MATHRISK - Mathematical Risk handling
5 OC - Optimisation et commande

Given a process with independent increments $X$ (not necessarily a martingale) and a large class of square integrable r.v. $H=f(X_T)$, $f$ being the Fourier transform of a finite measure $\mu$, we provide explicit Kunita-Watanabe and Föllmer-Schweizer decompositions. The representation is expressed by means of two significant maps: the expectation and derivative operators related to the characteristics of $X$. We also provide an explicit expression for the variance optimal error when hedging the claim $H$ with underlying process $X$. Those questions are motivated by finding the solution of the celebrated problem of global and local quadratic risk minimization in mathematical finance.

Mots clés: Lévy processes; Characteristic functions; Processes with independent increments; global and local quadratic risk minimization; expectation and derivative operators.; expectation and derivative operators; Föllmer-Schweizer decomposition; Kunita-Watanabe decomposition

Domaines: Probabilités [math.PR]

Stéphane Goutte, Nadia Oudjane, Francesco Russo. On some expectation and derivative operators related to integral representations of random variables with respect to a PII process. Stochastic Analysis and Applications, 2013, 31, pp.108--141. ⟨10.1080/07362994.2013.741395⟩. ⟨hal-00665852⟩