Journal of the
Korean Mathematical Society JKMS

Let $X = (X_t , t \in [0,T]$ be a generalized Brownian motion (gBm) determined by mean function $a(t)$ and variance function $ b(t)$. Let $L^2(\tilde\mu)$ denote the Hilbert space of square integrable functionals of $\tilde X = (X_t- a(t), \, t\in [0,T])$. In this paper we consider a class of nonlinear functionals of $X$ of the form $ F(\,\cdot + a) $ with $ F\in L^2(\tilde \mu)$ and discuss their analysis. Firstly, it is shown that such functionals do not enjoy, in general, the square integrability and Malliavin differentiability. Secondly, we establish regularity conditions on $F$ for which $ F( \,\cdot + a) $ is in $L^2(\tilde\mu)$ and has its Malliavin derivative. Finally we apply these results to compute the price and the hedging portfolio of a contingent claim in our financial market model based on a gBm $X$.

Keywords: generalized Brownian motion, Malliavin derivative, Black-Scholes model, Hedging portfolio

MSC numbers: 28C20, 60J65, 60H07