Generalized Brownian motions with application to finance
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Abstract
Let $X = (X_t , t \in \T)$ be a generalized Brownian motion(\GBm) determined by mean function $a(t)$ and variance function $ b(t)$. Let $\NonLinearI$ 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 $\NonLinearI$ 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 $X$.
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