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http://dx.doi.org/10.4134/JKMS.2006.43.2.357

GENERALIZED BROWNIAN MOTIONS WITH APPLICATION TO FINANCE  

Chung, Dong-Myung (Department of Mathematics Sogang University)
Lee, Jeong-Hyun (Courant Institute of Mathematical Sciences New York University)
Publication Information
Journal of the Korean Mathematical Society / v.43, no.2, 2006 , pp. 357-371 More about this Journal
Abstract
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 $L^2({\mu})$ denote the Hilbert space of square integrable functionals of $X\;=\;(X_t - a(t),\; t {in} [0, T])$. In this paper we consider a class of nonlinear functionals of X of the form F(. + a) with $F{in}L^2({\mu})$ 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(.+ a) is in $L^2({\mu})$ 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;
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