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Integral transformations and anticipative calculus for fractional Brownian motions. (English) Zbl 1072.60044

Mem. Am. Math. Soc. 825, 127 p. (2005).
The author gives direct and elementary proofs of two representation formulas for fractional Brownian motion (fBm) and introduces a class of integro-differential transformations \(A_H\) associated with one representation of fBm. The boundedness of \(A_H\) is used to estimate the rate of convergence of the approximation of fBm by polygonal approximation of standard Brownian motion (sBm). This approximation is the best in the sense that it minimizes the mean square error. The author also introduces another class of integro-differential transformations \(\Gamma_{H.T}\) associated with the representation of the fBm. This transformation plays a fundamental role in the definition of stochastic integral, Itô formula, Girsanov type formula, conditioning and so on. The author introduces a probability structure preserving mapping induced by the transformation \(\Gamma_{H.T}\) and defines the stochastic integral for fBm by pulling back to the sBm case. This definition is very general and a broad class of stochastic processes is integrable. The condition for the existence of stochastic integral, Meyer’s inequality, and \(L_p\) estimate of the stochastic integral are obtained by using the idea of probability structure preserving mapping. In particular, one obtains Radon-Nikodym derivative of nonlinear (random) translation of fBm over finite interval. One also obtains an integration by parts formula for general stochastic integral and an Itô type formula for some stochastic integral. The conditioning, Clark derivative and continuity of stochastic integral are also studied. As an application of the stochastic calculus developed in this paper, we solve a stochastic optimal control problem where the utility functional is quadratic and the controlled system is a linear stochastic differential equation driven by a fBm of any Hurst parameter. The optimal control is explicitly obtained by solving a Riccati type equation. The author mentions that some results of this paper may be extended to general Gaussian processes by using the reproducing kernel Hilbert space.

MSC:

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G30 Continuity and singularity of induced measures
60G15 Gaussian processes
26A33 Fractional derivatives and integrals
44A05 General integral transforms
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