Estimating General Diffusion Type Process Driven by Fractional Brownian Motion
نویسنده
چکیده
Conventions and Notations (1) The measure space (M;M; d ) is interpreted as having M as the underly set, M the sigma algebra on M , and d the mesaure on M: If M is a metric space then M is taken as the sigma algebra generated by all its metric-open sets. (2) For two measure spaces (M;M; d ) ; (M1;M1; d 1) and (M M1;M M1; d d 1) is their product measure space. All measurs in the paper are considered complete. (3) Let the triple ( ;F ; P ) be a proability space and (Rn;B (Rn) ; dm) the Euclidean-metric mesure space where dm is the Lebesgue measure on Rn. Let (S;U ; d ) be theire production space, that is S = [0;1) ; U = B ([0;1)) F ; d = dm dP . (4) A ltration of a measure space (M;M; d ) is a non-decreasing family of sub-sigma algebras fM : 2 Kg of M where K is some index set. Filtrations in this paper are augmented with d -mull sets ofM . (5) As a basic assumption, all stochastic process are Rn-valued, measurable random functions fX (t; !)g from U =(S;U ; d ) to E = (Rn;B (Rn) ; dm) and are adapted to the ltration fFt : t 2 [0;1)g of ( ;F ; P ) : The ltration fFt : t 2 [0;1)g is augmented with P null sets of (to avoid measurability when passing to a limit). (6) Conventions and Notations (7) The measure space (M;M; d ) is interpreted as having M as the underly set, M the sigma algebra on M , and d the mesaure on M: If M is a metric space then M is taken as the sigma algebra generated by all its metric-open sets. (8) For two measure spaces (M;M; d ) ; (M1;M1; d 1) and (M M1;M M1; d d 1) is their product measure space. All measurs in the paper are considered complete. Date : Feburary 26, 2009. 1991 Mathematics Subject Classi cation. Primary 05C38, 15A15; Secondary 05A15, 15A18.
منابع مشابه
Existence and Measurability of the Solution of the Stochastic Differential Equations Driven by Fractional Brownian Motion
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