zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Integral transformations and anticipative calculus for fractional Brownian motions. (English) Zbl 1072.60044
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 Γ 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 Γ 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 Ricatti 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:
60H05Stochastic integrals
60H07Stochastic calculus of variations and the Malliavin calculus
60G30Continuity and singularity of induced measures (stochastic processes)
60G15Gaussian processes
26A33Fractional derivatives and integrals (real functions)
44A05General integral transforms