Stochastic Integration for Fractional Brownian Motion in a Hilbert Space

Duncan, Tyrone E.; Jakubowski, Jacek; Pasik-Duncan, B.

doi:10.1142/s0219493706001645

Cited by 50 publications

(38 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For this purpose, we generalise the definition of a fractional Brownian motion in R n to Banach spaces. This definition is consistent with others in the literature, in particular the one in [9] for Hilbert spaces. …”

Section: Example 45 For a Set D ∈ B(rsupporting

confidence: 90%

“…For real valued integrands the construction is accomplished for example in [5] and for Hilbert space valued integrands in [9,11,22].…”

Section: Wiener Integrals For Hilbert Space Valued Integrandsmentioning

confidence: 99%

“…For comparison, we apply our methods to an example often considered in the literature and typically formulated in a Hilbert space setting. There are several works in the literature devoted to a similar or related problem, such as Brzeźniak et al [7] for the case of a Banach space, and Grecksch and Anh [13], Duncan and coauthors in a series of papers [9,10,11,12], Tindel et al [28], Maslowski and Nualart [17], Gubinelli et al [14] in the Hilbert space. Among these the papers [7] by Brzeźniak et al and [28] by Tindel et al are the most related to our work, and thus, it might be worth to comment on them in more detail in the sequel: in [7] the authors consider an abstract Cauchy problems in Banach spaces driven by a cylindrical Liouville fBm.…”

Section: Introductionmentioning

confidence: 99%

“…When restricted to the Hilbert space case, our approach is to some extent similar to the one in [9,11,12]. But our method has the advantage of providing not only sufficient, but also necessary conditions for the existence of a solution, which turns out to be both mild and weak.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Cylindrical fractional Brownian motion in Banach spaces

Issoglio

Riedle

2014

Stochastic Processes and their Applications

View full text Add to dashboard Cite

Abstract. In this article we introduce cylindrical fractional Brownian motions in Banach spaces and develop the related stochastic integration theory. Here a cylindrical fractional Brownian motion is understood in the classical framework of cylindrical random variables and cylindrical measures. The developed stochastic integral for deterministic operator valued integrands is based on a series representation of the cylindrical fractional Brownian motion, which is analogous to the Karhunen-Loève expansion for genuine stochastic processes. In the last part we apply our results to study the abstract stochastic Cauchy problem in a Banach space driven by cylindrical fractional Brownian motion.

show abstract

Section: Example 45 For a Set D ∈ B(rsupporting

confidence: 90%

“…For real valued integrands the construction is accomplished for example in [5] and for Hilbert space valued integrands in [9,11,22].…”

Section: Wiener Integrals For Hilbert Space Valued Integrandsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cylindrical fractional Brownian motion in Banach spaces

Issoglio

Riedle

2014

Stochastic Processes and their Applications

View full text Add to dashboard Cite

show abstract

“…This definition uses the methods in [1,6,11,30]. An alternative, equivalent method is given in [10].…”

Section: −1mentioning

confidence: 99%

Semilinear Stochastic Equations in a Hilbert Space with a Fractional Brownian Motion

Duncan¹,

Maslowski²,

Pasik-Duncan³

2009

SIAM J. Math. Anal.

Self Cite

View full text Add to dashboard Cite

Abstract. The solutions of a family of semilinear stochastic equations in a Hilbert space with a fractional Brownian motion are investigated. The nonlinear term in these equations has primarily only a growth condition assumption. An arbitrary member of the family of fractional Brownian motions can be used in these equations. Existence and uniqueness for both weak and mild solutions are obtained for some of these semilinear equations. The weak solutions are obtained by a measure transformation that verifies absolute continuity with respect to the measure for the solution of the associated linear equation. Some examples of stochastic differential and partial differential equations are given that satisfy the assumptions for the solutions of the semilinear equations.Key words. semilinear stochastic equations, fractional Brownian motion, stochastic partial differential equations, absolute continuity of measures AMS subject classifications. 60H15, 60G18, 60G15 DOI. 10.1137/08071764X1. Introduction. Fractional Brownian motion denotes a family of Gaussian processes with continuous sample paths that are indexed by the Hurst parameter H ∈ (0, 1) and that have properties that appear empirically in a wide variety of physical phenomena, such as hydrology, economic data, telecommunications, and medicine. Since some physical phenomena are naturally modeled by stochastic partial differential equations and the randomness can be described by a fractional Gaussian noise, it is important to study the problems of the solutions of stochastic differential equations in a Hilbert space with a fractional Brownian motion. A significant family of these stochastic equations is the set of semilinear equations, so it is important to investigate the existence and the uniqueness of the solutions of the equations and the sample path properties of the solutions. If primarily only some growth assumptions are made on the nonlinear terms in the semilinear equations, then it is natural to investigate weak solutions, especially those that arise by an absolutely continuous transformation of the measure of the solution of the associated linear stochastic equation.The study of the solutions of stochastic equations in an infinite-dimensional space with a (cylindrical) fractional Brownian motion (for example, stochastic partial differential equations) has been relatively limited. For the Hurst parameter H ∈ (1/2, 1), linear and semilinear equations with an additive fractional Gaussian noise, the formal derivative of a fractional Brownian motion, are considered in [8,13,15,28]. Random dynamical systems described by such stochastic equations and their fixed points are studied in [22]. A pathwise (or nonprobabilistic) approach is used in [21] to study a parabolic equation with a fractional Gaussian noise where the stochastic term is a nonlinear function of the solution. Strong solutions of bilinear evolution equations

show abstract

Solutions of linear and semilinear distributed parameter equations with a fractional Brownian motion

Duncan

Maslowski

Pasik-Duncan

2008

Adaptive Control & Signal

Self Cite

View full text Add to dashboard Cite

In this paper, some linear and semilinear distributed parameter equations (equations in a Hilbert space) with a (cylindrical) fractional Brownian motion are considered. Solutions and sample path properties of these solutions are given for the stochastic distributed parameter equations. The fractional Brownian motions are indexed by the Hurst parameter H ∈ (0, 1). For H = 1 2 the process is Brownian motion. Solutions of these linear and semilinear equations are given for each H ∈ (0, 1) with the assumptions differing for the cases H ∈ (0, 1 2 ) and H ∈ ( 1 2 , 1). For the linear equations, the solutions are mild solutions and limiting Gaussian measures are characterized. For the semilinear equations, the solutions are either mild or weak. The weak solutions are obtained by transforming the measure of the associated linear equation by a Radon-Nikodym derivative (likelihood function). An application to identification is given by obtaining a strongly consistent family of estimators for an unknown parameter in a linear equation with distributed noise or boundary noise.

show abstract

Stochastic Integration for Fractional Brownian Motion in a Hilbert Space

Cited by 50 publications

References 17 publications

Cylindrical fractional Brownian motion in Banach spaces

Cylindrical fractional Brownian motion in Banach spaces

Semilinear Stochastic Equations in a Hilbert Space with a Fractional Brownian Motion

Solutions of linear and semilinear distributed parameter equations with a fractional Brownian motion

Contact Info

Product

Resources

About