Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Tudor, Constantin

doi:10.1081/sap-200026448

Cited by 5 publications

(7 citation statements)

References 17 publications

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“…The next result establishes the existence of the solution u = {u t,x ; (t, x) ∈ R + × R d }, as a collection of random variables in L 2 (Ω). As in [18] (see also [6], [21], [25], [30], [32], [35]), one can find a closed formula for the kernels f n (·, t, x) appearing in the Wiener chaos expansion (6) of u t,x .…”

Section: Existence Of the Solutionmentioning

confidence: 98%

“…(We used (35) and the fact that W (ϕ + ψ) = W (ϕ) + W (ψ) a.s. for any ϕ, ψ ∈ HP, which can be checked in L 2 (Ω), using the fact that W is an isometry between HP and L 2 (Ω).) Relation (39) follows, since E[(u ε,δ t,…”

Section: Hpmentioning

confidence: 99%

“…where C([0, ∞), R d ) denotes the space of continuous functions x : [0, ∞) → R d , and we used (35) for the last equality. Hence E…”

Section: Relationship With the Local Timementioning

confidence: 99%

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Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Balan

Tudor²

2009

J Theor Probab

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International audienceWe consider the stochastic heat equation with multiplicative noise $u_t=\frac{1}{2}\Delta u+ u \diamond \dot{W}$ in $\bR_{+} \times \bR^d$, where $\diamond$ denotes the Wick product, and the solution is interpreted in the mild sense. The noise $\dot W$ is fractional in time (with Hurst index $H \geq 1/2$), and colored in space (with spatial covariance kernel $f$). We prove that if $f$ is the Riesz kernel of order $\alpha$, or the Bessel kernel of order $\alpha1/2$), respectively $d<2+\alpha$ (if $H=1/2$), whereas if $f$ is the heat kernel or the Poisson kernel, then the equation has a solution for any $d$. We give a representation of the $k$-th order moment of the solution, in terms of an exponential moment of the ``convoluted weighted'' intersection local time of $k$ independent $d$-dimensional Brownian motions

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Section: Existence Of the Solutionmentioning

confidence: 98%

Section: Hpmentioning

confidence: 99%

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Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Balan

Tudor²

2009

J Theor Probab

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“…Remark 2.3. There has been some study on fractional SDEs (see, for example, [27], [24], and the references therein). However, we could not find any result on the type of equation (2.19) in the literature.…”

Section: Sdes With Fbmmentioning

confidence: 99%

Stochastic Control for Linear Systems Driven by Fractional Noises

Hu¹,

Zhou²

2005

SIAM J. Control Optim.

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This paper is concerned with optimal control of stochastic linear systems involving fractional Brownian motion (FBM). First, as a prerequisite for studying the underlying control problems, some new results on stochastic integrals and stochastic differential equations associated with FBM are established. Then, three control models are formulated and studied. In the first two models, the state is scalar-valued and the control is taken as Markovian. Either the problems are completely solved based on a Riccati equation (for model 1, where the cost is a quadratic functional on state and control variables) or optimality is characterized (for model 2, where the cost is a power functional). The last control model under investigation is a general one, where the system involves the Stratonovich integral with respect to FBM, the state is multidimensional, and the admissible controls are not limited to being Markovian. A new Riccati-type equation, which is a backward stochastic differential equation involving both FBM and normal Brownian motion, is introduced. Optimal control and optimal value of the model are explicitly obtained based on the solution to this Riccati-type equation.

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“…However, similar to the Brownian case, one of the main difficulties for the Skorokhod-type SDEs is that the traditional Picard iteration is no longer effective and consequently the problem becomes rather subtle when the coefficients are nonlinear and/or random. Several extended Skorokhod integrals have been defined to circumvent such difficulties, with which some special forms of SDEs have been studied (see, for example, [10,14,19]). However, in most of the existing literature, the diffusion coefficient σ has to be very carefully specified so that the subtle restrictions on the stochastic integrals are satisfied.…”

Section: Introductionmentioning

confidence: 99%

Stochastic differential equations driven by fractional Brownian motions

Jien¹,

Ma²

2009

Bernoulli

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In this paper, we study the existence and uniqueness of a class of stochastic differential equations driven by fractional Brownian motions with arbitrary Hurst parameter $H\in (0,1)$. In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wiener spaces, and the coefficients are allowed to be random and even anticipating. The main technique used in this work is an adaptation of the anticipating Girsanov transformation of Buckdahn [Mem. Amer. Math. Soc. 111 (1994)] for the Brownian motion case. By extending a fundamental theorem of Kusuoka [J. Fac. Sci. Univ. Tokyo Sect. IA Math. 29 (1982) 567--597] using fractional calculus, we are able to prove that the anticipating Girsanov transformation holds for the fractional Brownian motion case as well. We then use this result to prove the well-posedness of the SDE.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ169 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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Fractional Bilinear Stochastic Equations with the Drift in the First Fractional Chaos

Cited by 5 publications

References 17 publications

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

Stochastic Control for Linear Systems Driven by Fractional Noises

Stochastic differential equations driven by fractional Brownian motions

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