Results 1  10
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18
On some exponential functionals of Brownian motion
 Adv. Appl. Prob
, 1992
"... Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, expl ..."
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Cited by 97 (10 self)
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Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.
Nondegeneracy of Wiener functionals arising from rough differential equations
 Trans. Amer. Math. Soc
"... Abstract. Malliavin Calculus is about Sobolevtype regularity of functionals on Wiener space, the main example being the Itô map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to (possibly stochastic) differential equations. We combi ..."
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Cited by 15 (5 self)
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Abstract. Malliavin Calculus is about Sobolevtype regularity of functionals on Wiener space, the main example being the Itô map obtained by solving stochastic differential equations. Rough path analysis is about strong regularity of solution to (possibly stochastic) differential equations. We combine arguments of both theories and discuss existence of a density for solutions to stochastic differential equations driven by a general class of nondegenerate Gaussian processes, including processes with sample path regularity worse than Brownian motion. 1.
Absolutely continuous flows generated by Sobolev class vector fields in finite and infinite dimensions
, 1996
"... We prove the existence of the global flow fU t g generated by a vector field A from a Sobolev class W 1;1 (¯) on a finite or infinite dimensional space X with a measure ¯, provided ¯ is sufficiently smooth, and rA and jffi ¯ Aj (where ffi ¯ A is the divergence with respect to ¯) are exponentially ..."
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Cited by 8 (2 self)
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We prove the existence of the global flow fU t g generated by a vector field A from a Sobolev class W 1;1 (¯) on a finite or infinite dimensional space X with a measure ¯, provided ¯ is sufficiently smooth, and rA and jffi ¯ Aj (where ffi ¯ A is the divergence with respect to ¯) are exponentially integrable. In addition, the measure ¯ is shown to be quasiinvariant under fU t g. In the case X = IR n and ¯ = pdx, where p 2 W 1;1 loc (IR n ) is a locally uniformly positive probability density, a sufficient condition is: exp(ckrAk)+ exp(cj(A; rp p )j) 2 L 1 (¯) for all c. In the infinite dimensional case we get analogous results for measures differentiable along sufficiently many directions. Examples of measures which fit our framework, important for applications, are symmetric invariant measures of infinite dimensional diffusions and Gibbs measures. Typically, in both cases such measures are essentially nonGaussian. Our result in infinite dimensions significantly extends pr...
Schied: Rademacher’s theorem on configuration spaces and applications
, 1998
"... Abstract: We consider an L 2Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρLipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular ..."
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Cited by 7 (3 self)
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Abstract: We consider an L 2Wasserstein type distance ρ on the configuration space ΓX over a Riemannian manifold X, and we prove that ρLipschitz functions are contained in a Dirichlet space associated with a measure on ΓX satisfying certain natural assumptions. These assumptions are in particular fulfilled by the classical Poisson measures and by a large class of tempered grandcanonical Gibbs measures with respect to a superstable lower regular pair potential. As an application we prove a criterion in terms of ρ for a set to be exceptional. This result immediately implies, for instance, a quasisure version of the spatial ergodic theorem. We also show that ρ is optimal in the sense that it is the intrinsic metric of our Dirichlet form. 0. Introduction. Let ΓX be the configuration space over a Riemannian manifold X. In this paper, we consider a class of probability measures on ΓX, which in particular contains certain Ruelle type Gibbs measures and mixed Poisson measures. Using a natural ‘nonflat ’ geometric structure of ΓX, recently analyzed in Albeverio, Kondratiev and Röckner (1996a),
Malliavin calculus for fractional delay equations, in "Journal of Theoretical Probability
"... Abstract. In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show ..."
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Cited by 3 (0 self)
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Abstract. In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a Hölder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H> 1/2 has a C ∞density. To this purpose, we use Malliavin calculus based on the Fréchet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm. 1.
Stochastic differential equations driven by fractional Brownian motions
, 2009
"... 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 ∈ (0,1). In particular, the stochastic integrals appearing in the equations are defined in the Skorokhod sense on fractional Wie ..."
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Cited by 1 (0 self)
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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 ∈ (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 wellposedness of the SDE.
Sufficient
, 2006
"... conditions for the invertibility of adapted perturbations of identity on the Wiener space ..."
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conditions for the invertibility of adapted perturbations of identity on the Wiener space
Then
, 2000
"... measurepreserving point transformations on the Wiener space and their ergodicity ..."
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measurepreserving point transformations on the Wiener space and their ergodicity
Then
, 2000
"... measurepreserving point transformations on the Wiener space and their ergodicity ..."
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measurepreserving point transformations on the Wiener space and their ergodicity
PATHWISE DEFINITION OF SECOND ORDER SDES LLUÍS QUERSARDANYONS AND SAMY TINDEL
"... Abstract. In this article, a class of second order differential equations on [0, 1], driven by a γHölder continuous function for any value of γ ∈ (0, 1) and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks to Young integration technique ..."
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Abstract. In this article, a class of second order differential equations on [0, 1], driven by a γHölder continuous function for any value of γ ∈ (0, 1) and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks to Young integration techniques. We then study the differentiability of the solution with respect to the driving process and consider the case where the equation is driven by a fractional Brownian motion, with two aims in mind: show that the solution we have produced coincides with the one which would be obtained with Malliavin calculus tools, and prove that the law of the solution is absolutely continuous with respect to the Lebesgue measure. 1.