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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 (9 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.
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 7 (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...
Change of Measures and Their Logarithmic Derivatives Under Smooth Transformations
, 1996
"... this paper is a new method of producing formulae for transformations of (in general nonGaussian) smooth measures on locally convex spaces under nonlinear smooth transformations of these spaces. The method is essentially based on the results of our preceding paper [5]. A formula is given which invol ..."
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Cited by 2 (2 self)
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this paper is a new method of producing formulae for transformations of (in general nonGaussian) smooth measures on locally convex spaces under nonlinear smooth transformations of these spaces. The method is essentially based on the results of our preceding paper [5]. A formula is given which involves the determinant known from the finite dimensional transformation formula and which for smooth vector fields and the special case of Brownian motion (resp. Gaussian measures) takes the form of the GirsanovMaruyama (resp. Ramer) formula.
ON THE ACTION OF THE GROUP OF DIFFEOMORPHISMS OF A SURFACE ON SECTIONS OF THE DETERMINANT LINE BUNDLE
, 2000
"... Let Σ denote a closed oriented surface. There is a natural action of the group Diff + (Σ) on sections of the chiral determinant line over the space of gauge equivalence classes of connections. The question we address is whether this action is unitarizable. We introduce a SDiffequivariant regulariza ..."
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Cited by 1 (1 self)
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Let Σ denote a closed oriented surface. There is a natural action of the group Diff + (Σ) on sections of the chiral determinant line over the space of gauge equivalence classes of connections. The question we address is whether this action is unitarizable. We introduce a SDiffequivariant regularization, and we prove the existence of, and explicitly compute, the limit as the regularization is removed. The SDiff unitary representations that arise, both by regularization and after removing the regularization, appear to be new. 0. Introduction. Let Σ denote a closed oriented surface, and let D denote the group of orientationpreserving diffeomorphisms of Σ. Let K denote a connected compact Lie group, A the space of Kconnections in the trivial bundle P =(Σ×K→Σ), and C the space of gauge equivalence classes of K
Quasiinvariance for Lévy processes under anticipating shifts
"... We prove a Girsanov theorem for the combination of a Brownian motion on IR + and a Poisson random measure on IR + \Theta [\Gamma1; 1] d under random anticipating transformations of paths and configurations. The factorization of the density function via CarlemanFreholm determinants and divergence ..."
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We prove a Girsanov theorem for the combination of a Brownian motion on IR + and a Poisson random measure on IR + \Theta [\Gamma1; 1] d under random anticipating transformations of paths and configurations. The factorization of the density function via CarlemanFreholm determinants and divergence operators appears as an extension of the martingale factorization in the adapted jump case. Key words: Quasi invariance, L'evy processes, Poisson random measures. Mathematics Subject Classification (1991). Primary: 60B11, 60H07, 60G15, 60G57. Secondary: 28C20, 46G12. 1