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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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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.
On the Browniandirected polymer in a Gaussian random environment
 J. Funct. Anal
, 2005
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and
, 2011
"... Abstract: Any negative moment of an increasing Lamperti process (Yt,t ≥ 0) is a completely monotone function of t. This property enticed us to study systematically, for a given Markov process (Yt,t ≥ 0), the functions f such that the expectation of f(Yt) is a completely monotone function of t. We ca ..."
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Abstract: Any negative moment of an increasing Lamperti process (Yt,t ≥ 0) is a completely monotone function of t. This property enticed us to study systematically, for a given Markov process (Yt,t ≥ 0), the functions f such that the expectation of f(Yt) is a completely monotone function of t. We call these functions temporally completely monotone (for Y). Our description of these functions is deduced from the analysis made by Ben Saad and Janßen, in a general framework, of a dual notion, that of completely excessive measures. Finally, we illustrate our general description in the cases when Y is a Lévy process, a Bessel process, or an increasing Lamperti process.
Martingale Property and Pricing for Timehomogeneous Diffusion Models in Finance
"... I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ..."
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I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public.
unknown title
, 2004
"... In a recent paper, Salminen and Yor (2004b) relate the distribution of the Dufresne’s reflected perpetuity I + � +∞ ..."
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In a recent paper, Salminen and Yor (2004b) relate the distribution of the Dufresne’s reflected perpetuity I + � +∞
Some martingales associated to reflected Lévy processes
, 2003
"... We introduce and describe several classes of martingales based on reflected Levy processes. We show how these martingales apply to various problems, in particular in fluctuation theory as an alternative to the use of excursion methods. Emphasis is given to the case of spectrally negative processes. ..."
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We introduce and describe several classes of martingales based on reflected Levy processes. We show how these martingales apply to various problems, in particular in fluctuation theory as an alternative to the use of excursion methods. Emphasis is given to the case of spectrally negative processes. 1
Perpetual integral functionals of diffusions and their numerical computations
"... Summary. In this paper we study perpetual integral functionals of diffusions. Our interest is focused on cases where such functionals can be expressed as first hitting times for some other diffusions. In particular, we generalize the result in [24] in which onesided functionals of Brownian motion w ..."
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Summary. In this paper we study perpetual integral functionals of diffusions. Our interest is focused on cases where such functionals can be expressed as first hitting times for some other diffusions. In particular, we generalize the result in [24] in which onesided functionals of Brownian motion with drift are connected with first hitting times of reflecting diffusions. Interpreting perpetual integral functionals as hitting times allows us to compute numerically their distributions by applying numerical algorithms for hitting times. Hereby, we discuss two approaches: • numerical inversion of the Laplace transform of the first hitting time, • numerical solution of the PDE associated with the distribution function of the first hitting time. For numerical inversion of Laplace tranforms we have implemented the Euler algorithm developed by Abate and Whitt. However, perpetuities lead often to diffusions for which the explicit forms of the Laplace transforms of first hitting times are not available. In such cases, and also otherwise, algorithms for numerical solutions of PDE’s can be evoked. In particular, we analyze the Kolmogorov PDE of some diffusions appearing in our work via the Crank–Nicolson scheme. AMS Classification: 60J65, 60J60, 62E25. 1
Et: = exp
"... Abstract We obtain the Laplace transform and integrability properties of the integral over R+ of the call quantity associated with geometric Brownian motion with negative drift, thus adding a new element to the list of already studied Brownian perpetuities. Key words BesselMcDonald functions, Sturm ..."
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Abstract We obtain the Laplace transform and integrability properties of the integral over R+ of the call quantity associated with geometric Brownian motion with negative drift, thus adding a new element to the list of already studied Brownian perpetuities. Key words BesselMcDonald functions, SturmLiouville equation, perpetuities.
ELECTRONIC COMMUNICATIONS in PROBABILITY A NOTE ON A.S. FINITENESS OF PERPETUAL IN TEGRAL FUNCTIONALS OF DIFFUSIONS
, 2005
"... In this note we use the boundary classification of diffusions in order to derive a criterion for the convergence of perpetual integral functionals of transient realvalued diffusions. We present a second approach, based on Khas’minskii’s lemma, which is applicable also to spectrally negative Lévy p ..."
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In this note we use the boundary classification of diffusions in order to derive a criterion for the convergence of perpetual integral functionals of transient realvalued diffusions. We present a second approach, based on Khas’minskii’s lemma, which is applicable also to spectrally negative Lévy processes. In the particular case of transient Bessel processes, our criterion agrees with the one obtained via Jeulin’s convergence lemma. 1