scholarly journals Another Look at the Hartman-Watson Distributions

2019 ◽  
Vol 53 (4) ◽  
pp. 1269-1297 ◽  
Author(s):  
Jacek Jakubowski ◽  
Maciej Wiśniewolski
Keyword(s):  
Brownian Motion ◽  
Bessel Functions ◽  
Special Function ◽  
Mathematical Finance ◽  
New Approach ◽  
Exponential Functional ◽  
Integral Convolution ◽  
Hyperbolic Cosine ◽  
Parabolic Pde ◽  
Explicit Representations

Abstract The article deals with the Hartman-Watson distributions and presents a new approach to them by investigating a special function u. The function u is strictly related to the distribution of the exponential functional of Brownian motion appearing in the mathematical finance framework. The study of the latter leads to new explicit representations for the function u. One of them is through a new parabolic PDE. Integral relations of convolution type between Hartman-Watson distributions and modified Bessel functions are presented. It turns out that u can be represented as an integral convolution of itself and the modified Bessel function K0. Finally, excursion theory and a subordinator connected to the hyperbolic cosine of Brownian motion are involved in order to obtain yet another representation for u. Possible applications of the resulting explicit formulas are discussed, among others Monte Carlo evaluations of u.

scholarly journals On some exponential functionals of Brownian motion

1992 ◽  
Vol 24 (03) ◽  
pp. 509-531 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bessel Functions ◽  
Exit Time ◽  
Fixed Time ◽  
Mathematical Finance ◽  
Time Interval ◽  
Bessel Processes ◽  
Fixed Time Interval ◽  
Last Exit Time ◽  
Subordination Result

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

scholarly journals Windings of planar processes, exponential functionals and Asian options

2018 ◽  
Vol 50 (3) ◽  
pp. 726-742 ◽  
Author(s):  
Wissem Jedidi ◽  
Stavros Vakeroudis
Keyword(s):  
Brownian Motion ◽  
Exit Time ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Mathematical Finance ◽  
Asian Options ◽  
Product Representation ◽  
Exponential Functional ◽  
Planar Brownian Motion ◽  
Divisibility Properties

Abstract Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

scholarly journals On some exponential functionals of Brownian motion

1992 ◽  
Vol 24 (3) ◽  
pp. 509-531 ◽  
Author(s):  
Marc Yor
Keyword(s):  
Brownian Motion ◽  
Bessel Functions ◽  
Exit Time ◽  
Fixed Time ◽  
Mathematical Finance ◽  
Time Interval ◽  
Bessel Processes ◽  
Fixed Time Interval ◽  
Last Exit Time ◽  
Subordination Result

In this paper, distributional questions which arise in certain mathematical finance models are studied: the distribution of the integral over a fixed time interval [0, T] of the exponential of Brownian motion with drift is computed explicitly, with the help of computations previously made by the author for Bessel processes. The moments of this integral are obtained independently and take a particularly simple form. A subordination result involving this integral and previously obtained by Bougerol is recovered and related to an important identity for Bessel functions. When the fixed time T is replaced by an independent exponential time, the distribution of the integral is shown to be related to last-exit-time distributions and the fixed time case is recovered by inverting Laplace transforms.

Brownian motion with quadratic killing and some implications

1986 ◽  
Vol 23 (04) ◽  
pp. 893-903 ◽  
Author(s):  
Michael L. Wenocur
Keyword(s):  
Brownian Motion ◽  
Weibull Distribution ◽  
Special Function ◽  
Function Theory ◽  
Killing Rate

Brownian motion subject to a quadratic killing rate and its connection with the Weibull distribution is analyzed. The distribution obtained for the process killing time significantly generalizes the Weibull. The derivation involves the use of the Karhunen–Loève expansion for Brownian motion, special function theory, and the calculus of residues.

Interpolation Methods Applied on Biomolecules and Condensed Matter Brownian Motion

2021 ◽  
Author(s):  
Sanja Aleksic ◽  
Bojana Markovic ◽  
Vojislav V. Mitic ◽  
Dusan Milosevic ◽  
Mimica Milosevic ◽  
...  
Keyword(s):  
Brownian Motion ◽  
Condensed Matter ◽  
Particle Systems ◽  
Fractal Nature ◽  
New Approach ◽  
Motion Data ◽  
Joint Properties ◽  
Physical Particle ◽  
High Level ◽  
Advanced Development

Biophysical and condensed matter systems connection is of great importance nowadays due to the need for a new approach in microelectronic biodevices, biocomputers or biochips advanced development. Considering that the living and nonliving systems’ submicroparticles are identical, we can establish the biunivocally correspondent relation between these two particle systems, as a biomimetic correlation based on Brownian motion fractal nature similarities, as the integrative property. In our research, we used the experimental results of bacterial motion under the influence of energetic impulses, like music, and also some biomolecule motion data. Our goal is to define the relation between biophysical and physical particle systems, by introducing mathematical analytical forms and applying Brownian motion fractal nature characterization and fractal interpolation. This work is an advanced research in the field of new solutions for high-level microelectronic integrations, which include submicrobiosystems like part of even organic microelectronic considerations, together with some physical systems of particles in solid-state solutions as a nonorganic part. Our research is based on Brownian motion minimal joint properties within the integrated biophysical systems in the wholeness of nature.

Integral functionals under the excursion measure

2020 ◽  
Vol 57 (1) ◽  
pp. 137-155
Author(s):  
Maciej Wiśniewolski
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Stochastic Analysis ◽  
New Method ◽  
Theoretic Approach ◽  
Integral Functionals ◽  
New Approach ◽  
Brownian Motion With Drift ◽  
Classical Potential ◽  
Excursion Measure

AbstractA new approach to the problem of finding the distribution of integral functionals under the excursion measure is presented. It is based on the technique of excursion straddling a time, stochastic analysis, and calculus on local time, and it is done for Brownian motion with drift reflecting at 0, and under some additional assumptions for some class of Itó diffusions. The new method is an alternative to the classical potential-theoretic approach and gives new specific formulas for distributions under the excursion measure.

‘Killing Mie Softly’: Analytic Integrals for Complex Resonant States

2020 ◽  
Vol 73 (2) ◽  
pp. 119-139
Author(s):  
R C Mcphedran ◽  
B Stout
Keyword(s):  
Bessel Functions ◽  
Open Systems ◽  
Scattering Problems ◽  
Resonant States ◽  
Complex Modes ◽  
Resonant Modes ◽  
Unit Function ◽  
Explicit Representations ◽  
Quantum Mechanical Scattering ◽  
Mechanical Scattering

Summary We consider integrals of products of Bessel functions and of spherical Bessel functions, combined with a Gaussian factor guaranteeing convergence at infinity. These integrals arise in wave and quantum mechanical scattering problems of open systems containing cylindrical or spherical scatterers, particularly when those problems are considered in the framework of complex resonant modes. Explicit representations are obtained for the integrals, building on those in the 1992 paper by McPhedran, Dawes and Scott. Attention is paid to those sums with a distributive part arising as the Gaussian tends towards the unit function. In this limit, orthogonality and normalisability of complex modes are investigated.

A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order

1951 ◽  
Vol 47 (4) ◽  
pp. 699-712 ◽  
Author(s):  
F. W. J. Olver
Keyword(s):  
Differential Equation ◽  
Bessel Functions ◽  
Numerical Technique ◽  
Third Order ◽  
New Approach ◽  
The Third ◽  
Zeros Of Bessel Functions ◽  
Zeros Of Functions ◽  
Linear Differential ◽  
Image Position

In a recent paper (1) I described a method for the numerical evaluation of zeros of the Bessel functions Jn(z) and Yn(z), which was independent of computed values of these functions. The essence of the method was to regard the zeros ρ of the cylinder functionas a function of t and to solve numerically the third-order non-linear differential equation satisfied by ρ(t). It has since been successfully used to compute ten-decimal values of jn, s, yn, s, the sth positive zeros* of Jn(z), Yn(z) respectively, in the ranges n = 10 (1) 20, s = 1(1) 20. During the course of this work it was realized that the least satisfactory feature of the new method was the time taken for the evaluation of the first three or four zeros in comparison with that required for the higher zeros; the direct numerical technique for integrating the differential equation satisfied by ρ(t) becomes unwieldy for the small zeros and a different technique (described in the same paper) must be employed. It was also apparent that no mere refinement of the existing methods would remove this defect and that a new approach was required if it was to be eliminated. The outcome has been the development of the method to which the first part (§§ 2–6) of this paper is devoted.

Sign in / Sign up

Export Citation Format

Share Document