The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

The Laplace Transform of Hitting Times of Integrated Geometric Brownian Motion

In this note we compute the Laplace transform of hitting times, to fixed levels, of integrated geometric Brownian motion. The transform is expressed in terms of the gamma and confluent hypergeometric functions. Using a simple Itô transformation and standard results on hitting times of diffusion processes, the transform is characterized as the solution to a linear second-order ordinary differential equation which, modulo a change of variables, is equivalent to Kummer's equation.

Sur les sous-groupes planaires des groupes des dispersions des équations difiérentielles linéaires du deuxième ordre

SynopsisA group consisting of real continuous functions of one real variable on the interval j = (−∞, ∞) is called planar if through each point of the plane j × j there passes just one element s ∈ .Every differential oscillatory equation (Q): y″ = Q(t)y (t ∈ j = (−∞, ∞), Q ∈ C(0)) admits functions, called the dispersions of (Q), that transform (Q) into itself. These dispersions are integrals of Kummer's equation (QQ): −{X, t} + Q(X)X′2(t) = Q(t) and form a three-parameter group , known as the dispersion group of (Q). The increasing dispersions of (Q) form a three-parameter group invariant in . The centre of the group is an infinite cyclic group , whose elements, the central dispersions of (Q), describe the position of conjugate points of (Q).The present paper contains new results concerning the algebraic structure of the group . It provides information on the following: (1) the existence and properties of planar subgroups of a given group and (2) the existence and properties of the groups containing a given planar group . The results obtained are: the planar subgroups of a given group form a system depending on two constants, SQ, such that for all ∈SQ. The equations (Q) whose groups contain the given planar group form a system dependent on one constant, QS, such that for all (Q)∈QS.

Three-Dimensional, Steady-State Heat Conduction in Cylinders of General Anisotropic-Media

This paper provides the analytical solution of three-dimensional steady-state heat conduction in solid and hollow cylinders of general anisotropic-media. By the use of Fourier transforms and a change of variables the partial differential equation is reduced to Kummer’s equation. Some calculated results for a solid cylinder are shown and discussed. A parameter γ is found to represent the coupling effect of three-dimensional anisotropy. For small values of γ, an approximate solution is recommended. The inequality σ > 0 which was found in an earlier paper is further discussed.

Derivation of Green-type, transitional and uniform asymptotic expansions from differential equations III. The confluent hypergeometric function U ( a , c , z ) for large | c |

The methods introduced by Jorna (1964 a , b ) are applied to Kummer’s equation, and Green-type, transitional and uniform expansions derived for solutions of the type denoted in Slater (1960) by U ( a , c , z ) which are valid for large | c |. The subsidiary function in the uniform expansion is essentially a parabolic cylinder function of order ½ a . The general exponential integral is studied as a special case. Here the uniform expansion involves as subsidiary function the extensively tabulated error integral.