A STABILITY THEOREM FOR THE SOLUTIONS OF THE INTEGRAL EQUATION FOR SELF-ADJOINT (SELF-RECIPROCAL) PROBABILITY DENSITIES

1994 ◽  
pp. 263-268
Keyword(s):  
Integral Equation ◽  
Stability Theorem ◽  
Probability Densities

On an integral equation for first-passage-time probability densities

1984 ◽  
Vol 21 (02) ◽  
pp. 302-314 ◽  
Author(s):  
L. M. Ricciardi ◽  
L. Sacerdote ◽  
S. Sato
Keyword(s):  
Integral Equation ◽  
Diffusion Process ◽  
Continuous Time ◽  
First Passage Time ◽  
Passage Time ◽  
Time Function ◽  
Standard Wiener Process ◽  
First Passage ◽  
Probability Densities ◽  
Bounded Derivative

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

On an integral equation for first-passage-time probability densities

1984 ◽  
Vol 21 (2) ◽  
pp. 302-314 ◽  
Author(s):  
L. M. Ricciardi ◽  
L. Sacerdote ◽  
S. Sato
Keyword(s):  
Integral Equation ◽  
Diffusion Process ◽  
Continuous Time ◽  
First Passage Time ◽  
Passage Time ◽  
Time Function ◽  
Standard Wiener Process ◽  
First Passage ◽  
Probability Densities ◽  
Bounded Derivative

We prove that for a diffusion process the first-passage-time p.d.f. through a continuous-time function with bounded derivative satisfies a Volterra integral equation of the second kind whose kernel and right-hand term are probability currents. For the case of the standard Wiener process this equation is solved in closed form not only for the class of boundaries already introduced by Park and Paranjape [15] but also for all boundaries of the type S(I) = a + bt ‘/p (p ∼ 2, a, b E ∼) for which no explicit analytical results have previously been available.

A new integral equation for the evaluation of first-passage-time probability densities

1987 ◽  
Vol 19 (04) ◽  
pp. 784-800 ◽  
Author(s):  
A. Buonocore ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Numerical Procedure ◽  
Passage Time ◽  
Continuous Functions ◽  
Time Dependent ◽  
First Passage ◽  
One Dimensional ◽  
Probability Densities

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

A note on the Volterra integral equation for the first-passage-time probability density

1995 ◽  
Vol 32 (03) ◽  
pp. 635-648 ◽  
Author(s):  
R. Gutiérrez Jáimez ◽  
P. Román Román ◽  
F. Torres Ruiz
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Volterra Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Closed Form Solutions ◽  
Actual Use ◽  
Probability Densities

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

A new integral equation for the evaluation of first-passage-time probability densities

1987 ◽  
Vol 19 (4) ◽  
pp. 784-800 ◽  
Author(s):  
A. Buonocore ◽  
A. G. Nobile ◽  
L. M. Ricciardi
Keyword(s):  
Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Numerical Procedure ◽  
Passage Time ◽  
Continuous Functions ◽  
Time Dependent ◽  
First Passage ◽  
One Dimensional ◽  
Probability Densities

The first-passage-time p.d.f. through a time-dependent boundary for one-dimensional diffusion processes is proved to satisfy a new Volterra integral equation of the second kind involving two arbitrary continuous functions. Use of this equation is made to prove that for the Wiener and the Ornstein–Uhlenbeck processes the singularity of the kernel can be removed by a suitable choice of these functions. A simple and efficient numerical procedure for the solution of the integral equation is provided and its convergence is briefly discussed. Use of this equation is finally made to obtain closed-form expressions for first-passage-time p.d.f.'s in the case of various time-dependent boundaries.

Stochastic asymptotic exponential stability of stochastic integral equations

1972 ◽  
Vol 9 (01) ◽  
pp. 169-177
Author(s):  
Chris P. Tsokos ◽  
M. A. Hamdan
Keyword(s):  
Integral Equation ◽  
Exponential Stability ◽  
Stochastic Integral ◽  
Stability Theorem ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Classical Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R +. The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

Stochastic asymptotic exponential stability of stochastic integral equations

1972 ◽  
Vol 9 (1) ◽  
pp. 169-177 ◽  
Author(s):  
Chris P. Tsokos ◽  
M. A. Hamdan
Keyword(s):  
Integral Equation ◽  
Exponential Stability ◽  
Stochastic Integral ◽  
Stability Theorem ◽  
Random Solution ◽  
Stochastic Integral Equation ◽  
Classical Stability ◽  
Stochastic Integral Equations ◽  
Exponentially Stable ◽  
Image Position

The object of this paper is to study the stochastic asymptotic exponential stability of a stochastic integral equation of the form A random solution of the stochastic integral equation is considered to be a second order stochastic process satisfying the equation almost surely. The random solution, y(t, ω) is said to be. stochastically asymptotically exponentially stable if there exist some β > 0 and a γ > 0 such that for t∈ R+.The results of the paper extend the results of Tsokos' generalization of the classical stability theorem of Poincaré-Lyapunov.

A note on the Volterra integral equation for the first-passage-time probability density

1995 ◽  
Vol 32 (3) ◽  
pp. 635-648 ◽  
Author(s):  
R. Gutiérrez Jáimez ◽  
P. Román Román ◽  
F. Torres Ruiz
Keyword(s):  
Integral Equation ◽  
Closed Form ◽  
Volterra Integral Equation ◽  
Diffusion Processes ◽  
First Passage Time ◽  
Passage Time ◽  
First Passage ◽  
Closed Form Solutions ◽  
Actual Use ◽  
Probability Densities

In this paper we prove the validity of the Volterra integral equation for the evaluation of first-passage-time probability densities through varying boundaries, given by Buonocore et al. [1], for the case of diffusion processes not necessarily time-homogeneous. We study, specifically those processes that can be obtained from the Wiener process in the sense of [5]. A study of the kernel of the integral equation, in the same way as that by Buonocore et al. [1], is done. We obtain the boundaries for which closed-form solutions of the integral equation, without having to solve the equation, can be obtained. Finally, a few examples are given to indicate the actual use of our method.

Sign in / Sign up

Export Citation Format

Share Document