Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (2) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density.In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

Smooth first-passage densities for one-dimensional diffusions

1987 ◽  
Vol 24 (02) ◽  
pp. 370-377 ◽  
Author(s):  
E. J. Pauwels
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Time Scale ◽  
First Passage ◽  
One Dimensional ◽  
Drift Coefficient

The purpose of this paper is to show that smoothness conditions on the diffusion and drift coefficient of a one-dimensional stochastic differential equation imply the existence and smoothness of a first-passage density. In order to be able to prove this, we shall show that Brownian motion conditioned to first hit a point at a specified time has the same distribution as a Bessel (3)-process with changed time scale.

Stochastic variational inequality and reflected BSDEs with jumps and RCLL obstacle

2017 ◽  
Vol 0 (0) ◽  
Author(s):  
Abou Sene ◽  
Aboubakary Diakhaby
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Variational Inequality ◽  
Stochastic Differential Equation ◽  
Point Process ◽  
Poisson Point Process ◽  
Backward Stochastic Differential Equation ◽  
One Dimensional ◽  
Stochastic Variational Inequality ◽  
Reflected Bsdes

AbstractIn this paper, we consider a class of one-dimensional reflected Backward Stochastic Differential Equation (BSDE for short) when the noise is driven by a Brownian motion and an independent Poisson point process. Using a stochastic variational inequality, we characterize its solution.

A Stochastic Differential Game with Safe and Risky Choices

1988 ◽  
Vol 2 (1) ◽  
pp. 31-39
Author(s):  
J. M. McNamara
Keyword(s):  
Differential Equation ◽  
Diffusion Coefficient ◽  
Stochastic Differential Equation ◽  
Differential Game ◽  
The State ◽  
Stochastic Differential Game ◽  
Final Time ◽  
One Dimensional ◽  
Zero Sum ◽  
Drift Coefficient

This paper considers a two-person zero-sum stochastic differential game. The dynamics of the game are given by a one-dimensional stochastic differential equation whose diffusion coefficient may be controlled by the players. The drift coefficient is held constant and cannot be controlled. Player l's objective is to maximize the probability that the state at final time, T, is positive, while Player 2's objective is to maximize the probability that the state is negative.

scholarly journals Generalized BSDE driven by a Lévy process

2006 ◽  
Vol 2006 ◽  
pp. 1-25 ◽  
Author(s):  
Mohamed El Otmani
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Existence And Uniqueness ◽  
Backward Stochastic Differential Equation ◽  
Existence Of A Solution ◽  
One Dimensional ◽  
Uniqueness Of The Solution ◽  
Independent Brownian Motion ◽  
Teugels Martingales

We study the solution of one-dimensional generalized backward stochastic differential equation driven by Teugels martingales and an independent Brownian motion. We prove existence and uniqueness of the solution when the coefficient verifies some conditions of Lipschitz. If the coefficient is left continuous, increasing, and bounded, we prove the existence of a solution.

scholarly journals Numerical Implementation of Stochastic Operational Matrix Driven by a Fractional Brownian Motion for Solving a Stochastic Differential Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
R. Ezzati ◽  
M. Khodabin ◽  
Z. Sadati
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Fractional Brownian Motion ◽  
Operational Matrix ◽  
Standard Brownian Motion ◽  
One Dimensional ◽  
Block Pulse Functions ◽  
Stochastic Operational Matrix ◽  
Stochastic Linear System

An efficient method to determine a numerical solution of a stochastic differential equation (SDE) driven by fractional Brownian motion (FBM) with Hurst parameterH∈(1/2,1)andnindependent one-dimensional standard Brownian motion (SBM) is proposed. The method is stated via a stochastic operational matrix based on the block pulse functions (BPFs). With using this approach, the SDE is reduced to a stochastic linear system ofmequations andmunknowns. Then, the error analysis is demonstrated by some theorems and defnitions. Finally, the numerical examples demonstrate applicability and accuracy of this method.

On Approximate Large Deviations for 1D Diffusion

2003 ◽  
Vol 10 (2) ◽  
pp. 381-399
Author(s):  
A. Yu. Veretennikov
Keyword(s):  
Differential Equation ◽  
Exact Solution ◽  
Large Deviations ◽  
Stochastic Differential Equation ◽  
Approximation Scheme ◽  
Sufficient Conditions ◽  
Rate Function ◽  
Euler Approximation ◽  
One Dimensional

Abstract We establish sufficient conditions under which the rate function for the Euler approximation scheme for a solution of a one-dimensional stochastic differential equation on the torus is close to that for an exact solution of this equation.

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