On the solution of a one-dimensional stochastic differential equation with singular drift coefficient

2004 ◽  
Vol 56 (5) ◽  
pp. 774-789
Author(s):  
A. M. Kulik
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
One Dimensional ◽  
Singular Drift ◽  
Drift Coefficient

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.

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.

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.

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.

scholarly journals Homogenization for deterministic maps and multiplicative noise

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130201 ◽  
Author(s):  
Georg A. Gottwald ◽  
Ian Melbourne
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Discrete Time ◽  
Continuous Time ◽  
Multiplicative Noise ◽  
Stochastic Integrals ◽  
Deterministic System ◽  
One Dimensional ◽  
Stable Lévy Process ◽  
Fast Flow

A recent paper of Melbourne & Stuart (2011 A note on diffusion limits of chaotic skew product flows. Nonlinearity 24 , 1361–1367 (doi:10.1088/0951-7715/24/4/018)) gives a rigorous proof of convergence of a fast–slow deterministic system to a stochastic differential equation with additive noise. In contrast to other approaches, the assumptions on the fast flow are very mild. In this paper, we extend this result from continuous time to discrete time. Moreover, we show how to deal with one-dimensional multiplicative noise. This raises the issue of how to interpret certain stochastic integrals; it is proved that the integrals are of Stratonovich type for continuous time and neither Stratonovich nor Itô for discrete time. We also provide a rigorous derivation of super-diffusive limits where the stochastic differential equation is driven by a stable Lévy process. In the case of one-dimensional multiplicative noise, the stochastic integrals are of Marcus type both in the discrete and continuous time contexts.

The Smoluchowski–Kramers approximation for the stochastic Liénard equation by mean-field

1991 ◽  
Vol 23 (2) ◽  
pp. 303-316 ◽  
Author(s):  
Kiyomasa Narita
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Diffusion Process ◽  
Langevin Equation ◽  
Mean Field ◽  
Two Dimensional ◽  
Large Parameter ◽  
Rigorous Evaluation ◽  
One Dimensional ◽  
Limit Diffusion

The oscillator of the Liénard type with mean-field containing a large parameter α < 0 is considered. The solution of the two-dimensional stochastic differential equation with mean-field of the McKean type is taken as the response of the oscillator. By a rigorous evaluation of the upper bound of the displacement process depending on the parameter α, a one-dimensional limit diffusion process as α → ∞is derived and identified. Then our result extends the Smoluchowski–Kramers approximation for the Langevin equation without mean-field to the McKean equation with mean-field.

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