scholarly journals An explicit representation of the transition densities of the skew Brownian motion with drift and two semipermeable barriers

2016 ◽  
Vol 22 (1) ◽  
pp. 1-23 ◽  
Author(s):  
David Dereudre ◽  
Sara Mazzonetto ◽  
Sylvie Roelly
Keyword(s):  
Brownian Motion ◽  
Sampling Method ◽  
Transition Density ◽  
Explicit Representation ◽  
Rejection Sampling ◽  
Skew Brownian Motion ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Transition Densities ◽  
The One

AbstractIn this paper, we obtain an explicit representation of the transition density of the one-dimensional skew Brownian motion with (a constant drift and) two semipermeable barriers. Moreover, we propose a rejection sampling method to simulate this density in an

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

1992 ◽  
Vol 29 (04) ◽  
pp. 996-1002 ◽  
Author(s):  
R. J. Williams
Keyword(s):  
Brownian Motion ◽  
Asymptotic Variance ◽  
Hitting Time ◽  
Local Times ◽  
Compact Interval ◽  
Reflected Brownian Motion ◽  
Direct Derivation ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Variance Parameters

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

A limit theorem for certain disordered random systems

1994 ◽  
Vol 26 (04) ◽  
pp. 1022-1043 ◽  
Author(s):  
Xinhong Ding
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Limit Theorem ◽  
Random Systems ◽  
Joint Probability ◽  
Dynamical Behaviour ◽  
One Dimensional ◽  
Linear Stochastic Differential Equation ◽  
The One ◽  
Image Position

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in : where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝ p -valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation where P(t, dx dy) is the joint probability law of ξ and X(t).

scholarly journals Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

2015 ◽  
Vol 47 (1) ◽  
pp. 210-230 ◽  
Author(s):  
Hongzhong Zhang
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Time Horizon ◽  
Three Dimensional ◽  
Bessel Processes ◽  
Occupation Times ◽  
Exponential Time ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

scholarly journals Exact sampling of diffusions with a discontinuity in the drift

2016 ◽  
Vol 48 (A) ◽  
pp. 249-259 ◽  
Author(s):  
Omiros Papaspiliopoulos ◽  
Gareth O. Roberts ◽  
Kasia B. Taylor
Keyword(s):  
Brownian Motion ◽  
Local Time ◽  
Rejection Sampling ◽  
Exact Methods ◽  
Exact Sampling ◽  
One Dimensional ◽  
Sample Paths ◽  
Finite Dimensional

AbstractWe introduce exact methods for the simulation of sample paths of one-dimensional diffusions with a discontinuity in the drift function. Our procedures require the simulation of finite-dimensional candidate draws from probability laws related to those of Brownian motion and its local time, and are based on the principle of retrospective rejection sampling. A simple illustration is provided.

Spectral properties of trinomial trees

2007 ◽  
Vol 463 (2083) ◽  
pp. 1681-1696 ◽  
Author(s):  
Aleksandar Mijatović
Keyword(s):  
Brownian Motion ◽  
Random Walk ◽  
Convergence Rate ◽  
Integral Representation ◽  
Spectral Properties ◽  
Transition Density ◽  
Convergence Estimate ◽  
Discrete Probability ◽  
Brownian Motion With Drift ◽  
Probability Kernel

In this paper, we prove that the probability kernel of a random walk on a trinomial tree converges to the density of a Brownian motion with drift at the rate O ( h 4 ), where h is the distance between the nodes of the tree. We also show that this convergence estimate is optimal in which the density of the random walk cannot converge at a faster rate. The proof is based on an application of spectral theory to the transition density of the random walk. This yields an integral representation of the discrete probability kernel that allows us to determine the convergence rate.

On some properties of sticky Brownian motion

2020 ◽  
pp. 2150037
Author(s):  
Haoyan Zhang ◽  
Pingping Jiang
Keyword(s):  
Brownian Motion ◽  
Laplace Transform ◽  
Explicit Expression ◽  
Lebesgue Measure ◽  
Transition Density ◽  
Positive Lebesgue Measure ◽  
First Hitting Time ◽  
Skew Brownian Motion ◽  
Basic Probability ◽  
Random Jump

In this paper, we investigate a generalization of Brownian motion, called sticky skew Brownian motion, which has two interesting characteristics: stickiness and skewness. This kind of processes spends a lot more time at its sticky points so that the time they spend at the sticky points has positive Lebesgue measure. By using time change, we obtain an SDE for the sticky skew Brownian motion. Then, we present the explicit relationship between symmetric local time and occupation time. Some basic probability properties, such as transition density, are studied and we derive the explicit expression of Laplace transform of transition density for the sticky skew Brownian motion. We also consider the first hitting time problems over a constant boundary and a random jump boundary, respectively, and give some corollaries based on the results above.

scholarly journals On the transition densities for reflected diffusions

2005 ◽  
Vol 37 (2) ◽  
pp. 435-460 ◽  
Author(s):  
Vadim Linetsky
Keyword(s):  
Square Root ◽  
Uhlenbeck Process ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Spectral Expansions ◽  
Ornstein Uhlenbeck Process ◽  
Transition Densities ◽  
Sturm Liouville ◽  
Analytical Expressions ◽  
Reflecting Barriers

Diffusion models in economics, finance, queueing, mathematical biology, and electrical engineering often involve reflecting barriers. In this paper, we study the analytical representation of transition densities for reflected one-dimensional diffusions in terms of their associated Sturm-Liouville spectral expansions. In particular, we provide explicit analytical expressions for transition densities of Brownian motion with drift, the Ornstein-Uhlenbeck process, and affine (square-root) diffusion with one or two reflecting barriers. The results are easily implementable on a personal computer and should prove useful in applications.

A limit theorem for certain disordered random systems

1994 ◽  
Vol 26 (4) ◽  
pp. 1022-1043 ◽  
Author(s):  
Xinhong Ding
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Limit Theorem ◽  
Random Systems ◽  
Joint Probability ◽  
Dynamical Behaviour ◽  
One Dimensional ◽  
Linear Stochastic Differential Equation ◽  
The One ◽  
Image Position

Many disordered random systems in applications can be described by N randomly coupled Ito stochastic differential equations in : where is a sequence of independent copies of the one-dimensional Brownian motion W and ( is a sequence of independent copies of the ℝp-valued random vector ξ. We show that under suitable conditions on the functions b, σ, K and Φ the dynamical behaviour of this system in the N → (limit can be described by the non-linear stochastic differential equation where P(t, dx dy) is the joint probability law of ξ and X(t).

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