The Complex Brownian Motion as a Weak Limit of Processes Constructed from a Poisson Process

2001 ◽  
pp. 149-158 ◽  
Author(s):  
X. Bardina
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Weak Limit

scholarly journals A four-dimensional random motion at finite speed

2006 ◽  
Vol 43 (4) ◽  
pp. 1107-1118 ◽  
Author(s):  
Alexander D. Kolesnik
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Transition Density ◽  
Random Motion ◽  
Characteristic Functions ◽  
Conditional Distributions ◽  
Continuous Component ◽  
Absolutely Continuous ◽  
Finite Speed ◽  
Uniform Law

We consider the random motion of a particle that moves with constant finite speed in the space ℝ4 and, at Poisson-distributed times, changes its direction with uniform law on the unit four-sphere. For the particle's position, X(t) = (X1(t), X2(t), X3(t), X4(t)), t > 0, we obtain the explicit forms of the conditional characteristic functions and conditional distributions when the number of changes of directions is fixed. From this we derive the explicit probability law, f(x, t), x ∈ ℝ4, t ≥ 0, of X(t). We also show that, under the Kac condition on the speed of the motion and the intensity of the switching Poisson process, the density, p(x,t), of the absolutely continuous component of f(x,t) tends to the transition density of the four-dimensional Brownian motion with zero drift and infinitesimal variance σ2 = ½.

Optimal Control of a Model for a System Subject to Continuous Wear

1988 ◽  
Vol 2 (3) ◽  
pp. 321-328 ◽  
Author(s):  
Laurence A. Baxter ◽  
Eui Yong Lee
Keyword(s):  
Optimal Control ◽  
Brownian Motion ◽  
Poisson Process ◽  
Average Cost ◽  
Infinite Horizon ◽  
Arrival Rate ◽  
The State ◽  
Negative Drift ◽  
Random Amount

The state of a system is modelled by Brownian motion with negative drift and an absorbing barrier at the origin. A repairman arrives according to a Poisson process of rate λ. If the state of the system at arrival of the repairman does not exceed a certain threshold, he/she increases it by a random amount, otherwise no action is taken. Costs are assigned to each visit of the repairman, to each repair, and to the system being in state 0. It is shown that there exists a unique arrival rate λ which minimizes the average cost per unit time over an infinite horizon.

Weak limit results for the extremes of a class of shot noise processes

1995 ◽  
Vol 32 (03) ◽  
pp. 707-726 ◽  
Author(s):  
Patrick Homble ◽  
William P. McCormick
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Impulse Response Function ◽  
Weak Limit ◽  
Intensity Function ◽  
Important Class ◽  
Homogeneous Poisson Process ◽  
Periodic Intensity ◽  
Non Homogeneous Poisson Process ◽  
Over Time

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

scholarly journals Estimation of the integral of a stochastic process

1978 ◽  
Vol 18 (1) ◽  
pp. 83-93 ◽  
Author(s):  
Noel Cressie
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Poisson Process ◽  
Covariance Structure ◽  
Regression Equation ◽  
Linear Drift ◽  
Independent Increments ◽  
Sampling Points ◽  
Optimum Sampling

Consider the class of stochastic processes with stationary independent increments and finite variances; notable members are brownian motion, and the Poisson process. Now for Xt any member of this class of processes, we wish to find the optimum sampling points of Xt, for predicting . This design question is shown to be directly related to finding sampling points of Yt for estimating β in the regression equation, Yt = β + Xt. Since processes with stationary independent increments have linear drift, the regression equation for Yt is the first type of departure we might look for; namely quadratic drift, and unchanged covariance structure.

Heavy traffic analysis of a queueing system with bounded capacity for two types of customers

1999 ◽  
Vol 36 (4) ◽  
pp. 1155-1166 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Upper Bound ◽  
Queueing System ◽  
Service System ◽  
Heavy Traffic ◽  
Arrival Times ◽  
Brownian Motion With Drift ◽  
Heavy Traffic Analysis ◽  
Capacity Bound

We study a service system with a fixed upper bound for its workload and two independent inflows of customers: frequent ‘small’ ones and occasional ‘large’ ones. The workload process generated by the small customers is modelled by a Brownian motion with drift, while the arrival times of the large customers form a Poisson process and their service times are exponentially distributed. The workload process is reflected at zero and at its upper capacity bound. We derive the stationary distribution of the workload and several related quantities and compute various important characteristics of the system.

The busy cycle of the reflected superposition of Brownian motion and a compound Poisson process

2001 ◽  
Vol 38 (1) ◽  
pp. 255-261 ◽  
Author(s):  
David Perry ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Closed Form ◽  
Queueing System ◽  
Heavy Traffic ◽  
Compound Poisson Process ◽  
Compound Poisson ◽  
Maximum Workload ◽  
Busy Cycle

We consider a reflected superposition of a Brownian motion and a compound Poisson process as a model for the workload process of a queueing system with two types of customers under heavy traffic. The distributions of the duration of a busy cycle and the maximum workload during a cycle are determined in closed form.

Weak approximation for a class of Gaussian processes

2000 ◽  
Vol 37 (02) ◽  
pp. 400-407 ◽  
Author(s):  
Rosario Delgado ◽  
Maria Jolis
Keyword(s):  
Brownian Motion ◽  
Poisson Process ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Stochastic Integral ◽  
Hurst Parameter ◽  
Standard Wiener Process ◽  
Weak Approximation ◽  
The Law ◽  
Continuous Gaussian Process

We prove that, under rather general conditions, the law of a continuous Gaussian process represented by a stochastic integral of a deterministic kernel, with respect to a standard Wiener process, can be weakly approximated by the law of some processes constructed from a standard Poisson process. An example of a Gaussian process to which this result applies is the fractional Brownian motion with any Hurst parameter.

Weak limit results for the extremes of a class of shot noise processes

1995 ◽  
Vol 32 (3) ◽  
pp. 707-726 ◽  
Author(s):  
Patrick Homble ◽  
William P. McCormick
Keyword(s):  
Poisson Process ◽  
Shot Noise ◽  
Impulse Response Function ◽  
Weak Limit ◽  
Intensity Function ◽  
Important Class ◽  
Homogeneous Poisson Process ◽  
Periodic Intensity ◽  
Non Homogeneous Poisson Process ◽  
Over Time

Shot noise processes form an important class of stochastic processes modeling phenomena which occur as shocks to a system and with effects that diminish over time. In this paper we present extreme value results for two cases — a homogeneous Poisson process of shocks and a non-homogeneous Poisson process with periodic intensity function. Shocks occur with a random amplitude having either a gamma or Weibull density and dissipate via a compactly supported impulse response function. This work continues work of Hsing and Teugels (1989) and Doney and O'Brien (1991) to the case of random amplitudes.

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