Approximation Of Brownian Motion Research Articles

Arithmetic modeling of Brownian motion and polyadic approximations

Arithmetic modeling of Brownian motion and polyadic approximations

Linear response theory of the heat of transport of electrolyte solutions: Electrostatic contributions

The electrostatic contribution to the heat of transport of very dilute electrolyte solutions is determined from the linear response theory. The model considered is that of charged hard spheres immersed in a medium of solvent molecules which screen the Coulomb potential between ions by the dielectric constant D. Furthermore, we assume that the fluctuation in phase that determines the momentum of the ion is just the Brownian motion. The Brownian motion approximation simplifies the calculation and enables us to evaluate the time integral. The Brownian motion approximation also effectively decouples the various diffusion fluxes and thus the contributions of the cross coefficients to the heat of transport are not calculated in this treatment. The limiting slope for the heat of transport predicted in an earlier work by Helfand and Kirkwood [J. Chem. Phys. 32, 857 (1960)] agrees with ours in the limit where the hard sphere diameter σ→0.

A Class of Approximations of Brownian Motion

Let B(t) = (B\t)i B(t), • • , B(t)} be a ^-dimensional Brownian motion and let {Bn(i) = (Bn (t),Bn\f),-~,Bn*(i))} be a sequence of approximations to B(t). We assume that the sample paths of Bn(t) are continuous and piecewise smooth for each n and Bn(t) converges to B(t). Let u(x) be a twice continuously differentiate function on R whose partial derivatives of order <^2 are all bounded. In the one-dimensional case E.Wong and M. Zakai [5] showed that I u(Bn(s))dBn(s) conJo verges to I u (B (s)) odB(s) where the symbol o denotes the symmetric Jo stochastic integral of Stratonovich (K. Ito [2]). They also dealt with the convergence of the more general functional of Bn(-)9 ([6]). In the two-dimensional case P. Levy [3] showed that S(t;n)= (B^(s)Jo dBJ(s) — B»(s)dBJ(s))/2 converges to the stochastic integral S(t) = (\B(s)°dff(s) -B(s}°dB(s')}/2 if (Bn(t)} is a sequence of polygonal Jo approximations to B ( f ) . E. J. McShane [4], on the other hand, gave an example of the sequence {Bn(t}} of approximations to B(t) such that S(t;n} converges to S(f)-\-t/n. In this paper we treat systematically a class of approximations of Brownian motion including McShane's example. In Section 2 we state the main results of the paper. We consider a sequence of Stieltjes integrals of the form In(ii) — u(Bn(s))dBn (5). First we will give some Jo conditions under which In(u) converges in the quadratic-mean sense. It

Approximations to the Average Run Length in Cumulative Sum Control Charts

An approximation to the average run length for cumulative sum control charts is derived using the analogy between this procedure and the sequential probability ratio test for normal observations. This approximation is also derived by using a Brownian motion approximation to the cumulative sum. The Brownian motion approximation does not require the normality assumption. The analytical expression for the average run length obtained from the approximation is then used to determine the optimal choice of parameters to minimize the average run length at a specified deviation flom control, subject to a fixed average run length when in control.

Diffusion approximation for multiprogrammed computer systems

This paper presents results of diffusion approximations which may be used in the analysis of multiprogrammed computer systems. First of all we give a mathematical proof for the best CPU utilization under different priority rules in the case of one CPU and more DTU-s. With the help of the Brownian motion approximation we get an explicit result for a model considered by Gaver and Shedler in priority queues, without using the Wald identity. Some examples are given when the diffusion approximation is more complicated than the Brownian motion one.

Chapman-Enskog-Hilbert Expansion for the Ornstein-Uhlenbeck Process and the Approximation of Brownian Motion

Chapman-Enskog-Hilbert Expansion for the Ornstein-Uhlenbeck Process and the Approximation of Brownian Motion

Perturbed Semigroup Limit Theorems with Applications to Discontinuous Random Evolutions

For $\varepsilon > 0$ small, let ${U^\varepsilon }(t)$ and $S(t)$ be strongly continuous semigroups of linear contractions on a Banach space $L$ with infinitesimal operators $A(\varepsilon )$ and $B$ respectively, where $A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o()$ as $\varepsilon \to 0$. Let $\{ B(u);u \geqslant 0\}$ be a family of linear operators on $L$ satisfying $B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2})$ as $\varepsilon \to 0$. Assume that $A(\varepsilon ) + {\varepsilon ^{ - 1}}B()$ is the infinitesimal operator of a strongly continuous contraction semigroup ${T_\varepsilon }(t)$ on $L$ and that for each $f \in L,{\lim _{\lambda \to 0}}\lambda \int _0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf$ exists. We give conditions under which ${T_\varepsilon }(t)$ converges as $\to 0$ to the semigroup generated by the closure of $P({A^{(1)}} + {\Pi ^{(1)}})$ on $\mathcal {R}(P) \cap \mathcal {D}({A^{(1)}}) \cap \mathcal {D}({\Pi ^{(1)}})$. If $P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f$, and we let $\hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h$, then we show that ${T_\varepsilon }(t/\varepsilon )f$ converges as $\varepsilon \to 0$ to the strongly continuous contraction semigroup generated by the closure of ${V^{(2)}} + \hat V$. From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.

Chapman-Enskog-Hilbert expansion for the Ornstein-Uhlenbeck process and the approximation of Brownian motion

Let ( x ( t ) , υ ( t ) ) (x(t),\upsilon (t)) denote the joint Ornstein-Uhlenbeck position-velocity process. Special solutions of the backward equation of this process are studied by a technique used in statistical mechanics. This leads to a new proof of the fact that, as ε ↓ 0 , ε x ( t / ε 2 ) \varepsilon \downarrow 0,\varepsilon x(t/{\varepsilon ^2}) tends weakly to Brownian motion. The same problem is then considered for υ ( t ) \upsilon (t) belonging to a large class of diffusion processes.

Perturbed semigroup limit theorems with applications to discontinuous random evolutions

For ε &gt; 0 \varepsilon &gt; 0 small, let U ε ( t ) {U^\varepsilon }(t) and S ( t ) S(t) be strongly continuous semigroups of linear contractions on a Banach space L L with infinitesimal operators A ( ε ) A(\varepsilon ) and B B respectively, where A ( ε ) = A ( 1 ) + ε A ( 2 ) + o ( ) A(\varepsilon ) = {A^{(1)}} + {\varepsilon A^{(2)}} + o() as ε → 0 \varepsilon \to 0 . Let { B ( u ) ; u ⩾ 0 } \{ B(u);u \geqslant 0\} be a family of linear operators on L L satisfying B ( ε ) = B + ε Π ( 1 ) + ε 2 Π ε ( 2 ) + o ( ε 2 ) B(\varepsilon ) = B + {\varepsilon \Pi ^{(1)}} + {\varepsilon ^2}{\Pi ^{\varepsilon (2)}} + o({\varepsilon ^2}) as ε → 0 \varepsilon \to 0 . Assume that A ( ε ) + ε − 1 B ( ) A(\varepsilon ) + {\varepsilon ^{ - 1}}B() is the infinitesimal operator of a strongly continuous contraction semigroup T ε ( t ) {T_\varepsilon }(t) on L L and that for each f ∈ L , lim λ → 0 λ ∫ 0 ∞ e − λ t S ( t ) f d t ≡ P f f \in L,{\lim _{\lambda \to 0}}\lambda \int _0^\infty {{e^{ - \lambda t}}} S(t)fdt \equiv Pf exists. We give conditions under which T ε ( t ) {T_\varepsilon }(t) converges as → 0 \to 0 to the semigroup generated by the closure of P ( A ( 1 ) + Π ( 1 ) ) P({A^{(1)}} + {\Pi ^{(1)}}) on R ( P ) ∩ D ( A ( 1 ) ) ∩ D ( Π ( 1 ) ) \mathcal {R}(P) \cap \mathcal {D}({A^{(1)}}) \cap \mathcal {D}({\Pi ^{(1)}}) . If P ( A ( 1 ) + Π ( 1 ) ) f = 0 , B h = − ( A ( 1 ) + Π ( 1 ) ) f P({A^{(1)}} + {\Pi ^{(1)}})f = 0,Bh = - ({A^{(1)}} + {\Pi ^{(1)}})f , and we let V ^ f = P ( A ( 1 ) + Π ( 1 ) ) h \hat Vf = P({A^{(1)}} + {\Pi ^{(1)}})h , then we show that T ε ( t / ε ) f {T_\varepsilon }(t/\varepsilon )f converges as ε → 0 \varepsilon \to 0 to the strongly continuous contraction semigroup generated by the closure of V ( 2 ) + V ^ {V^{(2)}} + \hat V . From these results we obtain new limit theorems for discontinuous random evolutions and new characterizations of the limiting infinitesimal operators of the discontinuous random evolutions. We apply these results in a model for the approximation of physical Brownian motion and in a model of the content of an infinite capacity dam.

Nonparametric Testing Under Progressive Censoring*

Summary Progressive censoring schemes (allowing a continuous monitoring of experimentation until a terminal decision is reached) are often adopted in clinical trials and life testing problems. In this paper, a general class of rank order tests for progressive censoring is proposed. Along with a basic martinga:le property and a Brownian motion approximation for a related rank order process, asymptotic distribution theory of the proposed statistics is developed. Asymptotic performance characteristics of the proposed tests (in the light of Bahadur efficiency and the stochastic smallness of the stopping variables) are studied.