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• ### Rates of convergence in a central limit theorem for stochastic processes defined by differential equations with a small parameter

1992

Let μ be a positive finite Borel measure on the real line R. For t ≥ 0 let et · E1 and E2 denote, respectively, the linear spans in L2(R, μ) of {eisx, s > t} and {eisx, s < 0}. Let θ: R → C such that ∥θ∥ = 1, denote by αt(θ, μ) the angle between θ · et · E1 and E2. The problems considered here...

• ### Strong Convergence in the Stochastic Averaging Principle.

1994

In this note we consider the almost sure convergence (as ϵ→0) of solution Xϵ(·), defined over the interval 0 ≤ τ ≤ 1, of the random ordinary differential equation View the MathML source Here {F(x, t, ω), t ≥ 0} is a strong mixing process for each x and (x, t) → F(x, t, ω) is subject to regularity...

• ### A law of the iterated logarithm for stochastic processes defined by differential equations with a small parameter

1994

Consider the following random ordinary differential equation: X˙ϵ(τ)=F(Xϵ(τ),τ/ϵ,ω)subject toXϵ(0)=x0, where {F(x,t,ω),t≥0} are stochastic processes indexed by x in Rd, and the dependence on x is sufficiently regular to ensure that the equation has a unique solution Xϵ(τ,ω) over the interval...

• ### Strong approximation for cross-covariances of linear variables with long-range dependence.

1995

Suppose {εk, −∞ < k < ∞} is an independent, not necessarily identically distributed sequence of random variables, and {cj}∞j=0, {dj}∞j=0 are sequences of real numbers such that Σjc2j < ∞, Σjd2j < ∞. Then, under appropriate moment conditions on {εk, −∞ < k < ∞}, View the MathML source, View the...

• ### Averaging for Fundamental Solutions of Parabolic Equations.

1997

Herein, an averaging theory for the solutions to Cauchy initial value problems of arbitrary order,ε-dependent parabolic partial differential equations is developed. Indeed, by directly developing bounds between the derivatives of the fundamental solution to such an equation and derivatives of the...

• ### Invariance Principles for Parabolic Equations with Random Coefficients

1997

A general Hilbert-space-based stochastic averaging theory is brought forth herein for arbitrary-order parabolic equations with (possibly long range dependent) random coefficients. We use regularity conditions onView the MathML sourcewhich are slightly stronger than those required to prove...

• ### Convergence of Markov chain approximations to stochastic reaction diffusion equations.

2002

In the context of simulating the transport of a chemical or bacterial contaminant through a moving sheet of water, we extend a well-established method of approximating reaction-diffusion equations with Markov chains by allowing convection, certain Poisson measure driving sources and a larger...

• ### Markov chain approximations to filtering equations for reflecting diffusion processes.

2004

Herein, we consider direct Markov chain approximations to the Duncan–Mortensen–Zakai equations for nonlinear filtering problems on regular, bounded domains. For clarity of presentation, we restrict our attention to reflecting diffusion signals with symmetrizable generators. Our Markov chains are...

• ### Rates for branching particle approximations of continuous discrete filters.

2005

Herein, we analyze an efficient branching particle method for asymptotic solutions to a class of continuous-discrete filtering problems. Suppose that t→Xt is a Markov process and we wish to calculate the measure-valued process t→μt(⋅)≐P{Xt∈⋅|σ{Ytk, tk≤t}}, where tk=kɛ and Ytk is a distorted,...

• ### On extending classical filtering equations.

2008

In this paper, we give a direct derivation of the Duncan–Mortensen–Zakai filtering equation, without assuming right continuity of the signal, nor its filtration, and without the usual finite energy condition. As a consequence, the Fujisaki–Kallianpur–Kunita equation is also derived. Our results...

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