The Rate of Convergence for the Smoluchowski-Kramers Approximation for Stochastic Differential Equations with FBM
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The Rate of Convergence for the Smoluchowski-Kramers Approximation for Stochastic Differential Equations with FBM Ta Cong Son1 Received: 15 July 2020 / Accepted: 18 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we study the rate inthe Smoluchowski–Kramers approximation for the solution t of the equation X t = x + BtH + 0 b(X s )ds where {BtH , t ∈ [0, T ]} is a fractional Brownian motion with Hurst parameter H ∈ 21 , 1 . Based on the techniques of Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence. Keywords Smoluchowski–Kramers approximation · Fractional Brownian motion · Total variation distance · Malliavin calculus Mathematics Subject Classification 60G22 · 60H07 · 91G30
1 Introduction Let us start with a physical motivation, the motion of a mass μ in the field b(X ) + σ (X ) B˙ with the friction proportional to the velocity is defined by the Newton law: μ
μ
μ
μ
μ
μ
μ X¨ t = b(t, X t ) + σ (t, X t ) B˙ t − α X˙ t ; X 0 = x0 , X˙ 0 = y0 .
(1.1)
Here b(X ) is the deterministic part of the force, σ (X ) is the intensity of the noise, the term B˙ t μ is the standard Gaussian white noise in R. The term α X˙ t describes the resistance (friction) to the motion. First, we assume that the friction coefficient α is a fixed positive constant. Then, μ without loss of generality, one can put α = 1. It is well known that, for 0 < μ 0
lim P
μ0
μ max |X t − X t | > δ = 0.
0≤t≤T
(1.3)
for any 0 < T < ∞ and δ > 0. Statement (1.3) is called Smoluchowski-Kramers approxμ imation of X t by X t . This statement justifies the description of the motion of a small particle by the first order equation (1.2) instead of the second order equation (1.1) (see to this purpose [7,11,13]). Because of its applications, the Smoluchowski-Kramers approximation has been studied intensively by several authors. Among others, we mention [3–5,7,8] for Smoluchowski-Kramers approximation results of various stochastic equations. Brahim and Ciorian in [2] studied the Smoluchowski-Kramers approximation for stochastic equation driven by fractional Brownian motion. Very recently, Tan and Dung in [15] gave an explicit estimate for the convergence rate in the Kolmogorov distance between the laws of μ X t and X t . Motivated by this observation, the aim of the present paper is to study the rate of convergence in total variation distance for the Smoluchowski–Kramers approximation for stochastic differential equations driven by fractional Brownian motion. The paper is organized as follows. In Sect. 2, we recall some concepts of stochastic calculus with respect to fractional Brownian motion, Malliavin calculus and give a general bound on total variation distance between two random variables. Our main results are then stated and proved in Sect. 3.
2 Preliminaries 2.1 Malliavin Calculus Let us recall some elements of stochastic calculus of variations (for more details see [10]). We suppose that (Bt )t∈[0,T ] is defined on a compl
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