Asymptotic expansions for SDE’s with small multiplicative noise

Stochastic Newton equation in strong potential limit

“…Then for any u ∈ (0, s 0 ∧ ξ ), we have by (A.5) that 1 2 λ −1 |v u | 2 ≥ 2bg(r 2 )u + η ≥ η/2, hence |v u | ≥ η 1/2 λ 1/2 for any u ∈ (0, s 0 ∧ ξ ). In particular, since v · is continuous, this implies that v u has the same sign for all u ∈ (0, s 0 ∧ ξ ).…”

“…This type of problem -add some perturbation given by Brownian motion to a Newton mechanical system, and consider the limit of the solution -has been studied by many authors, for example, Kesten-Papanicolaou [4] considered the limit of the distribution of the solution, when the force converges to 0, and Albeverio-Smii [1] considered the asymptotic expansion of the solution. However, in the literature, to the best knowledge of the author, there are not so many papers concerning with our problem of taking the potential to infinity.…”

Small Noise Expansion for the l'EVY Perturbed Vasicek Model

“…It has been shown in [1,3] that under hypothesis 2.1 on µ ǫ and σ ǫ , a solution X ǫ t of equation (1) can be represented as a power series with respect to the parameter ǫ, namely…”

“…Our analysis is based on [3], in the setting proposed in [9,Sec.6.2]. Let us underline that during recent years a wide range of small noise expansion techniques have been developed, particularly with respect to the so called Loval Volatility Models (LVMs), see, e.g., [4,8,11,13].…”

“…The paper is organized as follows: in Sect. 2 the we present the approach developed in [3] is presented and then applied, in Sect. 3, to provide order corrections to both the Vasicek model and its zero coupon bond price.…”