Parametrix Methods for One-Dimensional Reflected SDEs

Regularity of the Local Time of Diffusions on the Positive Real Line with Reflection at Zero

“…Tsuchiya [12] obtained, by using parametrix methodology, an approximation for E[f (X t (x))] and investigated the existence of the density of X t (x), see also [2]. The main purpose of the present paper is to generalize the methodology exposed by [12] and to give some approximation formula of the transition semigroup P t f (x, ℓ) := E[f (X t (x), ℓ + L t (x))] under mild regularity conditions on the coefficients of X t (x).…”

“…,λ)∈R2 :|x−ξ|+|ℓ−λ|>R}H 0 (ξ, 2āt)H 0 (λ, 2āt) dξ dλ } Here we also do the change of variable ξ = x−x ′ in the first term and (ξ, λ) = (x− ξ, ℓ−λ) in the second term. Since {ξ ∈ R :|ξ −x|+|ℓ| > R} ⊂ {ξ ∈ R : |ξ| > R/2} for any (x, ℓ) ∈ K,we have ∫ {ξ∈R: |ξ−x|+|ℓ|>R}H 0 (ξ, 2āt) dξ ≤ We have also that for any (x, ℓ) ∈ K {(ξ, λ) ∈ R 2 : |x − ξ| + |ℓ − λ| > R} ⊂ {(ξ, λ) ∈ R 2 : |ξ| + |λ| > R/2} ⊂ {(ξ, λ) ∈ R 2 : |ξ| > R/4} ∪ {(ξ, λ) ∈ R 2 : |λ| > R/4}, and hence ∫ ∫ {(ξ,ℓ ′ )∈R 2 :|x−ξ|+|ℓ−λ|>R}H 0 (ξ, 2āt)H 0 (λ, 2āt) dξ dλ ≤ ∫ {ξ∈R : |ξ|>R/4} H 0 (ξ, 2āt) dξ + ∫ {λ∈R : |λ|>R/4} H 0 (λ, 2āt) dλ ≤ 2 √ T ∫ {ξ∈R: |ξ|>R/8āT }H 0 (ξ, 1) dξ.…”

A generic construction for high order approximation schemes of semigroups using random grids

“…By using a change of measure and a rejection algorithm, Beskos and Roberts [9] have proposed such a method for one-dimensional diffusions. Recently, Bally and Kohatsu-Higa [7] have given a probabilistic representation of the parametrix method that opens the road to construct unbiased estimators for a wide class of Markov processes, including stopped or reflected diffusions [11,4]. By using a different approach, Henry-Labordère et al [13] have lately proposed unbiased estimators for SDEs that present nonetheless a similar structure as the ones obtained with the parametrix method.…”