Numerical approximation of irregular SDEs via Skorokhod embeddings

Ankirchner, Stefan; Kruse, Thomas; Urusov, Mikhail

doi:10.1016/j.jmaa.2016.03.055

Cited by 14 publications

(16 citation statements)

References 27 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[SS13,FH16]) and in numerical analysis (e.g. [GMO15,AKU16] There are two direct extensions of the Skorokhod embedding problem: generalizing the process and generalizing the deterministic initial condition δ 0 to an arbitrary distribution µ 0 . A natural motivation for the latter is the interest of constructing multi-marginal Skorokhod embeddings.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Skorokhod embeddings and Poisson equations

Döring¹,

Gonon²,

Prömel³

et al. 2019

Ann. Appl. Probab.

View full text Add to dashboard Cite

The classical Skorokhod embedding problem for a Brownian motion W asks to find a stopping time τ so that Wτ is distributed according to a prescribed probability distribution µ. Many solutions have been proposed during the past 50 years and applications in different fields emerged. This article deals with a generalized Skorokhod embedding problem (SEP): Let X be a Markov process with initial marginal distribution µ0 and let µ1 be a probability measure. The task is to find a stopping time τ such that Xτ is distributed according to µ1. More precisely, we study the question of deciding if a finite mean solution to the SEP can exist for given µ0, µ1 and the task of giving a solution which is as explicit as possible. If µ0 and µ1 have positive densities h0 and h1 and the generator A of X has a formal adjoint operator A * , then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation A * H = h1 − h0 and give a fairly explicit construction of the stopping time using the solution of the Poisson equation. For the class of Lévy processes we carry out the procedure and extend a result of Bertoin and Le Jan to Lévy processes without local times. ∞ 1 1 |η(u)| du < ∞, 0 if lim inf u→∞ |η(u)| ∈ (0, ∞], D if lim inf u→∞ |η(u)| = 0.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

On Skorokhod embeddings and Poisson equations

Döring¹,

Gonon²,

Prömel³

et al. 2019

Ann. Appl. Probab.

View full text Add to dashboard Cite

show abstract

“…As stated in Corollary 1.4, EMCEL approximations can be constructed for every general diffusion. In particular, they can be used in cases where η is not continuous and where (weak) Euler schemes do not converge (see, e.g., Section 5.4 in [3]). In Sections 7 and 8 we consider further irregular examples.…”

Section: Convergence Of the Weak Euler Schemementioning

confidence: 99%

A functional limit theorem for coin tossing Markov chains

Ankirchner

Kruse

Urusov

2020

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

We prove a functional limit theorem for Markov chains that, in each step, move up or down by a possibly state dependent constant with probability 1/2, respectively. The theorem entails that the law of every one-dimensional regular continuous strong Markov process can be approximated with such Markov chains arbitrarily well. It applies, in particular, to Markov processes that cannot be characterized as solutions to stochastic differential equations. We illustrate the theorem with Markov processes exhibiting sticky features, e.g., sticky Brownian motion and a Brownian motion slowed down on the Cantor set.

show abstract

“…In [2], the authors propose a new algorithm to approximate the laws of the solutions to a class of SDEs with irregular coefficients. The pathwise convergences of numerical methods with constant and adaptive step sizes for some highly non-linear SDEs are studied in [6] and [24], respectively.…”

Section: Motivationmentioning

confidence: 99%

The partially truncated Euler–Maruyama method and its stability and boundedness

Guo

Liu

Mao³

et al. 2017

Applied Numerical Mathematics

View full text Add to dashboard Cite

The partially truncated Euler-Maruyama (EM) method is proposed in this paper for highly nonlinear stochastic differential equations (SDEs). We will not only establish the finite-time strong L r -convergence theory for the partially truncated EM method, but also demonstrate the real benefit of the method by showing that the method can preserve the asymptotic stability and boundedness of the underlying SDEs.

show abstract

Numerical approximation of irregular SDEs via Skorokhod embeddings

Cited by 14 publications

References 27 publications

On Skorokhod embeddings and Poisson equations

On Skorokhod embeddings and Poisson equations

A functional limit theorem for coin tossing Markov chains

The partially truncated Euler–Maruyama method and its stability and boundedness

Contact Info

Product

Resources

About