Wasserstein convergence rates for random bit approximations of continuous Markov processes

Ankirchner, Stefan; Kruse, Thomas; Urusov, Mikhail

doi:10.1016/j.jmaa.2020.124543

Cited by 5 publications

(13 citation statements)

References 25 publications

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“…-measurable for all pairs (θ, ϑ) ∈ 2 withθ = ϑ. (2) Let M, p > 0 and q > t (with t from ( 1)) be such that E d X (X θ , X ϑ ) p ≤ Md (θ, ϑ) q for θ, ϑ ∈ .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In particular, if d X is complete, then the random process (X θ ) θ∈ has a modification which satisfies (2) such that all its paths are Hölder-continuous of all orders β ∈ ]0, (q − t)/ p[.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…The key assumption on the geometry of the parametric space is (1), where the value of t is important, as we need to have q > t in (3). 1 We remark that, if is a bounded subset of R m with the Euclidean metric d m,2 = d , then (1) is always satisfied with t = m, 2 More generally, a relatively compact subset of an m-dimensional connected Riemannian manifold always satisfies (1) with t = m (we provide more detail in Sect. 3).…”

Section: Remarkmentioning

confidence: 99%

“…4 and 5 require that the expectation is finite. As another example of this kind, we mention that the proof of Theorem 6.1 in [2] would not work without finiteness of such an expectation (see formula (106) in [2]).…”

Section: Remarkmentioning

confidence: 99%

See 3 more Smart Citations

A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes

Krätschmer

Urusov

2022

J Theor Probab

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This paper deals with moduli of continuity for paths of random processes indexed by a general metric space $$\Theta $$ Θ with values in a general metric space $${{\mathcal {X}}}$$ X . Adapting the moment condition on the increments from the classical Kolmogorov–Chentsov theorem, the obtained result on the modulus of continuity allows for Hölder-continuous modifications if the metric space $${{\mathcal {X}}}$$ X is complete. This result is universal in the sense that its applicability depends only on the geometry of the space $$\Theta $$ Θ . In particular, it is always applicable if $$\Theta $$ Θ is a bounded subset of a Euclidean space or a relatively compact subset of a connected Riemannian manifold. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result, a criterion is presented to guarantee uniform tightness of random processes with continuous paths. This is applied to find central limit theorems for Banach-valued random processes.

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“…-measurable for all pairs (θ, ϑ) ∈ 2 withθ = ϑ. (2) Let M, p > 0 and q > t (with t from ( 1)) be such that E d X (X θ , X ϑ ) p ≤ Md (θ, ϑ) q for θ, ϑ ∈ .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

Section: Remarkmentioning

confidence: 99%

See 2 more Smart Citations

A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes

Krätschmer

Urusov

2022

J Theor Probab

Self Cite

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“…Note that for there exists a such that for all it holds that The upper estimate is given in Lemma 5.1. For the lower estimate follows from [2, Proposition 5.3]. We get the lower estimate for by choosing and such that Then it holds by the log-convexity of norms (see for example [35, Lemma 1.11.5]) that Since for it holds that and we have for any that …”

Section: Resultsmentioning

confidence: 99%

Mean square rate of convergence for random walk approximation of forward-backward SDEs

Geiß¹,

Labart

Luoto³

2020

Adv. Appl. Probab.

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Let (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

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Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation

Meier

Zhang

2023

European Journal of Operational Research

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Wasserstein convergence rates for random bit approximations of continuous Markov processes

Cited by 5 publications

References 25 publications

A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes

A Kolmogorov–Chentsov Type Theorem on General Metric Spaces with Applications to Limit Theorems for Banach-Valued Processes

Mean square rate of convergence for random walk approximation of forward-backward SDEs

Simulation of multidimensional diffusions with sticky boundaries via Markov chain approximation

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