Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q -- L $ρ$ Drift Coefficient and Additive Noise

Jourdain, Benjamin; Menozzi, Stéphane

doi:10.48550/arxiv.2105.04860

Cited by 3 publications

(5 citation statements)

References 38 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…From here, one can deduce estimate (5.12) for discrete times 𝑠, 𝑡 ∈ 𝐷 𝑛 . Gaussian upper bounds for the kernel density of 𝑄 𝑛 𝑠,𝑡 , which are valid for all times 𝑡 > 𝑠, 𝑠 ∈ 𝐷 𝑛 , are also established in [BHZ20,JM21] under Lipschitz regularity of 𝑎. Some related estimates are also obtained in [GK96,GK21] under the additional condition 𝛼 > 𝑑/𝑝.…”

mentioning

confidence: 85%

“…Remark 2.5. Similar truncated vector elds to (1.5) with the values 𝜒 = 1/2 and 𝜒 = 𝑑/(2𝑝) +1/𝑞 were considered in [JM21], in which a weak rate of convergence of order 1 2 − 𝑑 2𝑝 − 1 𝑞 was obtained. When 𝜒 1/2, we choose 𝜌 = 2 which ensures that (1/𝑛) 𝜒 (𝜌−1) (1/𝑛) 1/2 and hence, Corollary 2.4(b) yields the strong rate…”

Section: 𝑛 𝜒 𝑑mentioning

confidence: 99%

“…At the moment of writing, we are aware of two publications on the topic. Jourdain and Menozzi consider in [JM21] the case 𝜎 is the identity matrix and show that the tamed Euler-Maruyama scheme with truncated drifts converges in probability laws at the rate 1 2 − 𝑑 2𝑝 − 1 𝑞 . Gyöngy and Krylov in [GK21] recently show that the tamed Euler-Maruyama scheme with truncated drifts converges in probability to the exact solution, albeit without any rate.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Taming singular stochastic differential equations: A numerical method

Lê¹,

Ling²

2021

Preprint

View full text Add to dashboard Cite

A. We consider a generic and explicit tamed Euler-Maruyama scheme for multidimensional time-inhomogeneous stochastic di erential equations with multiplicative Brownian noise. The di usive coe cient is uniformly elliptic, Hölder continuous and weakly di erentiable in the spatial variables while the drift satis es the Ladyzhenskaya-Prodi-Serrin condition, as considered by Krylov and Röckner (2005). In the discrete scheme, the drift is tamed by replacing it by an approximation. A strong rate of convergence of the scheme is provided in terms of the approximation error of the drift in a suitable and possibly very weak topology. A few examples of approximating drifts are discussed in detail. The parameters of the approximating drifts can vary and-under suitable conditions-be ne-tuned to achieve the standard 1/2-strong convergence rate with a logarithmic factor.

show abstract

mentioning

confidence: 85%

Section: 𝑛 𝜒 𝑑mentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Taming singular stochastic differential equations: A numerical method

Lê¹,

Ling²

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…• Compute estimates on the density of (X m t ) which are uniform in m, using a Duhamel expansion and a normalization method first introduced by [MPZ21] (Brownian setting with unbounded Hölder drift) and then exploited in [JM21] (Brownian setting with L q − L p drift) and [MZ22] (unbounded drift, stable driven with multiplicative isotropic noise).…”

Section: Main Theoremmentioning

confidence: 99%

Heat kernel estimates for stable-driven SDEs with distributional drift

Fitoussi¹

2023

Preprint

View full text Add to dashboard Cite

We consider the formal SDE) is a time-inhomogeneous Besov drift and Z t is a symmetric d-dimensional α-stable process, α ∈ (1, 2), whose spectral measure is absolutely continuous w.r.t. the Lebesgue measure on the sphere. Above, L r and B β p,q respectively denote Lebesgue and Besov spaces. We show that, when β >, the martingale solution associated with the formal generator of (E) admits a density which enjoys two-sided heat kernel bounds as well as gradient estimates w.r.t. the backward variable. Our proof relies on a suitable mollification of the singular drift aimed at using Duhamel expansion. We then use a normalization method combined with Besov space properties (thermic characterization, duality and product rules) to derive estimates.

show abstract

“…Another question apart from well-posedness of these singular McKean-Vlasov equations is how to solve them numerically (in a certain sense). Let us recall that even for standard SDEs with singular or irregular drift, where existence/uniqueness is known for quite some time, the convergence of the corresponding Euler scheme with non-vanishing rate has been established only very recently [BDG19,JM21]. The situation with the singular McKean-Vlasov equations presented above is much more complicated and very few results are available in the literature.…”

Section: Introductionmentioning

confidence: 99%

RKHS regularization of singular local stochastic volatility McKean-Vlasov models

Bayer¹,

Belomestny²,

Butkovsky³

et al. 2022

Preprint

View full text Add to dashboard Cite

Motivated by the challenges related to the calibration of financial models, we consider the problem of solving numerically a singular McKean-Vlasov equationwhere W is a Brownian motion and v is an adapted diffusion process. This equation can be considered as a singular local stochastic volatility model. Whilst such models are quite popular among practitioners, unfortunately, its well-posedness has not been fully understood yet and, in general, is possibly not guaranteed at all. We develop a novel regularization approach based on the reproducing kernel Hilbert space (RKHS) technique and show that the regularized model is wellposed. Furthermore, we prove propagation of chaos. We demonstrate numerically that a thus regularized model is able to perfectly replicate option prices due to typical local volatility models. Our results are also applicable to more general McKean-Vlasov equations.

show abstract

Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q -- L $ρ$ Drift Coefficient and Additive Noise

Cited by 3 publications

References 38 publications

Taming singular stochastic differential equations: A numerical method

Taming singular stochastic differential equations: A numerical method

Heat kernel estimates for stable-driven SDEs with distributional drift

RKHS regularization of singular local stochastic volatility McKean-Vlasov models

Contact Info

Product

Resources

About