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Numerical methods for SDEs with singular coefficients - Lecture 2

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Auteurs : Jourdain, Benjamin (Auteur de la conférence)
CIRM (Editeur )

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Résumé : We will consider the discretization of the stochastic differential equation$$X_t=X_0+W_t+\int_0^t b\left(s, X_s\right) d s, t \in[0, T]$$where the drift coefficient $b:[0, T] \times \mathbb{R}^d \rightarrow \mathbb{R}^d$ is measurable and satisfies the integrability condition : $\|b\|_{L^q\left([0, T], L^\rho\left(\mathbb{R}^d\right)\right)}<\infty$ for some $\rho, q \in(0,+\infty]$ such that$$\rho \geq 2 \text { and } \frac{d}{\rho}+\frac{2}{q}<1 .$$Krylov and Röckner established strong existence and uniqueness under this condition.Let $n \in \mathbb{N}^*, h=\frac{T}{n}$ and $t_k=k h$ for $k \in \left [ \left [0,n \right ] \right ]$. Since there is no smoothing effect in the time variable, we introduce a sequence $\left(U_k\right)_{k \in \left [ \left [0,n-1 \right ] \right ]}$ independent from $\left(X_0,\left(W_t\right)_{t \geq 0}\right)$ of independent random variables which are respectively distributed according to the uniform law on $[k h,(k+1) h]$. The resulting scheme Euler is initialized by $X_0^h=X_0$ and evolves inductively on the regular time-grid $\left(t_k=k h\right)_{k \in \left [ \left [0,n \right ] \right ]}$ by:$$X_{t_{k+1}}^h=X_{t_k}^h+W_{t_{k+1}}-W_{t_k}+b_h\left(U_k, X_{t_k}^h\right) h$$where $b_h$ is some truncation of the drift function $b$. When $b$ is bounded, one of course chooses $b_h=b$. Then the order of weak convergence in total variation distance is $1 / 2$, as proved in [1]. It improves to 1 up to some logarithmic correction under some additional uniform in time bound on the spatial divergence of the drift coefficient. In the general case (1), we will see that for suitable truncations $b_h$, the difference between the transition densities of the stochastic differential equation and its Euler scheme is bounded from above by $C h^{\frac{1}{2}\left(1-\left(\frac{d}{\rho}+\frac{2}{q}\right)\right)}$ multiplied by some centered Gaussian density, as proved in [2].

Mots-Clés : diffusion processes; singular drift; Euler scheme; weak error analysis

Codes MSC :
60H10 - Stochastic ordinary differential equations
60H35 - Computational methods for stochastic equations
65C05 - Monte Carlo methods
65C30 - Stochastic differential and integral equations

    Informations sur la Vidéo

    Réalisateur : Hennenfent, Guillaume
    Langue : Anglais
    Date de Publication : 26/01/2024
    Date de Captation : 11/01/2024
    Sous Collection : Research School
    Catégorie arXiv : Probability
    Domaine(s) : Analyse Numérique & Calcul Formel ; Probabilités & Statistiques
    Format : MP4 (.mp4) - HD
    Durée : 01:30:48
    Audience : Chercheurs ; Etudiants Science Cycle 2 ; Doctoral Students, Post-Doctoral Students
    Download : https://videos.cirm-math.fr/2024-01-11_Jourdain_2.mp4

Informations sur la Rencontre

Nom de la Rencontre : Stochastic and Deterministic Analysis for Irregular Models / Analyse stochastique et déterministe pour les modèles irréguliers
Organisateurs de la Rencontre : Olivera, Christian ; Richard, Alexandre ; Russo, Francesco ; Tomašević, Milica
Dates : 08/01/2024 - 12/01/2024
Année de la rencontre : 2024
URL de la Rencontre : https://conferences.cirm-math.fr/2993.html

Données de citation

DOI : 10.24350/CIRM.V.20122403
Citer cette vidéo: Jourdain, Benjamin (2024). Numerical methods for SDEs with singular coefficients - Lecture 2. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20122403
URI : http://dx.doi.org/10.24350/CIRM.V.20122403

Voir Aussi

Bibliographie

  • BENCHEIKH, Oumaima et JOURDAIN, Benjamin. Convergence in Total Variation of the Euler--Maruyama Scheme Applied to Diffusion Processes with Measurable Drift Coefficient and Additive Noise. SIAM Journal on Numerical Analysis, 2022, vol. 60, no 4, p. 1701-1740. - https://doi.org/10.1137/20M1371774

  • JOURDAIN, Benjamin et MENOZZI, Stéphane. Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with LQ--L $\rho $ Drift Coefficient and Additive Noise. arXiv preprint arXiv:2105.04860, 2021. - https://doi.org/10.48550/arXiv.2105.04860

  • KRYLOV, Nicolai V. et RÖCKNER, Michael. Strong solutions of stochastic equations with singular time dependent drift. Probability theory and related fields, 2005, vol. 131, p. 154-196. - http://dx.doi.org/10.1007/s00440-004-0361-z

  • LÊ, Khoa et LING, Chengcheng. Taming singular stochastic differential equations: A numerical method. arXiv preprint arXiv:2110.01343, 2021. - https://doi.org/10.48550/arXiv.2110.01343



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