Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise
We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q − L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and Röckner [26] proved existence of a unique strong solution, is met. We show weak con...
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Veröffentlicht in: | The Annals of applied probability 2024-02, Vol.34 (1B) |
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description | We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q − L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and Röckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 − (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density. |
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We show weak convergence with order 1 2 (1 − (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.</description><identifier>ISSN: 1050-5164</identifier><identifier>EISSN: 2168-8737</identifier><identifier>DOI: 10.1214/23-AAP2006</identifier><language>eng</language><publisher>Institute of Mathematical Statistics (IMS)</publisher><subject>Mathematics ; Probability</subject><ispartof>The Annals of applied probability, 2024-02, Vol.34 (1B)</ispartof><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><orcidid>0000-0001-5069-6067 ; 0000-0001-5069-6067</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>230,315,782,786,887,27931,27932</link.rule.ids><backlink>$$Uhttps://hal.science/hal-03223426$$DView record in HAL$$Hfree_for_read</backlink></links><search><creatorcontrib>Jourdain, Benjamin</creatorcontrib><creatorcontrib>Menozzi, Stéphane</creatorcontrib><title>Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise</title><title>The Annals of applied probability</title><description>We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q − L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and Röckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 − (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.</description><subject>Mathematics</subject><subject>Probability</subject><issn>1050-5164</issn><issn>2168-8737</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNqVyrtOwzAUgGELgUS4LDzBWRkMtpMm7RilRR0KKpc9spJjclASR7YT1BGxdObFeAeeBJB4AaZP-vUzdiHFlVQyuVYxz_OtEiI9YJGS6ZzPszg7ZJEUM8FnMk2O2Yn3L0KIRbLIIvZe2H5C94x9hfCgA4I1EBqE1dii47fajTvdaXisGuwQ8mFoCWsIFpZkzOjJ9rB1tkLv0cMrhQY2cA9f-48fP99g6cgEKCwaQxVhH0D3NeR1TYEmhDtLHs_YkdGtx_M_T9nlzeqpWPNGt-XgqNNuV1pN5TrflL9NxErFiUonFf_n_QZCJloi</recordid><startdate>20240201</startdate><enddate>20240201</enddate><creator>Jourdain, Benjamin</creator><creator>Menozzi, Stéphane</creator><general>Institute of Mathematical Statistics (IMS)</general><scope>1XC</scope><scope>VOOES</scope><orcidid>https://orcid.org/0000-0001-5069-6067</orcidid><orcidid>https://orcid.org/0000-0001-5069-6067</orcidid></search><sort><creationdate>20240201</creationdate><title>Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise</title><author>Jourdain, Benjamin ; Menozzi, Stéphane</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-hal_primary_oai_HAL_hal_03223426v23</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Mathematics</topic><topic>Probability</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Jourdain, Benjamin</creatorcontrib><creatorcontrib>Menozzi, Stéphane</creatorcontrib><collection>Hyper Article en Ligne (HAL)</collection><collection>Hyper Article en Ligne (HAL) (Open Access)</collection><jtitle>The Annals of applied probability</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Jourdain, Benjamin</au><au>Menozzi, Stéphane</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise</atitle><jtitle>The Annals of applied probability</jtitle><date>2024-02-01</date><risdate>2024</risdate><volume>34</volume><issue>1B</issue><issn>1050-5164</issn><eissn>2168-8737</eissn><abstract>We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q − L ρ drift coefficient when the condition d ρ + 2 q < 1, under which Krylov and Röckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 − (d ρ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.</abstract><pub>Institute of Mathematical Statistics (IMS)</pub><doi>10.1214/23-AAP2006</doi><orcidid>https://orcid.org/0000-0001-5069-6067</orcidid><orcidid>https://orcid.org/0000-0001-5069-6067</orcidid><oa>free_for_read</oa></addata></record> |
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title | Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q − L ρ Drift Coefficient and Additive Noise |
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