We consider time-homogeneous uniformly nondegenerate stochastic differential games in domains and propose constructing $varepsilon$-optimal strategies and policies by using adjoint Markov strategies and adjoint Markov policies which are actually time-homogeneous Markov, however, relative not to the original process but to a couple of processes governed by a system consisting of the main original equation and of an adjoint stochastic equations of the same type as the main one. We show how to find $varepsilon$-optimal strategies and policies in these classes by using the solvability in Sobolev spaces of not the original Isaacs equation but of its appropriate modification. We also give an example of a uniformly nondegenerate game where our assumptions are not satisfied and where we conjecture that there are no not only optimal Markov but even $varepsilon$-optimal adjoint (time-homogeneous) Markov strategies for one of the players.
{"title":"On the Adjoint Markov Policies in Stochastic Differential Games","authors":"N. Krylov","doi":"10.31390/COSA.13.1.01","DOIUrl":"https://doi.org/10.31390/COSA.13.1.01","url":null,"abstract":"We consider time-homogeneous uniformly nondegenerate stochastic differential games in domains and propose constructing $varepsilon$-optimal strategies and policies by using adjoint Markov strategies and adjoint Markov policies which are actually time-homogeneous Markov, however, relative not to the original process but to a couple of processes governed by a system consisting of the main original equation and of an adjoint stochastic equations of the same type as the main one. We show how to find $varepsilon$-optimal strategies and policies in these classes by using the solvability in Sobolev spaces of not the original Isaacs equation but of its appropriate modification. We also give an example of a uniformly nondegenerate game where our assumptions are not satisfied and where we conjecture that there are no not only optimal Markov but even $varepsilon$-optimal adjoint (time-homogeneous) Markov strategies for one of the players.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46683456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a Stochastic 2d Cahn-Hilliard-Navier-Stokes System Driven by Jump Noise","authors":"G. Deugoue, T. T. Medjo","doi":"10.31390/COSA.13.1.05","DOIUrl":"https://doi.org/10.31390/COSA.13.1.05","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48245268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the joint law of (Xt(x), Lt(x)) where Xt(x) is the solution of a one dimensional stochastic differential equation on (0,+∞) with reflection at zero, and Lt(x) is its local time. In particular, we give some representation formula of the distribution of (Xt(x), Lt(x)), and we investigate the regularity of the joint density with respect to the local time argument under ellipticity and mild regularity conditions on coefficients of Xt(x).
{"title":"Regularity of the Local Time of Diffusions on the Positive Real Line with Reflection at Zero","authors":"Masafumi Hayashi","doi":"10.31390/COSA.13.1.02","DOIUrl":"https://doi.org/10.31390/COSA.13.1.02","url":null,"abstract":"We study the joint law of (Xt(x), Lt(x)) where Xt(x) is the solution of a one dimensional stochastic differential equation on (0,+∞) with reflection at zero, and Lt(x) is its local time. In particular, we give some representation formula of the distribution of (Xt(x), Lt(x)), and we investigate the regularity of the joint density with respect to the local time argument under ellipticity and mild regularity conditions on coefficients of Xt(x).","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45855175","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Slaves, Slaveholders, and a Kentucky Community’s Struggle toward Freedom","authors":"Tom Barber","doi":"10.31390/cwbr.21.4.04","DOIUrl":"https://doi.org/10.31390/cwbr.21.4.04","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41424192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lincoln’s Spies: Their Secret War to Save a Nation","authors":"M. P. Johnson","doi":"10.31390/cwbr.21.4.11","DOIUrl":"https://doi.org/10.31390/cwbr.21.4.11","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42714445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Exponential estimates for exit from a ball of radius r by time T for solutions of the two-dimensional stochastic Navier-Stokes equations are first derived, and then studied in the context of Freidlin-Wentzell type large deviations principle. The existence of a similar estimate is discussed for a suitable class of stochastic evolution equations with multiplicative noise.
{"title":"Exponential Inequalities for Exit Times for Stochastic Navier-Stokes Equations and a Class of Evolutions","authors":"Po-Han Hsu, P. Sundar","doi":"10.31390/COSA.12.3.07","DOIUrl":"https://doi.org/10.31390/COSA.12.3.07","url":null,"abstract":"Exponential estimates for exit from a ball of radius r by time T for solutions of the two-dimensional stochastic Navier-Stokes equations are first derived, and then studied in the context of Freidlin-Wentzell type large deviations principle. The existence of a similar estimate is discussed for a suitable class of stochastic evolution equations with multiplicative noise.","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45851127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Slaves, Slaveholders, and a Kentucky Community’s Struggle toward Freedom","authors":"Jennifer Harbour","doi":"10.31390/cwbr.21.4.10","DOIUrl":"https://doi.org/10.31390/cwbr.21.4.10","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":"2 5","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41261419","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Church in the Wild: Evangelicals in Antebellum America","authors":"S. McBride","doi":"10.31390/cwbr.21.4.13","DOIUrl":"https://doi.org/10.31390/cwbr.21.4.13","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42166487","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Spying on the South: An Odyssey Across the American Divide","authors":"D. Crofts","doi":"10.31390/cwbr.21.4.06","DOIUrl":"https://doi.org/10.31390/cwbr.21.4.06","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41419856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Selling the Sights: The Invention of the Tourist in American Culture","authors":"S. Adams","doi":"10.31390/cwbr.21.4.12","DOIUrl":"https://doi.org/10.31390/cwbr.21.4.12","url":null,"abstract":"","PeriodicalId":53434,"journal":{"name":"Communications on Stochastic Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42322259","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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