Strong completeness and semi-flows for stochastic differential equations with monotone drift

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• 作者：Michael Scheutzow ; ^{ms@math.tu-berlin.de" class="auth_mail" title="E-mail the corresponding author} ; Susanne Schulze ^{Susanne_Richter85@web.de" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Stochastic flow ; Stochastic semi-flow ; Stochastic differential equation ; Monotonicity ; Strong completeness ; Strong Δ-completeness
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：446
• 期：2
• 页码：1555-1570
• 全文大小：417 K

文摘

It is well-known that a stochastic differential equation (sde) on a Euclidean space driven by a (possibly infinite-dimensional) Brownian motion with Lipschitz coefficients generates a stochastic flow of homeomorphisms. If the Lipschitz condition is replaced by an appropriate one-sided Lipschitz condition (sometimes called monotonicity condition) and the number of driving Brownian motions is finite, then existence and uniqueness of global solutions for each fixed initial condition is also well-known. In this paper we show that under a slightly stronger one-sided Lipschitz condition the solutions still generate a stochastic semiflow which is jointly continuous in all variables (but which is generally neither one-to-one nor onto). We also address the question of strong Δ-completeness which means that there exists a modification of the solution which if restricted to any set class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16305522&_mathId=si1.gif&_user=111111111&_pii=S0022247X16305522&_rdoc=1&_issn=0022247X&md5=20acaa69f96836a747fc7c3fc664cc50" title="Click to view the MathML source">A⊂R^dclass="mathContainer hidden">class="mathCode">$A\subset {\mathbf{R}}^d$ of dimension Δ is almost surely continuous with respect to the initial condition.