Backward stochastic differential equations driven by optional semimartingales with conditional reflecting and regulated barrier

Abstract We introduce a new type of reflected backward stochastic differential equations (BSDEs) driven by optional semimartingale process with lower optional barrier and so-called regulated trajectories for which the reflection constraint is imposed on its main solution component, denoted as Y by convention, but in terms of its conditional expectation <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>𝔼</m:mi> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:msub> <m:mi>Y</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy="false">|</m:mo> <m:msub> <m:mi mathvariant="script">𝒢</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy="false">]</m:mo> </m:mrow> </m:mrow> </m:math> {\mathbb{E}[Y_{t}|\mathcal{G}_{t}]} on a general sub-filtration <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:msub> <m:mi mathvariant="script">𝒢</m:mi> <m:mi>t</m:mi> </m:msub> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> {\{\mathcal{G}_{t}\}} . This paper is devoted to the question of existence and uniqueness of strong solutions of the conditional RBSDE under Lipschitz conditions and by combining the Snell envelope method with Skorokhod lemma. Thus the connection between optimal stopping problems and linear conditional RBSDEs is given. Moreover, an example of applications to mathematical finance is presented.