This is a joint work by the author and Z. Qian. In this work, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale on the Sierpinski gasket. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynm...

This is an article by the author. In this item, we consider stochastic control problems on the Sierpinski gasket. An order comparison lemma is derived using heat kernel estimate for Brownian motion on the gasket. Using the order comparison lemma and techniques of BSDEs, we establish a Pontryagin stochastic maximum principle for these control problems. It turns out that the stochastic maximum principle on the Sierpinski gasket involves two necessity equations in contrast to its counterpart on ...

This is a joint work by the author and Z. Qian. In this article, we initiate a study of some non-linear parabolic equations on Sierpinski gasket, motivated by modellings of fluid flows along fractals (which can be considered as models of simplified rough porous media). Unlike the regular space case, such parabolic type equations involving non-linear convection terms must take a different form, due to the fact that convection terms must be singular to the “linear part” determined the heat semi...

This is an article by the author. In this article, we study the asymptotics of tail probability for the suprema of a class of stochastic processes which satisfy the controlled conditional increment condition and have $\alpha$-exponential marginal decays. These conditions are quite weak compared to other assumptions (e.g. being jointly Gaussian, or satisfying Markov property) considered in literature, and are satisfied by a many processes commonly encountered. We start with establishing a Doob...