Stochastic optimal transport with at most quadratic growth cost

Abstract: We consider a class of stochastic optimal transport, SOT for short, with given two endpoint marginals in the case where a cost function exhibits at most quadratic growth. We first study the upper and lower estimates, the short--time asymptotics, the zero--noise limits, and the explosion rate as time goes to infinity of SOT. We also show that the value function of SOT is equal to zero or infinity in the case where a cost function exhibits less than linear growth. As a by--product, we characterize the finiteness of the value function of SOT by that of the Monge--Kantorovich problem with the same two endpoint marginals. As an application, we show the existence of a continuous semimartingale, with given initial and terminal distributions, of which the drift vector is $r$th integrable for $r\in [1,2)$. We also consider the same problem for Schr\"odinger's problem where $r=2$. This paper is a continuation of our previous work.