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. As an application, we show the existence of a continuous semimartingale, with given initial and terminal distributions, of which the drift vector is rth integrable for $r\in [1,2)$. We also consider the same problem for Schrödinger’s problem where $r=2$. This paper is a continuation of our previous work.

Appendix A: Proof of (17)

In this section we prove (17) in Remark 1.4 for readers’ convenience.

From Theorem 1.2, p(0, x; T, y) is positive and continuous under (A), and the following holds:

$$\begin{aligned} \varphi _2(0,x):=\int _{\mathbb {R}^d}p(0,x ;T,y) \nu _2(dy)>0, x\in \mathbb {R}^d. \end{aligned}$$

First, we prove the following:

$$\begin{aligned} q(y)dy=\frac{p(0,x_0 ;T,y)}{\varphi _2(0,x_0)}\nu _2(dy). \end{aligned}$$

(A.1)

Indeed, from (5),

$$\begin{aligned} \delta _{x_0}(dx)&=\nu _1(dx)\int _{\mathbb {R}^d}p(0,x ;T,y) \nu _2(dy)=\varphi _2(0,x)\nu _1(dx),\nonumber \\ q(y)dy&=\nu _2(dy)\int _{\mathbb {R}^d}p(0,x ;T,y)\nu _1(dx), \end{aligned}$$

(A.2)

from which the following holds:

$$\begin{aligned} q(y)dy&=\nu _2(dy)\int _{\mathbb {R}^d}p(0,x ;T,y) \frac{1}{\varphi _2(0,x)}\delta _{x_0}(dx). \end{aligned}$$

(A.1)–(A.2) imply the following:

$$\begin{aligned} \delta _{x_0}(dx)=\nu _1(dx)\varphi _2(0,x_0)\int _{\mathbb {R}^d} \frac{p(0,x ;T,y)q(y)}{p(0,x_0 ;T,y)}dy. \end{aligned}$$

(A.3)

(A.1) and (A.3) imply (17).