  For a given Lagrangian $L:[0,T]\times M\times M^\ast\rightarrow \mathbb{R}_+$
and probability measures $\mu\in\mathcal{P}(M^\ast)$, $\nu\in \mathcal{P}(M)$,
we introduce the stochastic ballistic transportation problems
\begin{align}\tag{$\star$}
  \underline{B}(\mu,\nu):=\inf\left\{\mathbb{E}\left[\langle V,X_0\rangle
+\int_0^T L(t,X,\beta(t,X))\,dt\right]\middle\rvert V\sim\mu,X_T\sim
\nu\right\}\\\tag{$\star\star$}
  \overline{B}(\nu,\mu):=\sup\left\{\mathbb{E}\left[\langle V,X_T\rangle
-\int_0^T L(t,X,\beta(t,X))\,dt\right]\middle\rvert V\sim\mu,X_0\sim
\nu\right\} \end{align} where $X$ is a diffusion process with drift $\beta$.
This cost is based on the stochastic optimal transport problem presented by
Mikami and the deterministic ballistic transport introduced by Ghoussoub. We
obtain a Kantorovich-style duality result that reformulates this problem in
terms of solutions to the Hamilton-Jacobi-Bellman equation \begin{equation*}
  \frac{\partial\phi}{\partial t}+\frac{1}{2}\Delta \phi+H(t,x,\nabla\phi)=0,
\end{equation*} and show how optimal processes may be thereby attained.
