Consider the following dynamics
\begin{align} dX_{s} &= a(s,X_{s},Y_{s},Z_{s})ds + \sigma(s,X_{s},Y_{s},Z_{s})dW_s \\ X_{t}&=x \, (\in\mathbb{R}^{n}) \end{align} and the associated payoff functional \begin{align} J(t,x,Y,Z) = \mathbb{E}[\int_{t}^{T}f(s,X_{s},Y_{s},Z_{s})ds + g(X_{T})] \end{align} My question, is how to write the variational inequality concerning the problem \begin{align} V(t,x) = \inf_{z}\sup_{y}J(t,x,y,z) \end{align}