with respect to the random parameter $\omega=(\sigma, H)\in\Omega$, unless this is necessary.

Throughout the paper, differential equations and inequalities are to be understood in the viscosity sense (see [@CIL] for the general background). We next recall the appropriate notion of viscosity solution for quasilinear equations like  which may be singular at $\xi=0$.

\[d.viscsol\] Given $U\subseteq {\mathbb{R}^d}\times (0,\infty)$ and a function $u\in C(U)$, we say that $u$ is a *subsolution* (respectively, *supersolution*) of the equation $$\label{}
\partial_t u - \operatorname{tr}\left( A \left( Du,x \right) D^2u  \right) + H(Du,x) = 0 \quad \mbox{in} \ U$$ if, for every $\phi\in C^2(U)$ and $(x_0,t_0) \in U$ such that $$\label{}
(x,t) \mapsto u(x,t) - \phi(x,t) \quad \mbox{has a local maximum (resp., minimum) at} \ (x_0,t_0),$$ we have $$\label{}
\partial_t \phi(x_0,t_0) - \operatorname{tr}^*\left( A \left( D\phi(x_0,t_0),x \right) D^2\phi(x_0,t_0)  \right) + H(D\phi(x_0,t_0),x) \leq 0$$ (resp., $$\label{}
\partial_t \phi(x_0,t_0) - \operatorname{tr}_*\left( A \left( D\phi(x_0,t_0),x \right) D^2\phi(x_0,t_0)  \right) + H(D\phi(x_0,t_0),x) \geq 0 \, \mbox{)}.$$ Here, for any symmetric matrix $X\in {\mathbb{R}}^{d\times d}$, $\operatorname{tr}^*(A(\xi,x)X)$ and $\operatorname{tr}_*(A(\xi,x)X)$ stand for the upper and lower semicontinuous envelopes of the map $(\xi,x) \to \operatorname{tr}(A(\xi,x)X)$, which agree with $ \operatorname{tr}(A(\xi,x)X)$ on the domain of $A$ and are defined at $\xi=0$ for any matrix $X$ by 