I am working with a derivative instrument that has given me this nonlinear second order parabolic PDE, roughly the Black-Scholes equation but with the addition of a non-linear term: \begin{align} & \mathcal{L}v(t,x) + C_1 \left[R - v\right]^{+} + C_2 \left[R - v\right]^{-}=0,\qquad \text{for}\quad(t,x)\in[0,T]\times[0,\infty) \\[1em] & v(T,x) = g(x), \quad x\in[0,\infty)\\[0.5em] & v(t, 0)=0, \quad t\in[0,T]\\[0.5em] & v(t,x)\to\infty \quad \text{as }x\to\infty. \end{align} $$$$ where $$\mathcal{L}v(t,x)=\frac{\partial v}{\partial t}(t,x) + r(t) x\frac{\partial v}{\partial x}(t,x) + \frac{1}{2}\sigma(t,x)^2 \frac{\partial^2 v}{\partial x^2}(t,x)-r(t)v(t,x).$$
Here, $C_1, C_2$ are positive constants and $(\cdot)^{+}:=\max(\cdot,0)$ and $(\cdot)^{-}:=\min(\cdot, 0)$. The functions $r:[0,T]\to[0,\infty)$, $R:[0,T]\to[0,\infty)\to[0,\infty)$ and $g:[0,\infty)\to[0,\infty)$ are continuous. For simplicity, $R$ can usually be taken to be a constant multiple of $g$.
I was wondering where I can find a existence and uniqueness result for the solution the the above PDE (classical preferably, but viscosity solutions are okay too). I am happy for additional conditions to be imposed on $R, g, r$ are they are usually quite simple.
Found an answer after searching around. This is a quasi-linear parabolic PDE. Pardoux and Peng 1992 (https://link.springer.com/chapter/10.1007/BFb0007334) give sufficient conditions for existence/uniqueness of a viscosity solution via the corresponding forwards backwards stochastic differential equation in chapter 4.