In my work I faced a system of PDE of the form \begin{align} \varphi^\prime_v(t,v)|_{v=X_t} = \sigma(t,X_t),\newline \varphi^\prime_t(t,v)|_{v=X_t} = b(t,X_t), \end{align}

where $t\in[0,T]$, $v\in\mathbf R$, $X_t$ is known continuous function of unbounded variation, $\varphi(t,v)$ is unknown, $\varphi^\prime_v(t,v) = \frac{\partial}{\partial v} \varphi(t,v)$, $\varphi^\prime_t(t,v) = \frac{\partial}{\partial t} \varphi(t,v)$ and there is an initial value $\varphi(0,X_0) = x_0$. Function $\varphi(t,X_t)$ is to be determined.

Roughly this is a system of PDE \begin{align} \varphi^\prime_v(t,v)= \sigma(t,v),\newline \varphi^\prime_t(t,v) = b(t,v), \end{align} defined on the line $(t,X_t)$.

I need to find a solution for this system, but yet haven't dealt with such equations and don't even know if they have any special name, so that I could search. I will be very happy if somebody gives me recommendations on textbooks which consider this kind of problems.