Let $u(x,t)$ be defined on $\Bbb R\times [0, T]$ such that: $$u_t + (a - bx) u_x + \frac12 c^2 u_{xx} = xu,\quad u(x,T)\equiv 1$$ in which $a,b,c>0$ are constants. How to solve this PDE? Is it possible to transform it into a 1D heat equation $u_t=\Delta u$? This resembles the Black-Scholes PDE but now the "riskfree rate" is a variable.
Motivation: this is the Black Scholes PDE corresponding to the zero bond price process under the Vasicek model.