Pardon,
why doesn't Mathematica solve this one:
DSolve[{sigma^2/2 D[P[r, t], {r, 2}] + (a (R - r) + q sigma) D[P[r, t], r] - r P[r, t] - D[P[r, t], t] == 0, P[r, 0] == 1}, P, {r, t}]
