10.4. Solution of the Black-Scholes PDE. Use the Gaussian integration by parts for- mula E[F'(Z)] = E[ZF(Z)], where Z is a standard Gaussian variable, to prove that the function f(t, x) in equa- tion (10.13) is the solution of the Black-Scholes PDE (10.11). (10.13) (t, x) = er(Tt)E[g(xeT1Z+(r=&)(T1))]. (10.11) 0/1 (t, x) + 1 ox012/12 (t, x) + rxf (t, x) rf(t, x) = 0, at f(T,x) = g(x).