I am trying to find a solution to this boundary value problem

h_t= 1/2 sigma^2 x^2 h_xx + rx h_x s.t.

Ah(b,t)+Bh_x(b,t)=g(t),

where A,B,sigma,r,b are constants and g(t) is a given function of time (h_x is the partial derivative w.r.t. the first component of h(x,t)).

I need an explicit solution for it, whatever it is (any solutions satisfying the PDE and the boundary condition will be fine). I tried to use these commands, but it did not give anything:

eqn = 1/2 x^2 (Sigma)^2 D(h(x, t), {x, 2}) + r x D(h(x, t), x) – D(h(x, t), {t}) == 0;

ibc = {A h(b, t) + B Derivative(1, 0)(h)(b, t) == g(t)};

sol = DSolveValue({eqn, ibc}, h(t, x), {t, x}) // FullSimplify

I think I need to use something else, but I am just a beginner in Mathematica.

Thank you in advance!