Suppose V follows the mean reverting process $$dV=η( ̅V-V)Vdt+σVdz$$ I want to find the optimal investment rule, and using Itos's lemma I get that the differential equation that F(V) must satisfy
$$ 1/2 σ^2 V^2 F''(V)+[r-μ+η( ̅V-V)]VF'(V) -rF = 0$$ F(V) must satisfy the boundary conditions
$$F(0)=0$$
$$F(V^* )=V^*-I$$
$$F' (V^* )=1$$
The solution is stated to be given by: $$F(V)=AV^θ H(2η/σ^2 V;θ,b)$$ The solution is a confluent hypergeometric function H(x;θ,b(θ)) How do I solve for V and A?
I have calculated the value for θ and b. But I don't know how to proceed. In the book it says that A is a constant to be determined. We find A as well as the critical value V* at which it is optimal to invest , from the remaining two boundry conditions.
And hence I am lost..
Because the confluent hypergeometric function is an infinite series, A and V* must be found numerically... What does this mean? Isn't there an analytical closed form solution here?
see http://web.mit.edu/rpindyck/www/Courses/RO_P2_Hand%20Outs.pdf from slide 42 down for the whole set of equations..
Hope someone can help me. Do not know how to solve this :/