I am trying to solve this optimal control problem :
$ V(x,t) = inf( E[\int_{0}^{1}(x(t)^2 - \frac{1}{2}u^2(t))dt + x(1)^2])$
subject to
$dx(t) = u(t)dW_t$
$x(0) = x_0 \in R $
$u(t) \in [-1,1] $
where E is the expected value and R is the set of the real number, and $W_t$ is a brownian motion.
in order to solve that problem, I have written the HJB equation, which claims that the value function V(x,t) is solution to the following pde :
$ -v_t + sup_{u}(-\frac{1}{2}u^2v_{xx} -x^2 + \frac{1}{2}u^2) =0$
$v(1,x)= x^2$
The $sup_{u}(-\frac{1}{2}u^2v_{xx} -x^2 + \frac{1}{2}u^2)$ is over $u \in [-1,1] $
And then at this step I am blocked because I just can't solve the sup, I have tried to make the assumption that v(x,t) can be written $v(x,t) =\frac{1}{2} \theta(t)x^2$ but it does not really help to get a solution to the equation. If you have any idea to solve this equation I would be grateful. In advance thank you very much for your help.
Actually, I have managed to find a solution to the equation, I post it for people who followed the topic and if it could interest some students :
$ -v_t + sup_{u}(-\frac{1}{2}u^2v_{xx} -x^2 + \frac{1}{2}u^2) =0$
$v(1,x)= x^2$
This system can be rewritten :
$ -v_t + -x^2+ sup_{u}(\frac{1}{2}u^2(1-v_{xx})) =0$
$v(1,x)= x^2$
We make the assumption that we can write $v(t,x)=\frac{1}{2} \theta(t)x^2$
then we have the following system
$ -(\frac{1}{2}\theta(t)' +1) x^2 + max(\frac{1}{2}(1-v_{xx}),0) =0$
$\theta(1)= 2$
The first equation has to be valid for all x then it lead to have
$\frac{1}{2}\theta(t)' +1 = 0$ and then $\theta(t)= -2t + C$ and based on the terminal condition we have C = 4
$\theta(t)= -2t + 4$
then we can check as t is in [0,1] that $max(\frac{1}{2}(1-v_{xx}),0) = 0$ and then the feedback control is u is null