This is a Maths Finance problem. pl give complete solution.

suppose dx(t) =a(t)dt+dW(t) with x(0) = x. with the drift u, as the control, consider the problem of maximizing the terminal value bX(T) with a running cost of the form au2(t); i.e. the "cost" function to be minimized is E o au ()dt - bx(CT) (T) Explicitly calculate the HJB equation for the value function V (t, x), find its solution in the form V(t,x)--bx+h(t) and the optimal control u'(t,X'()). suppose dx(t) =a(t)dt+dW(t) with x(0) = x. with the drift u, as the control, consider the problem of maximizing the terminal value bX(T) with a running cost of the form au2(t); i.e. the "cost" function to be minimized is E o au ()dt - bx(CT) (T) Explicitly calculate the HJB equation for the value function V (t, x), find its solution in the form V(t,x)--bx+h(t) and the optimal control u'(t,X'())