E.6.2 This exercise is similar to that above but with a slightly different setup. The agent wishes to liquidate It shares and her objective is to maximise expected terminal wealth which is denoted by xy (in the exercise above we wrote terminal wealth as . (S, - kvu) Vu du). The value function is H (t, x, S,q) = supEtz.S.a XY -Q4 (ST -QQ7) (6.41) where dXY = (S, -kve) vedt. (6.42) (a) Show that the HJB satisfied by the value function H(t, S, q, x) is 0 = (8, + jo?ass) H + sup ((-va, + (S - kv) vax) H}, (6.43) and the optimal liquidation rate in feedback form is OH - SO, H -2k 0. H (6.44)(b) To solve (6.43), use the terminal condition H(T, x, S, q) = x + q S - aq2 to propose the ansatz H(t, S, x, q) = x + h(t) q2+qs, (6.45) where h(t) is a deterministic function of time. Show that k h()=T-tta (6.46) and T-t+4