Here is a partial development.Define [$]g(t,V_t,X_t) = E_t[above][$], which can be rewritten [$]M_t = E_t[M_T][$], defining [$]M_t \equiv g(t,V_t,X_t)[$]. This shows [$]M_t[$] is a martingale, which means its Ito's rule expansion has no "dt" term. That expansionis [$]dg = (g_t + A g) dt + ...[$], so [$]g_t + A g = 0[$], where [$] A g[$] is the same operator on g as the operator on v in the "sup" in (2). Applying the sup to the construction left to you. p.s. This type of thing is textbook. See, for example, Merton's 'Continuous-time Finance', pg 128
Last edited by Alan
on January 13th, 2015, 11:00 pm, edited 1 time in total.