Optimal Investment with Transaction Costs and Stochastic Volatility Part I: Infinite Horizon

Proof. The continuity in the (x, y) variables follows from concavity in the same way as in [4] in Corollaries 7 3.2, 5.5 and 5.8. For the continuity in time t, note that it is the same as continuity in terminal time T . 8 Clearly, the integral term in [1] eq. (2.5) is continuous. Regarding the second term U (Liq(XT , YT ), note 9 that the same admissible strategy (C,L,M) that liquidates the position at time T and consumes nothing 10 after that, can be used for longer maturity T + ∆T , for ∆T > 0. Thus V̂ is upper semi-continuous in T . 11 The lower semi-continuity of V̂ follows from the the lower semi-continuity of the process Liq(X,Y ). 12 The continuity in the z variable, is the result of stability of strong solutions. Fix (t0, x0, y0, z0) such 13 that (x0, y0) ∈ S and ε0 > 0. For |z − z0| < 1, let (C000, L000,M 000) be a nearly-optimal 14 admissible strategy, such that 15