Convergence of Optimal Investment Problems in the Vanishing Fixed Cost Limit

We consider an optimal investment problem for an investor facing both constant and proportional transaction costs and study the limit as the constant cost tends to zero. Combining the stochastic Perron's method with stability arguments for viscosity solutions, we show that the value function converges to the value function of the problem with purely proportional costs. Moreover, using a Komlos-type argument, we show that forward-convex combinations of the optimal strategies in the problem with constant costs converge to an optimal strategy for the problem without a constant cost