Convex ordering for stochastic control: the (path dependent) swing contracts case

Abstract

We investigate propagation of convexity and convex ordering on a typical discrete-time stochastic optimal control problem, namely the pricing of swing option. The dynamics of the underlying asset is modelled by the Euler scheme of a Brownian diffusion with affine drift, and convex volatility. We prove that the value function associated to the stochastic optimal control problem is a convex function of the underlying asset price. We also introduce a domination criterion offering insights into the functional monotonicity of the value function with respect to parameters of the underlying dynamics. We particularly focus on the one-dimensional setting where, by means of Stein's formula and regularization techniques, we show that the convexity assumption for the volatility dynamics can be relaxed with a semi-convexity assumption. Finally, to validate our results, we also conduct numerical illustrations.