Convex ordering of solutions to one-dimensional SDEs

TL;DR

This paper investigates how convexity properties propagate through solutions of one-dimensional SDEs, establishing conditions under which convexity and comparison principles hold for solutions and their functionals.

Contribution

It introduces conditions on the diffusion coefficient's spatial convexity to extend convexity propagation from single-time to multi-time and path functionals, with a novel approximation approach.

Findings

Convexity propagation holds at one instant without additional assumptions.

Spatial convexity of the diffusion coefficient is needed for two-time convexity extension.

The method allows convexity-preserving Monte Carlo simulations.

Abstract

In this paper, we are interested in the propagation of convexity by the strong solution to a one-dimensional Brownian stochastic differential equation with coefficients Lipschitz in the spatial variable uniformly in the time variable and in the convex ordering between the solutions of two such equations. We prove that while these properties hold without further assumptions for convex functions of the processes at one instant only, an assumption almost amounting to spatial convexity of the diffusion coefficient is needed for the extension to convex functions at two instants. Under this spatial convexity of the diffusion coefficients, the two properties even hold for convex functionals of the whole path. For directionally convex functionals, the spatial convexity of the diffusion coefficient is no longer needed. Our method of proof consists in first establishing the results for time discretization schemes of Euler type and then transferring them to their limiting Brownian diffusions. We thus exhibit approximations which avoid {\em convexity arbitrages} by preserving convexity propagation and comparison and can be computed by Monte Carlo simulation.