Conic Martingales from Stochastic Integrals

TL;DR

This paper introduces conic martingales, a class of stochastic processes constrained within boundaries, providing a construction method, analytical tractability, and applications to modeling survival probabilities.

Contribution

It defines conic martingales, offers a construction approach, and characterizes a unique class with separable coefficients, applying them to survival probability modeling.

Findings

Identified a unique class of martingales with separable coefficients.

Provided a simple construction method for conic martingales.

Applied conic martingales to model stochastic survival probabilities.

Abstract

In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations. We then provide a simple way to construct and handle such processes. Specific attention is paid to martingales in $[0,1]$. One of these martingales proves to be analytically tractable. It is shown that up to shifting and rescaling constants, it is the only martingale (with the trivial constant, Brownian motion and Geometric Brownian motion) having a separable coefficient $\sigma(t,y)=g(t)h(y)$ and that can be obtained via a time-homogeneous mapping of Gaussian diffusions. The approach is exemplified to the modeling of stochastic conditional survival probabilities in the univariate (both conditional and unconditional to survival) and bivariate cases.