On the Martingale Property of Certain Local Martingales

2009/05/22 by Aleksandar Mijatović, Mijatovic, Aleksandar, Mikhail Urusov +1 · 3 citations
Economics, Econometrics and Finance · Mathematics · #60G44 (Primary) 60G48 #60H10 (Secondary) #Advanced Harmonic Analysis Research #FOS: Economics and business #FOS: Mathematics #General Finance (q-fin.GN) #Probability (math.PR) #Stochastic processes and financial applications #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.0905.3701

openalex publication_date 2009/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The stochastic exponential Z_t=exp\M_t-M₀-(1/2) _t\ of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where M_t=∫₀^t b(Y_u) dW_u and Y is a one-dimensional diffusion driven by a Brownian motion W. Furthermore, we provide a necessary and sufficient condition for Z to be a uniformly integrable martingale in the same setting. These conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of Y. As an application we provide a deterministic criterion for the absence of bubbles in a one-dimensional setting.

Cited by

• From CKLS Process to CIR-type and OU-type Processes: Using a Twice-differentiable Mapping and Generalized Girsanov's Theorem
• Estimation of the elasticity for CKLS model from high-frequency observations
• Double Local-to-Unity: Inference under Nearly Nonstationary Volatility

Related

• Martingale property of generalized stochastic exponentials
• When a Stochastic Exponential is a True Martingale. Extension of a Method of Bene^s
• Martingale Property in Terms of Semimartingale Problems
• A remark on conditions that a diffusion in the natural scale is a martingale
• A complete characterization of local martingales which are functions of Brownian motion and its maximum