On the martingale property of stochastic exponentials

  title={On the martingale property of stochastic exponentials},
  author={Bernard Wong and Chris C. Heyde},
  journal={Journal of Applied Probability},
  pages={654 - 664}
We present a necessary and sufficient condition for a stochastic exponential to be a true martingale. It is proved that the criteria for the true martingale property are related to whether a related process explodes. An alternative and interesting interpretation of this result is that the stochastic exponential is a true martingale if and only if under a ‘candidate measure’ the integrand process is square integrable over time. Applications of our theorem to problems arising in mathematical… 

The martingale property in the context of stochastic differential equations

This note studies the martingale property of a nonnegative, continuous local martingale Z, given as a nonanticipative functional of a solution to a stochastic differential equation. The condition

Brownian Super-exponents

We introduce a transform on the class of stochastic exponentials for d-dimensional Brownian motions. Each stochastic exponential generates another stochastic exponential under the transform. The new

On the martingale property of certain local martingales

The stochastic exponential $${Z_t= {\rm exp}\{M_t-M_0-(1/2)\langle M,M\rangle_t\}}$$ of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient

On changes of measure in stochastic volatility models

Pricing in mathematical finance often involves taking expected values under different equivalent measures. Fundamentally, one needs to first ensure the existence of ELMM, which in turn requires that

Exponential Martingales and Time integrals of Brownian Motion

We find a simple expression for the probability density of $\int \exp (B_s - s/2) ds$ in terms of its distribution function and the distribution function for the time integral of $\exp (B_s + s/2)$.

A note on a paper by Wong and Heyde

In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some

The affine transform formula for affine jump-diffusions with a general closed convex state space

We establish existence of exponential moments and the validity of the affine transform formula for affine jump-diffusions with a general closed convex state space. This extends known results for



A note on the existence of unique equivalent martingale measures in a Markovian setting

The existence of a unique equivalent measure up to an explosion time is proved by means of results that give a handle on situations where an equivalent martingale measure cannot exist.

No Arbitrage Condition for Positive Diffusion Price Processes

Using the Ray-Knight theorem we give conditions for anonnegative diffusion without drift to reach zero or not. These results also givenecessary and sufficient conditions for such a diffusion process

Statistics of random processes

1. Essentials of Probability Theory and Mathematical Statistics.- 2. Martingales and Related Processes: Discrete Time.- 3. Martingales and Related Processes: Continuous Time.- 4. The Wiener Process,

Diffusions, Markov processes, and martingales

This celebrated book has been prepared with readers' needs in mind, remaining a systematic treatment of the subject whilst retaining its vitality. The second volume follows on from the first,

Stochastic filtering theory

1 Stochastic Processes: Basic Concepts and Definitions.- 2 Martingales and the Wiener Process.- 3 Stochastic Integrals.- 4 The Ito Formula.- 5 Stochastic Differential Equations.- 6 Functionals of a

Complications with stochastic volatility models

  • C. Sin
  • Mathematics, Economics
    Advances in Applied Probability
  • 1998
We show a class of stock price models with stochastic volatility for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually

Complete Models with Stochastic Volatility

The paper proposes an original class of models for the continuous‐time price process of a financial security with nonconstant volatility. The idea is to define instantaneous volatility in terms of

Brownian Motion and Stochastic Calculus

This chapter discusses Brownian motion, which is concerned with continuous, Square-Integrable Martingales, and the Stochastic Integration, which deals with the integration of continuous, local martingales into Markov processes.

A Comparison of Two Quadratic Approaches to Hedging in Incomplete Markets

This paper provides comparative theoretical and numerical results on risks, values, and hedging strategies for local risk‐minimization versus mean‐variance hedging in a class of stochastic volatility

The valuation of options for alternative stochastic processes