Skip to content
Stochastic Calculus / Stochastic Integration and Itô's Formula

Extension to Local Martingales and Itô Processes

In practice, the integrand HtH_t in a stochastic integral is often not globally square-integrable: E[∫0THt2 dt]\mathbb{E}[\int_0^T H_t^2 \, dt] may be infinite. For example, in the geometric Brownian motion model, the integrand that arises in Itô's formula for ln⁡St\ln S_t is Ht=1/StH_t = 1/S_t, and when StS_t is log-normally distributed, E[∫0TSt−2 dt]\mathbb{E}[\int_0^T S_t^{-2} \, dt] is finite but one still needs a general framework that handles such integrands without re-checking square integrability at each step.
The solution is the local martingale: rather than requiring Zt=∫0tHs dWsZ_t = \int_0^t H_s \, dW_s to be a martingale globally, we only require it to be a martingale when stopped at a sequence of stopping times τn↑∞\tau_n \uparrow \infty. Every adapted continuous stochastic integral is a local martingale; whether it is a true martingale depends on an integrability condition.
The broader class of processes considered here are Itô processes (also called Itô processes with drift):
dXt=μt dt+σt dWt,dX_t = \mu_t \, dt + \sigma_t \, dW_t,

which is short-hand for Xt=X0+∫0tμs ds+∫0tσs dWsX_t = X_0 + \int_0^t \mu_s \, ds + \int_0^t \sigma_s \, dW_s. These are the natural objects for modelling stock prices, interest rates, and most continuous-time finance models.

Learning objectives. By the end of this note you should be able to:
  • State the definition of a local martingale and construct the localising sequence from the stochastic integral.
  • Recognise the doubling-strategy example as a local martingale that fails to be a martingale.
  • State a sufficient condition for a local martingale to be a true martingale.
  • Define an Itô process dXt=μt dt+σt dWtdX_t = \mu_t \, dt + \sigma_t \, dW_t and write its integral representation.
  • Identify the "martingale part" and "drift part" of an Itô process and explain why this decomposition is unique.

In this lesson

  1. The Failure of the Martingale Property
  2. Local Martingales
  3. When Is a Local Martingale a True Martingale?
  4. Itô Processes
  5. Integration with Respect to an Itô Process
  6. Financial Application: GBM as an Itô Process

Before you start

The full lesson, with worked examples and exercise solutions, is for members.

Read the full lesson

New to SelfMadeQuant.com? See membership options or explore the curriculum.