Local martingales

Definition

Local martingales are stochastic processes that generalize the concept of martingales by allowing for certain types of 'stopping times' which can handle the growth in variance over time. They satisfy a martingale-like property when examined over short time intervals, making them useful in the study of financial mathematics and stochastic calculus. Local martingales can be seen as a bridge between martingales and more complex processes, retaining essential properties while allowing for greater flexibility in modeling.

5 Must Know Facts For Your Next Test

1. Local martingales can be seen as a generalization of martingales, which allows for certain non-negative local martingale components.
2. They are particularly important in the theory of stochastic integration and are often used in the construction of Itô calculus.
3. Local martingales can fail to be true martingales due to the presence of large fluctuations in their trajectories, especially when not conditioned on stopping times.
4. Every local martingale can be approximated by a true martingale at appropriate stopping times, making them useful in mathematical finance.
5. The Doob-Meyer decomposition theorem states that every local submartingale can be expressed as the sum of a local martingale and a predictable increasing process.

Review Questions

• How do local martingales extend the concept of traditional martingales in terms of variance handling?
  • Local martingales extend traditional martingales by allowing for variance to grow over time through the use of stopping times. While traditional martingales require that future values remain constant in expectation given past values, local martingales can tolerate fluctuations within short time intervals. This ability to manage variance makes local martingales particularly useful in various applications like finance where underlying processes may exhibit volatility.
• Discuss the significance of stopping times in relation to local martingales and how they affect their properties.
  • Stopping times are crucial in defining local martingales because they allow for the analysis of processes at specific points in time when events occur. In contrast to classic martingales that are constrained by fixed expectations, local martingales leverage stopping times to manage potential discrepancies in their behavior over time. This feature ensures that despite possible divergence from classic martingale behavior, one can still find short intervals where local martingales behave predictably, which is essential for applications such as pricing derivatives.
• Evaluate how the Doob-Meyer decomposition theorem provides insight into local martingales' behavior within stochastic processes.
  • The Doob-Meyer decomposition theorem is key in understanding local martingales because it states that any local submartingale can be decomposed into a local martingale and an increasing predictable process. This decomposition highlights how local martingales function within a broader framework of stochastic processes, providing a way to separate the random fluctuations from deterministic trends. By applying this theorem, one can analyze financial models more effectively, revealing deeper insights into asset pricing and risk management by differentiating between inherent randomness and expected growth.

Related terms

Martingale:

A stochastic process that maintains its expected future value, given all past information, meaning that the expected value of the next observation is equal to the current observation.

Stopping Time:

A random variable that represents a specific time at which a certain event occurs, allowing for decision-making based on past information up to that time.

Brownian Motion:

A continuous-time stochastic process that models random motion, commonly used in finance and physics, serving as a fundamental building block for many other stochastic processes.