Recognizing Martingale Conditions and Their Limits

Summary

The document asks whether three conditional-expectation statements are equivalent ways to characterize a martingale: conditioning a later value on the current information, conditioning each value on the initial information, and conditioning the terminal value on current information. The questioner proposes that integrability and adaptation might make all three sufficient, but gives no attempted counterexample or solution.

These formulations invite comparison with the defining martingale property, which requires integrable, filtration-adapted values and the conditional expectation of each future value to equal the current value. A condition about the terminal value alone need not establish that property at every intermediate time, while conditioning on the initial information is also weaker than conditioning on the full information available at each time. The document itself supplies no proof or examples, so it serves as a probability-theory question rather than a complete treatment. Its relevance to quantitative finance is through martingale models for prices and other stochastic processes.

Key ideas

• A martingale requires integrability, adaptation, and a conditional expectation condition across time.
• The three stated expectation conditions differ in which information and time points they involve.
• A terminal-time condition alone may not establish the required relation for every future time.
• The document poses the equivalence question but provides no solution or counterexamples.

Tags

# Recognizing a Martingale


# Recognizing a Martingale












Under which conditions is the stochastic process $\{X_t\}_{t=0,1,...,T}$ a martingale? Demonstrate and explain clearly for each case below. If it is not necessarily a martingale, provide a counterexample.

$\mathbb{E}_t[X_s] = X_t$ for $0 \leq t \leq s \leq T$

$\mathbb{E}_0[X_t] = X_0$ for $0 \leq t \leq T$

$\mathbb{E}_t[X_T] = X_t$ for $0 \leq t \leq T$

I am having a bit of a hard time figuring this out. My hunch is that, as long as (1) $\mathbb{E}[|X_t|] < \infty$ and (2) $X_t$ is adapted to a filtration $\mathcal{F}_T$, then all of the stochastic processes listed above are Martingales because they are just equivalent ways of stating the first property of Martingales, that is, $\mathbb{E}[X_T|\mathcal{F}_t] = X_t$ for $0 \leq t \leq T$ . Therefore, a counterexample could be anything that does not satisfy properties (1) and (2). Is this correct? If not, why? Thanks!

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