Martingale Conditions for a Brownian Motion Quadratic Process

Summary

The document poses a martingale condition problem for a process formed from the square of standard Brownian motion, a linear Brownian term, a deterministic time term, and a constant. Its hint evaluates the conditional expectation between two times by separating the Brownian increment from the value already known at the earlier time. The increment has mean zero and variance equal to the elapsed time, which gives the conditional expectation of the squared Brownian term as the earlier square plus elapsed time. The linear term has conditional expectation equal to its earlier value, while the deterministic terms remain unchanged by conditioning.

Comparing this conditional expectation with the process at the earlier time yields the condition that cancels the elapsed-time contribution; the linear coefficient and constant do not affect that condition. The excerpt provides the derivation ingredients but stops before stating the final condition explicitly. The argument assumes standard Brownian motion and its natural information filtration, and does not discuss more general processes or filtrations.

Key ideas

• Condition on the information available at the earlier time and decompose Brownian motion into its past value and an independent increment.
• The conditional expectation of the squared Brownian value gains a term equal to the elapsed time.
• The conditional expectation of the linear Brownian term is its value at the earlier time.
• The deterministic time coefficient must cancel the elapsed-time contribution for the process to be a martingale.
• The linear and constant coefficients do not change the martingale condition.

Tags

# Determine the conditions for Brownian motion to be a Martingale


# Determine the conditions for Brownian motion to be a Martingale












Let $W_T$ denote normalised univariate Brownian motion and let

$X_t = W_t^2 + \alpha W_t + \beta t + \gamma$

where $\alpha, \beta$ and $\gamma$ are constants. Determine conditions on these constants such that $X_t$ is a Martingale.

## Answer by math (score 2, accepted)

https://quant.stackexchange.com/a/31758

Hint

Let $s<t$. $$E[W_t^2|F_s]=E[(W_t-W_s)^2|F_s]+2 E[(W_t-W_s)W_s|F_s]+E[W_s^2|F_s]$$ so $$E[W_t^2|F_s]=E[(W_t-W_s)^2]+2W_s E[(W_t-W_s)]+W_s^2=t-s+W_s^2$$

Also

$$E[\alpha W_t|F_s]=\alpha E[W_t-W_s|F_s]+\alpha E[ W_s|F_s]=\alpha E[W_t-W_s]+\alpha W_s=\alpha W_s$$

and

$$E[\beta t+\gamma |F_s]=\beta t +\gamma$$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.