Question

Set $$ p(x, t)=\frac{1}{\sqrt{t}} \exp \left(x^{2} / 2 t\right), \quad t>0 $$ Show that \(p(X(t), a+t)\) is a martingale for \(a>0\).

Short answer

To show that \(p(X(t), a+t)\) is a martingale for \(a>0\), it must fulfill the condition that for all \(s < t\), the conditional expectation \(E[p(X(t), a+t) | \mathcal{F}_s] = p(X(s), a+s)\). This is confirmed through calculating the differential of \(p(x,t)\) using Ito's Lemma and proving that there is no drift i.e., the dt term is absent.

Step-by-step solution

Recall martingale definition

Definition: A process \(X(t)\) is a martingale if for all \(s < t\), the expected value of \(X(t)\), given the information up to time \(s\) is equal to \(X(s)\). That is: \(E[X(t) | \mathcal{F}_s] = X(s)\) where \(\mathcal{F}_s\) is the sigma field generated by the process up to time \(s\). Using this, to prove that \(p(X(t), a+t)=p(x, a+t)\) is a martingale, it needs to be shown that it satisfies the martingale property.

Apply conditional expectation

Apply the conditional expectation and calculate \(E[p(X(t), a+t) | \mathcal{F}_{a}]\). We should use the fact that the function \(p(X(t), a+t)\) is known at time a, as per Doob martingale’s definition.

Apply Ito’s lemma

To compute \(dp(x,t)\) which is the differential of \(p(x,t)\), one would typically apply the Ito's lemma, which is a fundamental tool in stochastic calculus. Depending on the outcome of this calculation, one would then be able to make conclusions about the martingale property of \(p(X(t), a+t)\). Ito's lemma in its simplest form is given by \(df(X)=f'(X) dX+f''(X) d[X,X]\) where d[X,X] is quadratic variation of process X.

Decision based on computation

If on calculation, it results that the dt component is 0 and the only remaining terms are in dW (Wiener process increments). Then, \(p(X(t), a+t)\) is a martingale because martingales do not have drift terms (dt components).