Question

If \(\\{Y(t), t \geqslant 0\\}\) is a Martingale, show that $$ E[Y(t)]=E[Y(0)] $$

Short answer

By applying the martingale property \(E[Y(t)|\mathcal{F}_0] = Y(0)\) and utilizing the Tower Property (Law of Iterated Expectations), we can conclude that the expected value of a martingale process \(Y(t)\) is equal to the expected value of its initial state \(Y(0)\), as shown by the equation \(E[Y(t)] = E[Y(0)]\).

Step-by-step solution

Definition of a Martingale

Recall that a continuous-time stochastic process \(\{Y(t), t \geqslant 0\}\) is a martingale if it satisfies the following conditions: 1. \(Y(t)\) is adapted to the filtration \(\{\mathcal{F}_t\}\). 2. \(E[|Y(t)|] < \infty \) for all \(t \geq 0\). 3. \(E[Y(t)|\mathcal{F}_s]=Y(s)\) for any \(0 \leq s \leq t\). In this problem, we are specifically interested in the third condition.

Apply the Martingale Property

Given that \(\{ Y(t), t \geqslant 0 \}\) is a martingale, we can use the martingale property: \(E[Y(t)|\mathcal{F}_s] = Y(s)\) for any \(0 \leq s \leq t\). We'll set \(s = 0\), since we want to relate \(E[Y(t)]\) with \(E[Y(0)]\). Therefore, we have: $$ E[Y(t)|\mathcal{F}_0] = Y(0) $$

Compute the Unconditional Expectation

Now, we'll compute the unconditional expectation, by taking the expectation of the above equation with respect to the \(\sigma\)-algebra generated by the random variable \(Y(0)\): $$ E[E[Y(t)|\mathcal{F}_0]] = E[Y(0)] $$

Apply the Tower Property

We can apply the Tower Property (also known as the Law of Iterated Expectations) to the left side of the equation: $$ E[Y(t)] = E[Y(0)] $$ So, we have proven that the expected value of \(Y(t)\) is equal to the expected value of \(Y(0)\) for the martingale \(\{Y(t), t \geqslant 0\}\).