Martingales of restricted model are given martingales of the complete model.

Question

Let $(\Omega, F_{t_{=\{0,...,T\}}}, P)$ be the filtrated probability space. Let $P^*$ denote the set of equivalent martingale measures for this model. Now look at the restricted model $M \le T$. Denote by $\hat{P}$ the set of equivalent martingale measures of the restricted model. Now I need to show that for every $ \hat{Q} \in \hat{P}$ there exists a $Q^* \in P^*$ such that $Q^*_{\mid F_{M}} = \hat{Q}$. Can anyone give me a hint on how to do that?

Answer 1

Let $S$ be your given asset and take $Q^* \in P^*$

Let $\hat{Q} \in \hat{P}$ be your given measure s.t. $S$ is a $\left(\hat{Q},(\mathcal{F}_t)_{t\in\{0,\ldots,M\}}\right)$-martingale.

The idea now is to construct a new measure that equals $\hat{Q}$ till time $M$ and then "use" $Q^*$ later on.

For that define $\tilde{Q}$ by $$d\tilde{Q} = \frac{d\hat{Q}}{dQ^*}\Big|_{\mathcal{F}_M} dQ^*$$

Then $S$ is a $\left(\tilde{Q},(\mathcal{F}_t)_{t\in\{0,\ldots,T\}}\right)$-martingale, so $\tilde{Q}\in P^*$ and $$\frac{d\tilde{Q}}{d\hat{Q}}\Big|_{\mathcal{F}_M} = 1$$ hence $\tilde{Q} = \hat{Q}$ on $\mathcal{F}_M$.