Risk Neutral Pricing

Question

I'm studying risk neutral pricing and there's a thing I'm not understanding... Let S be a process with the sequent dynamic $\frac{dS}{S}= \mu dt + \sigma dW^P$ where $P$ is the phisical probability.
Now, if I change the measure from $P$ to the risk neutral measure $Q$ and $r$ is the risk-free rate I obtain the dynamic $\frac{dS}{S} = rdt + \sigma dW^Q.\\$ Now if I consider the new process $S^*=e^{-rt}S$, applying Ito's formula we obtain $dS^* = S^*\sigma dW^Q$ and this means that $S^*$ is a martingale, due to the fact it has no drift. I don't understand why a Process with no drift is a Martingale...thank you very much!!!

Answer 1

A process $S$ is a martingale if $E_t\left(S_T|S_t\right)=S_t$.

Now, if $S^*$ is governed by the Stochastic Differential Equation

$$dS_t^*=\sigma S_t^*dW_t^Q$$

then

$$S_T^*=S_t+\int_t^T\sigma S_s^*dW_s^Q$$

Taking the expectation of $S^*_T$, we have

$$E_t\left(S_T^*|S_t\right)=S_t+\int_t^T\sigma E_t\left(S_s^*dW_s^Q|S_t\right)$$

whereupon using the Law of Iterated Expectation yields

$$\begin{align} E_t\left(S_T^*|S_t\right)&=S_t+\int_t^T\sigma E_t\left(S_s^*dW_s^Q|S_t\right)\\\\ &=S_t+\int_t^T\sigma E_t\left(E_s\left(S_s^*dW_s^Q|S^*_s\right)|S_t\right)\\\\ &=S_t+\int_t^T\sigma E_t\left(S_sE_s\left(dW_s^Q|S^*_s\right)|S_t\right)\\\\ &=S_t \end{align}$$

Since $dW^Q$ has zero expectation.