(a) Let $\left(B_{t}: t \geqslant 0\right)$ be a Brownian motion and consider the process

$$Y_{t}=Y_{0} e^{\sigma B_{t}+\left(\mu-\frac{1}{2} \sigma^{2}\right) t}$$

for $Y_{0}>0$ deterministic. For which values of $\mu$ is $\left(Y_{t}: t \geqslant 0\right)$ a supermartingale? For which values of $\mu$ is $\left(Y_{t}: t \geqslant 0\right)$ a martingale? For which values of $\mu$ is $\left(1 / Y_{t}: t \geqslant 0\right)$ a martingale? Justify your answers.

(b) Assume that the riskless rates of return for Dollar investors and Euro investors are $r_{D}$ and $r_{E}$ respectively. Thus, 1 Dollar at time 0 in the bank account of a Dollar investor will grow to $e^{r_{D} t}$ Dollars at time $t$. For a Euro investor, the Dollar is a risky, tradable asset. Let $\mathbb{Q}_{E}$ be his equivalent martingale measure and assume that the EUR/USD exchange rate at time $t$, that is, the number of Euros that one Dollar will buy at time $t$, is given by

$$Y_{t}=Y_{0} e^{\sigma B_{t}+\left(\mu-\frac{1}{2} \sigma^{2}\right) t},$$

where $\left(B_{t}\right)$ is a Brownian motion under $\mathbb{Q}_{E}$. Determine $\mu$ as function of $r_{D}$ and $r_{E}$. Verify that $Y$ is a martingale if $r_{D}=r_{E}$.

(c) Let $r_{D}, r_{E}$ be as in part (b). Let now $\mathbb{Q}_{D}$ be an equivalent martingale measure for a Dollar investor and assume that the EUR/USD exchange rate at time $t$ is given by

$$Y_{t}=Y_{0} e^{\sigma B_{t}+\left(\mu-\frac{1}{2} \sigma^{2}\right) t}$$

where now $\left(B_{t}\right)$ is a Brownian motion under $\mathbb{Q}_{D}$. Determine $\mu$ as function of $r_{D}$ and $r_{E}$. Given $r_{D}=r_{E}$, check, under $\mathbb{Q}_{D}$, that is $Y$ is not a martingale but that $1 / Y$ is a martingale.

(d) Assuming still that $r_{D}=r_{E}$, rederive the final conclusion of part (c), namely the martingale property of $1 / Y$, directly from part (b).