This is a Finmath question.Problem $2$

Suppose that $S$ is the value of one share of a non$-$dividend$-$paying stock.

Let $r$ be the constant interest rate on the bank account $B_t=e^{rt}.$

Suppose that there exists a dynamic portfolio strategy, consisting of shares of the stock, and

units of the bank account $($and no other assets$),$ that satisfies the following three conditions:

First, the portfolio's total time$-t$ value is a stochastic process $L_t.$

Second, at every time$-t$ it holds

$\frac{L_t}{S_t}$

shares of stock. Here $>1$ is a constant. So we are constructing the portfolio in such a way that

a $1$ percent move in $S$ always causes a $$ percent move in the portfolio value $L.$

Third, the portfolio is self$-$financing.

$($a$)$ According to the first and second conditions, how many units of the bank account does the

portfolio hold at time $t?$

$($b$)$ Suppose that $S$ follows geometric Brownian motion

$d{S}_t=S_{t}dt+S_{t}d{W}_t$

where $$ and $>0$ are constants, and $W$ is a Brownian motion under physical probabilities.

Using the self$-$financing condition, show that $L$ is also a geometric Brownian motion.

$($Still working under physical probability measure$)$ What are its drift and volatility?

This $L$ is a model of a leveraged trading strategy with leverage ratio $.$Th