The random price \(X(t)\) of a risky security per unit at time \(t\) is

\[X(t)=5 e^{-0.01 t+B(t)+0.2|B(t)|}\]

where \(\{B(t), t \geq 0\}\) is the Brownian motion with volatility

\[\sigma=0.04\]

At time \(t=0\) a speculator acquires the right to buy the share at price \(\$ 5.1\) at any time point in the future, independently of the then current market value; i.e., the speculator owns an American call option with strike price \(x_{s}=\$ 5.1\) on the share. The speculator makes up his mind to exercise the option at that time point, when the mean difference between the actual share price \(x\) and the strike price is maximal.

(1) Is the stochastic process \(\{X(t), t \geq 0\}\) a geometric Brownian motion with drift?

(2) Is the share price on average increasing or decreasing?

(3) Determine the optimal actual share price \(x=x^{*}\).

(4) What is the probability that the investor will exercise the option?