The price of a unit of a share at time point \(t\) is \(X(t)=10 e^{D(t)}, t \geq 0\), where \(\{D(t), t \geq 0\}\) is a Brownian motion process with drift parameter \(\mu=-0.01\) and volatility \(\sigma=0.1\). At time \(t=0\) a speculator acquires an option, which gives him the right to buy a unit of the share at strike price \(x_{s}=10.5\) at any time point in the future, independently of the then current market value. It is assumed that this option has no expiry date. Although the drift parameter is negative, the investor hopes to profit from random fluctuations of the share price. He makes up his mind to exercise the option at that time point, when the expected difference between the actual share price \(x\) and the strike price \(x_{s}\) is maximal.

(1) What is the initial price of a unit of the share?

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

(3) Determine the corresponding share price which maximizes the expected profit of the speculator.