The capital \(X_{t}\) of an insurance company, measured in millions of dollars, follows the generalized Wiener process

\[d X_{t}=0.5 d t+2 d W_{t}\]

where unit time is one year. The insurance company goes bankrupt if capital gets negative. Find an initial capital, such that the probability that capital is negative at the end of a four-year period is no more than 5%.

Actually, bankruptcy may well occur before the four years, and we should study the distribution of hitting times, i.e., the random time at which a stochastic process assumes a given value. Since this is beyond the scope of this book, we assume that capital is only checked at the end of the time horizon.