Brownian motion is a central model in finance and other areas such as physics. In finance the price of one share of the risky asset, the stock, is modeled by exponential Brownian motion however by taking the log of stock prices, Brownian motion can be used as the basis of the calculations. Over a certain fixed length of time, a reasonably low risk, how much one can lose is as crucial to the success of any fund as a high profit. For this reason, investors are naturally interested in the maximum and the minimum, and also the maximum loss and maximum gain. Therefore it is a good idea to study their distributions and joint distributions. For the investors the results in this book explain the variability of the maximum and the minimum and of maximum gain and maximum loss of stock prices and in this book some useful observations and conjectures are collected on the risk and gain of stock prices.

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