In this paper, we study the optimal level of cash for the firm to hold. We model\nthe cash level with inflows and outflows due to deposits and withdrawals; in\nbetween, the cash level is a stochastic process where it signals a time to sell.\nAfter modeling the continuous jump, we implemented first step analysis method\nto find the probability of the event with initial cash and we were able to\ncalculate data driven by set of difference equations. These data are used to determine\nthe length of the period of the investment. Then, we adopt the probabilistic\ndecision model where it goes under mathematical optimization. This\nmodel let the investor to maximize the probability of success or to stop on one\nof the largest fortunes using the equation of the principle of optimality. Finally,\nto solve these optimal equations, we used the result of positive dynamic\nprogramming and we elaborated them by proves.
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