In this paper, we considered the different strategies that generate the optimal wealth on investment. The strategy examine depends on the utility function an investor is willing to adopt, say H* at time N in every 2n possible states; in an N period setting. Negative exponential, logarithm, square root and power utility functions were established, as the market structures changed according to a Markov chain through a martingale approach. The problem of maximization is solved via Lagrange method. The performance of the investment from day-to-day is driven by the ratio of the risk neutral probability and the probability of rising to falling.
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