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In this paper we
consider the problem of determining the optimal time to buy an asset in a position
of an uptrend or downtrend in the financial market and currency market as well
as other markets. Asset price is modeled as a geometric Brownian motion with
drift being a two-state Markov chain. Based on observations of asset prices,
investors want to detect the change points of price trends as accurately as
possible, so that they can make the decision to buy. Using filtering techniques
and stochastic analysis, we will develop the optimal boundary at which
investors implement their decisions when the posterior probability process
reaches a certain threshold.
To solve the selling problem which is resembled to the buying problem in , in this paper we solve the problem of determining the optimal time to sell a property in a location the drift of the asset drops from a high value to a smaller one at some random change-point. This change-point is not directly observable for the investor, but it is partially observable in the sense that it coincides with one of the jump times of some exogenous Poisson process representing external shocks, and these jump times are assumed to be observable. The asset price is modeled as a geometric Brownian motion with a drift that initially exceeds the discount rate, but with the opposite relation after an unobservable and exponentially distributed time and thus, we model the drift as a two-state Markov chain. Using filtering and martingale techniques, stochastic analysis transform measurement, we reduce the problem to a one-dimensional optimal stopping problem. We also establish the optimal boundary at which the investor should liquidate the asset when the price process hit the boundary at first time.