This article analyzes the behavior of a Brownian fluctuation process under a mixed strategic game setup. A variant of a compound Brownian motion has been newly proposed, which is called the Shifted Brownian Fluctuation Process to predict the turning points of a stochastic process. This compound process evolves until it reaches one step prior to the turning point. The Shifted Brownian Fluctuation Game has been constructed based on this new process to find the optimal moment of actions. Analytically tractable results are obtained by using the fluctuation theory and the mixed strategy game theory. The joint functional of the Shifted Brownian Fluctuation Process is targeted for transformation of the first passage time and its index. These results enable us to predict the moment of a turning point and the moment of actions to obtain the optimal payoffs of a game. This research adapts the theoretical framework to implement an autonomous trader for value assets including stocks and cybercurrencies.