  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.
