Dynamic theory of stochastic movement of systems

Nagasawa, M (2007). Dynamic theory of stochastic movement of systems. In: Jensen, B S; Palokangas, T. Stochastic economic dynamics. Frederiksberg: Copenhagen Business School Press, 133-164.

Abstract

The author studies the dynamic theory of stochastic processes. The dynamic theory concerns an evolution
equation which contains a potential function , and the diﬀusion matrix and drift vector. The case with no
potential term can be treated in the framework of the conventional theory of Markov processes of Kolmogorov
and Itˆo, which is a kinematic theory. The kinematic equation determines Markov (diﬀusion) processes, i.e.,the
movement of systems. By contrast, the author considers the equation of motion in the mechanics part of the
dynamic theory. The equation of motion contains the potential function of external forces. External forces
inﬂuence the movement of systems, but not in a direct way. The kinematic equation ﬁnally describes sample
paths of the movement of observing systems. In this paper the author clariﬁes the mathematical structures
which connect three notions, external force, induced drift vector and sample paths of the movement.

The author studies the dynamic theory of stochastic processes. The dynamic theory concerns an evolution
equation which contains a potential function , and the diﬀusion matrix and drift vector. The case with no
potential term can be treated in the framework of the conventional theory of Markov processes of Kolmogorov
and Itˆo, which is a kinematic theory. The kinematic equation determines Markov (diﬀusion) processes, i.e.,the
movement of systems. By contrast, the author considers the equation of motion in the mechanics part of the
dynamic theory. The equation of motion contains the potential function of external forces. External forces
inﬂuence the movement of systems, but not in a direct way. The kinematic equation ﬁnally describes sample
paths of the movement of observing systems. In this paper the author clariﬁes the mathematical structures
which connect three notions, external force, induced drift vector and sample paths of the movement.