Integration of the stochastic logistic equation via symmetry analysis

Abstract

We apply the recently developed theory of symmetry of stochastic differential equations to a stochastic version of the logistic equation, obtaining an explicit integration, i.e. an explicit formula for the process in terms of any single realization of the driving Wiener process.

Asymptotic symmetry and asymptotic solutions to Ito stochastic differential equations

“…Transforming back to the original variable x via ( 59) we obtain the solution to (57) in closed -albeit involved -form. The reader can see [75] for numerical experiments confirming the analytical procedure. ♦…”

“…Definition 7.2. (a) If the function J is such that dJ = 0 on solutions to the Ito equation (35), i.e., (75) is identically zero, we say that it is an invariant for the Ito equation (a phase space invariant if J = Ψ(x, t), and ( 78) is identically satisfied, a configurational invariant if J = Ψ(x)). (b) If this relation is valid only on the level set J = c, we say that J is a conditional invariant (a conditional phase space invariant if J = Ψ(x, t), a conditional configurational invariant if J = Ψ(x)) for the Ito equation.…”

Symmetry of the isotropic Ornstein-Uhlenbeck process in a force field

“…Symmetry of Stochastic Differential Equations (SDEs) has been studied only in relatively recent years [21][22][23][24][25][26][27][29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47]. By now, a sound formulation of the theory is available, and we know how to use symmetries to integrate -or reduce the order of -SDEs [29][30][31][32][33][34][35][36][37][38].…”

Developing a Framework for the Optimization Processes of Logistics Costs: A Hurwitz Criterion Approach

“…Logistic activities, like any other field, are associated with decision-making (Gaeta, 2019). Moreover, decisions often have to be made in conditions of uncertainty, that is, under conditions “in which execution operation is uncertain, or there is a conscious opposition of a competitor or not enough clear and precise objectives of the operations.” Of course, the presence of uncertainty in the moment of decision-making complicates the process of choosing the optimal solution because uncertainty is probably the most critical risk factor in economic activities (Wetherill & Weiss, 1962).…”

“…Logistic activities, like any other field, are associated with decision-making (Gaeta, 2019). Moreover, decisions often have to be made in conditions of uncertainty, that is, under conditions "in which execution operation is uncertain, or there is a conscious opposition of a competitor or not enough clear and precise objectives of the operations."…”