Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1019with zero correlation time, modelled in terms of white noise η j , is properlyconsidered as the limit of real noise with finite correlation time (see [Gardiner(1985)]). The path integral succinctly demonstrates the differencebetween the two: The Ito prescription corresponds to the the so–calledprepoint discretization of L, whereinθṀ(t) → M ρ+1 − M ρ and M(t) → M ρ .The Stratonovich prescription corresponds to the midpoint discretization ofL, whereinθṀ(t) → M ρ+1 − M ρ and M(t) → 1 2 (M ρ+1 + M ρ ).In terms of the functions appearing in the Fokker–Planck equation (6.20),the Ito prescription of the prepoint discretized Lagrangian, L I , is relativelysimple:L I (Ṁ G , M G , t) = 1 2 (Ṁ G − g G )g GG ′(Ṁ G′ − g G′ ) − V,however, this is deceptive because of its nonstandard calculus [Ingber(1997)].Now, as L possesses a variational principle, sets of contour graphs, atdifferent long–time epochs of the path–integral of P over its variables at allintermediate times, give a visually intuitive and accurate decision–aid toview the dynamic evolution of the scenario. For example, this Lagrangianapproach permits a quantitative assessment of the following concepts usuallyonly loosely defined:‘Momentum’ = Π G = LṀ G‘Mass’ = g GG ′ = LṀ G Ṁ G′‘Force’ = F = L M G‘F = ma’ : 0 = δL = L M G − ∂ t LṀ GThese physical entities provide another form of intuitive, but quantitativelyprecise, presentation of these analyzes. For example, daily newspapers usethis terminology to discuss the movement of security prices. Π G serve ascanonical momenta indicators (CMI) for these systems [Ingber (1997)].