Ivancevic_Applied-Diff-Geom

1036 Applied Differential Geometry: A Modern IntroductionThis means that, apart from volatility effects, the price z i at time t i willhave a value remarkably close to the expected value ¯z = z i−1 + A∆t, givenby the drift growth. In order to take care of the volatility effects, a possiblesolution is to estimate the integral of interest, i.e.,E[O i |S i−1 ] =∫ +∞−∞dz p(z|z i−1 )O i (e z ), (6.60)by inserting in (6.60) the analytical expression for the p(z|z i−1 ) transitionprobability{1p(z|z i−1 ) = √ exp − (z − z i−1 − A∆t) 2 }2π∆tσ2 2σ 2 ∆t{ }1= √ exp (z − ¯z)2−2π∆tσ2 2σ 2 ,∆ttogether with a Taylor expansion of the kernel function O i (e z ) = f(z)around the expected value ¯z. Hence, up to the second–order in z − ¯z, thekernel function becomeswhich inducesf(z) = f(¯z) + (z − ¯z)f ′ (¯z) + 1 2 f ′′ (¯z)(z − ¯z) 2 + O((z − ¯z) 3 ),E[O i |S i−1 ] = f(¯z) + σ22 f ′′ (¯z), + . . . ,since the first derivative does not give contribution to (6.60), being theintegral of an odd function over the whole z range. The second derivativecan be numerically estimated asf ′′ (¯z) = 1δ 2 [f(¯z + δ σ ) − 2f(¯z) + f(¯z − δ σ )],σwith δ σ = O(σ √ ∆t), as dictated by the dynamics of the stochastic process.6.3.7 Application: Nonlinear Dynamics of Complex NetsRecall that many systems in nature, such as neural nets, food webs,metabolic systems, co–authorship of papers, the worldwide web, etc. canbe represented as complex networks, or small–world networks (see, e.g.,[Watts and Strogatz (1998); Dorogovtsev and Mendes (2003)]). In particular,it has been recognized that many networks have scale–free topology;the distribution of the degree obeys the power law, P (k) ∼ k −γ . The study