GeometricBrownianMotion

Systems / SDEs / Noise-driven systems

Multiplicative noise — the log-normal process underlying Black–Scholes and population growth under uncertainty.

stochastic · SDE1 dimensionmultiplicative

sample path

Definition

\[ \begin{aligned} dx &= \left(\mu x\right)\,dt + \left(\sigma x\right)\,dW_{x} \end{aligned} \]

Parameters

Symbol	Default	Role
mu	0.1	drift rate
sigma	0.3	volatility

State variables: x

Properties

Lyapunov spectrum

TODO — SDE Lyapunov not computed at build time

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = \mu$
constant

Equilibria

TODO — equilibria of a stochastic system not enumerated

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.GeometricBrownianMotion()
traj = sys.integrate(final_time=100.0, dt=0.01)

Reference

Osborne (1959), Oper. Res. 7, 145-173

BibTeX

@misc{geometricbrownianmotion,
  title = {GeometricBrownianMotion system},
  note = {Osborne (1959), Oper. Res. 7, 145-173}
}