represents a geometric Brownian motion process with drift μ, volatility σ, and initial value x0.
- GeometricBrownianMotionProcess is also known as exponential Brownian motion and Rendleman–Bartter model.
- In mathematical finance, GeometricBrownianMotionProcess is used in Black–Scholes model for stock price modeling.
- GeometricBrownianMotionProcess is a continuous-time and continuous-state random process.
- The state of a geometric Brownian motion satisfies an Ito differential equation , where follows a standard WienerProcess.
- The state follows LogNormalDistribution[(μ-) t+Log[x0],σ].
- The parameter μ can be any real number, and σ and x0 any positive real numbers.
- GeometricBrownianMotionProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examplesopen allclose all
Basic Examples (3)
Basic Uses (8)
Process Slice Properties (5)
Univariate time slice follows a LogNormalDistribution:
Multi-time slice follows a LogMultinormalDistribution:
CentralMoment has no closed form for symbolic order:
FactorialMoment has no closed form for symbolic order:
Cumulant has no closed form for symbolic order:
Generalizations & Extensions (1)
Define a transformed GeometricBrownianMotionProcess:
Properties & Relations (6)
GeometricBrownianMotionProcess is not weakly stationary:
A geometric Brownian motion process is a special ItoProcess:
As well as StratonovichProcess:
It agrees with the algorithm for simulating corresponding GeometricBrownianMotionProcess: