Classic Noise Processes
This section describes the available NoiseProcess
types. Note that all keyword arguments are splatted into the NoiseProcess
constructor, and thus options like reset
are available on the pre-built processes.
DiffEqNoiseProcess.WienerProcess
— FunctionThe WienerProcess
, also known as Brownian motion, or the noise in the Langevin equation, is the stationary process with white noise increments and a distribution N(0,dt)
. The constructor is:
WienerProcess(t0,W0,Z0=nothing;kwargs...)
WienerProcess!(t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.WienerProcess!
— FunctionThe WienerProcess
, also known as Brownian motion, or the noise in the Langevin equation, is the stationary process with white noise increments and a distribution N(0,dt)
. The constructor is:
WienerProcess(t0,W0,Z0=nothing;kwargs...)
WienerProcess!(t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.RealWienerProcess
— FunctionThe RealWienerProcess
is a Brownian motion that is forced to be real-valued. While the normal WienerProcess
becomes complex valued if W0
is complex, this version is real valued for when you want to, for example, solve an SDE defined by complex numbers where the noise is in the reals.
RealWienerProcess(t0,W0,Z0=nothing;kwargs...)
RealWienerProcess!(t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.RealWienerProcess!
— FunctionThe RealWienerProcess
is a Brownian motion that is forced to be real-valued. While the normal WienerProcess
becomes complex valued if W0
is complex, this version is real valued for when you want to, for example, solve an SDE defined by complex numbers where the noise is in the reals.
RealWienerProcess(t0,W0,Z0=nothing;kwargs...)
RealWienerProcess!(t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.OrnsteinUhlenbeckProcess
— Functiona Ornstein-Uhlenbeck
process, which is a Wiener process defined by the stochastic differential equation
\[dX_t = \theta (\mu - X_t) dt + \sigma dW_t\]
The OrnsteinUhlenbeckProcess
is distribution exact (meaning, not a numerical solution of the stochastic differential equation, but instead follows the exact distribution properties). The constructor is:
OrnsteinUhlenbeckProcess(Θ,μ,σ,t0,W0,Z0=nothing;kwargs...)
OrnsteinUhlenbeckProcess!(Θ,μ,σ,t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.OrnsteinUhlenbeckProcess!
— FunctionA Ornstein-Uhlenbeck
process, which is a Wiener process defined by the stochastic differential equation
\[dX_t = \theta (\mu - X_t) dt + \sigma dW_t\]
The OrnsteinUhlenbeckProcess
is distribution exact (meaning, not a numerical solution of the stochastic differential equation, but instead follows the exact distribution properties). The constructor is:
OrnsteinUhlenbeckProcess(Θ,μ,σ,t0,W0,Z0=nothing;kwargs...)
OrnsteinUhlenbeckProcess!(Θ,μ,σ,t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.GeometricBrownianMotionProcess
— FunctionA GeometricBrownianMotion
process is a Wiener process with constant drift μ
and constant diffusion σ
. I.e. this is the solution of the stochastic differential equation
\[dX_t = \mu X_t dt + \sigma X_t dW_t\]
The GeometricBrownianMotionProcess
is distribution exact (meaning, not a numerical solution of the stochastic differential equation, but instead follows the exact distribution properties). It can be back interpolated exactly as well. The constructor is:
GeometricBrownianMotionProcess(μ,σ,t0,W0,Z0=nothing;kwargs...)
GeometricBrownianMotionProcess!(μ,σ,t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.GeometricBrownianMotionProcess!
— FunctionA GeometricBrownianMotion
process is a Wiener process with constant drift μ
and constant diffusion σ
. I.e. this is the solution of the stochastic differential equation
\[dX_t = \mu X_t dt + \sigma X_t dW_t\]
The GeometricBrownianMotionProcess
is distribution exact (meaning, not a numerical solution of the stochastic differential equation, but instead follows the exact distribution properties). It can be back interpolated exactly as well. The constructor is:
GeometricBrownianMotionProcess(μ,σ,t0,W0,Z0=nothing;kwargs...)
GeometricBrownianMotionProcess!(μ,σ,t0,W0,Z0=nothing;kwargs...)
DiffEqNoiseProcess.CorrelatedWienerProcess
— FunctionOne can define a CorrelatedWienerProcess
which is a Wiener process with correlations between the Wiener processes. The constructor is:
CorrelatedWienerProcess(Γ,t0,W0,Z0=nothing;kwargs...)
CorrelatedWienerProcess!(Γ,t0,W0,Z0=nothing;kwargs...)
where Γ
is the constant covariance matrix.
DiffEqNoiseProcess.CorrelatedWienerProcess!
— FunctionOne can define a CorrelatedWienerProcess
which is a Wiener process with correlations between the Wiener processes. The constructor is:
CorrelatedWienerProcess(Γ,t0,W0,Z0=nothing;kwargs...)
CorrelatedWienerProcess!(Γ,t0,W0,Z0=nothing;kwargs...)
where Γ
is the constant covariance matrix.
DiffEqNoiseProcess.SimpleWienerProcess
— FunctionThe SimpleWienerProcess
, also known as Brownian motion, or the noise in the Langevin equation, is the stationary process with white noise increments and a distribution N(0,dt)
. The constructor is:
SimpleWienerProcess(t0,W0,Z0=nothing;kwargs...)
SimpleWienerProcess(t0,W0,Z0=nothing;kwargs...)
Unlike WienerProcess, this uses the SimpleNoiseProcess and thus does not support adaptivity, but is slightly more lightweight.
DiffEqNoiseProcess.SimpleWienerProcess!
— FunctionThe SimpleWienerProcess
, also known as Brownian motion, or the noise in the Langevin equation, is the stationary process with white noise increments and a distribution N(0,dt)
. The constructor is:
SimpleWienerProcess(t0,W0,Z0=nothing;kwargs...)
SimpleWienerProcess(t0,W0,Z0=nothing;kwargs...)
Unlike WienerProcess, this uses the SimpleNoiseProcess and thus does not support adaptivity, but is slightly more lightweight.
DiffEqNoiseProcess.CompoundPoissonProcess
— Typehttps://www.math.wisc.edu/~anderson/papers/AndPostleap.pdf Incorporating postleap checks in tau-leaping J. Chem. Phys. 128, 054103 (2008); https://doi.org/10.1063/1.2819665
DiffEqNoiseProcess.CompoundPoissonProcess!
— Typehttps://www.math.wisc.edu/~anderson/papers/AndPostleap.pdf Incorporating postleap checks in tau-leaping J. Chem. Phys. 128, 054103 (2008); https://doi.org/10.1063/1.2819665
Bridges
DiffEqNoiseProcess.BrownianBridge
— FunctionA BrownianBridge
process is a Wiener process with a pre-defined start and end value. This process is distribution exact and back be back interpolated exactly as well. The constructor is:
BrownianBridge(t0,tend,W0,Wend,Z0=nothing,Zend=nothing;kwargs...)
BrownianBridge!(t0,tend,W0,Wend,Z0=nothing,Zend=nothing;kwargs...)
where W(t0)=W₀
, W(tend)=Wend
, and likewise for the Z
process if defined.
DiffEqNoiseProcess.BrownianBridge!
— FunctionA BrownianBridge
process is a Wiener process with a pre-defined start and end value. This process is distribution exact and back be back interpolated exactly as well. The constructor is:
BrownianBridge(t0,tend,W0,Wend,Z0=nothing,Zend=nothing;kwargs...)
BrownianBridge!(t0,tend,W0,Wend,Z0=nothing,Zend=nothing;kwargs...)
where W(t0)=W₀
, W(tend)=Wend
, and likewise for the Z
process if defined.
Missing docstring for DiffEqNoiseProcess.GeometricBrownianBridge
. Check Documenter's build log for details.
Missing docstring for DiffEqNoiseProcess.GeometricBrownianBridge!
. Check Documenter's build log for details.
Missing docstring for DiffEqNoiseProcess.CompoundPoissonBridge
. Check Documenter's build log for details.
Missing docstring for DiffEqNoiseProcess.CompoundPoissonBridge!
. Check Documenter's build log for details.