StochasticProcesses.jl

Numerical methods for stochastic processes

Getting started

Important

This is not a Julia package. You cannot install it with add StochasticProcesses.

Prerequisites

You will need a working installation of Julia, jupyterlab, jupytext, and IJulia to generate and run the notebooks. If you can run Julia notebooks on your machine, proceed to the next step.

• Install Julia. Using juliaup is recommended.
• Install jupyterlab and jupytext using anaconda or any other way you prefer.
• Install IJulia using Julia package manager.

Running the notebooks

• Clone/download the repository.
• Install the Julia dependencies by activating the project and then instantiating it.
• The notebooks are converted and stored under jl/ folder as plain .jl files using jupytext. To recreate the notebooks from these files run make notebooks and then move it to the base directory of the repository. You have to manually move them to avoid overwriting any notebooks you have previously generated.
• If you do not have make, you can convert them directly by running jupytext --to ipynb filename.jl.
• Now you can run jupyterlab and start running the notebooks.

Notation

General

• t: Time (usually an array of times)
• Δt: Time step
• tmax: Maximum time
• N: Number of time steps, calculated as tmax / Δt
• nens: Ensemble size (number of realizations)
• df: DataFrame
• pars: Set of parameters (usually a named tuple)

Random Processes

• W: Wiener process or Brownian motion
• X: A general random process
• Xan: Analytical value of X
• ρ: A random process that cannot be negative, such as a density

References

General references on stochastic processes

• Gardiner, Crispin W., Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences (2002).
• Van Kampen, N. G., Stochastic Processes in Physics and Chemistry (2007).
• Øksendal, B. K., Stochastic Differential Equations: An Introduction with Applications (2007).
• Van Kampen, N. G., Itô versus Stratonovich. Journal of Statistical Physics 24(1) (1981).

Numerical methods for Langevin equations

• Higham, Desmond J., An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. SIAM Review 43(3) (2001).
• Pechenik, Leonid, and Herbert Levine, Interfacial Velocity Corrections Due to Multiplicative Noise. Physical Review E 59(4) (1999).
• Dornic, Ivan, Hugues Chaté, and Miguel A. Muñoz, Integration of Langevin Equations with Multiplicative Noise and the Viability of Field Theories for Absorbing Phase Transitions. Physical Review Letters 94(10) (2005).
• Cox, S.M., and P.C. Matthews, Exponential Time Differencing for Stiff Systems. Journal of Computational Physics 176(2) (2002).

Gillespie algorithm and its applications

• Gillespie, Daniel T., A General Method for Numerically Simulating the Stochastic Time Evolution of Coupled Chemical Reactions. Journal of Computational Physics 22(4) (1976).
• Gillespie, Daniel T., Exact Stochastic Simulation of Coupled Chemical Reactions. The Journal of Physical Chemistry 81(25) (1977).
• Gillespie, Daniel T., Approximate Accelerated Stochastic Simulation of Chemically Reacting Systems. The Journal of Chemical Physics 115(4) (2001).
• Cao, Yang, Daniel T. Gillespie, and Linda R. Petzold, Efficient Step Size Selection for the Tau-Leaping Simulation Method. The Journal of Chemical Physics 124(4) (2006).
• Goldenfeld, Nigel, Tommaso Biancalani, and Farshid Jafarpour, Universal Biology and the Statistical Mechanics of Early Life. Philosophical Transactions of the Royal Society A 375(2109) (2017).