sdepack¶

Runge-Kutta numerical integration of scalar stochastic differential equations (SDEs) for Python.

Overview¶

sdepack is a Python library for solving scalar stochastic differential equations (SDEs) using stochastic Runge-Kutta methods. It allows you to easily and efficiently integrate SDEs driven by a Wiener process — from molecular dynamics and population genetics to option pricing and control systems.

A stochastic differential equation is a differential equation in which one or more terms are stochastic processes, producing a solution that is itself a random process.

sdepack targets scalar Itô SDEs of the general form:

where \(W(t)\) is a Wiener process (standard Brownian motion). Within the solvers, the SDE is parameterized as:

where \(F\) is the user-supplied drift callback, \(G\) is the user-supplied diffusion callback, and \(Q\) scales the noise intensity. For time-invariant routines, \(F\) and \(G\) do not depend explicitly on time.

Requirements¶

• NumPy

Example Usage¶

import numpy as np
import sdepack

# Ornstein-Uhlenbeck process: dX = -X dt + dW
x = np.zeros(1001, dtype=np.float64)
sdepack.rk4_ti_solve(
    lambda x: -x,    # drift F(X) = -X
    lambda x: 1.0,   # diffusion G(X) = 1
    x,
    0.0,             # t0
    10.0,            # tn
    1.0,             # x0
    1000,            # n steps
    1.0,             # Q (noise intensity)
    42,              # seed
)
print(x[:6])

Main Features¶

1. Seven stochastic Runge-Kutta solvers (orders 1–4).
2. Both time-invariant and time-variant SDE forms.
3. Plain Python callables for drift and diffusion — lambdas work too.
4. Seeded, fully reproducible trajectories via a built-in PRNG.
5. NumPy array output — integrates directly with matplotlib, scipy, and the scientific Python stack.

Available solvers:

References¶

• N. J. Kasdin, 1995, Runge-Kutta algorithm for the numerical integration of stochastic differential equations, Journal of Guidance, Control, and Dynamics, Vol. 18, No. 1, pp. 114–120.