Python SDE Simulator with C++ Backend

This project demonstrates how to accelerate Python code by implementing a computationally intensive task (SDE simulation) in C++ and exposing it to Python using pybind11.

Setup and Installation

1. Download the repo
2. Navigate to the project directory
3. Run

pip install .

4. You can run the example script which simulates and plots a process

python ./tests/simulate.py

Usage

You can use the simulator in python by importing it. The two main functions are euler_maruyama and milstein, which solve the SDE $dX_t = a(X_t, t) dt + b(X_t, t) dB_t$ using the Euler-Maruyama and Milstein schemes, respectively.

import sde_simulation

em = sde_simulation.euler_maruyama(x0, T, n_steps, a, b, B_increments)
mil = sde_simulation.milstein(x0, T, n_steps, a, b, db_dx, B_increments)

Where x0 is the initial value, T is the time horizon, n_steps is the number of steps, a is the drift function, b is the diffusion function, db_dx is the derivative of the diffusion function with respect to x, and B_increments (optional) is the sequence of Brownian increments. If B_increments is not provided, they will be generated using numpy.random with a standard normal distribution and variance equal to T/n_steps.

The function returns a numpy array of length n_steps + 1 containing the values of the SDE at each time step.

Example:

import sde_simulation
import numpy as np

# Define the SDE parameters
x0 = 1.0
T = 1.0
n_steps = 1000

a = lambda x, t: 0.1 * x  # Drift term
b = lambda x, t: 0.2    # Diffusion term
db_dx = lambda x, t: 0.0  # Derivative of diffusion term

# Generate Brownian increments
rng = np.random.default_rng(0)
B_increments = rng.normal(0, np.sqrt(T/n_steps), n_steps)

# Run the simulations
em_path = sde_simulation.euler_maruyama(x0, T, n_steps, a, b, B_increments)
mil_path = sde_simulation.milstein(x0, T, n_steps, a, b, db_dx, B_increments)

# Print the results
print("Euler-Maruyama path:", em_path)
print("Milstein path:", mil_path)