.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/probability/stochastic_processes/plot_01_brownian_motion.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_probability_stochastic_processes_plot_01_brownian_motion.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_probability_stochastic_processes_plot_01_brownian_motion.py:


Brownian motion
========================================================

Sample paths of standard Brownian motion spread like :math:`\sqrt{t}`:
at time :math:`t`, :math:`W(t) \sim \mathcal N(0, t)`, so about 95% of
paths lie within :math:`\pm 2\sqrt t`. The quadratic variation
:math:`\sum (\Delta W)^2` equals :math:`t`, the signature of a path that
is continuous but nowhere differentiable.

.. GENERATED FROM PYTHON SOURCE LINES 13-18

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.probability import brownian_motion








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Paths and the 2-sigma envelope
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-35

.. code-block:: Python


    result = brownian_motion(n_paths=2000, n_steps=1000, t_max=1.0, seed=0)
    t = result.times
    fig, ax = plt.subplots()
    ax.plot(t, result.paths[:30].T, lw=0.8)
    ax.plot(t, 2 * np.sqrt(t), "k--", t, -2 * np.sqrt(t), "k--")
    ax.set_xlabel("t")
    ax.set_ylabel("W(t)")
    ax.set_title(r"Brownian motion with $\pm 2\sqrt{t}$ envelope")

    inside = np.mean(np.abs(result.paths[:, -1]) <= 2.0)
    print(f"Var W(1) = {result.paths[:, -1].var():.4f} (theory 1)")
    print(f"fraction within 2 sd at t = 1: {inside:.4f} (theory 0.9545)")




.. image-sg:: /api/gallery/probability/stochastic_processes/images/sphx_glr_plot_01_brownian_motion_001.png
   :alt: Brownian motion with $\pm 2\sqrt{t}$ envelope
   :srcset: /api/gallery/probability/stochastic_processes/images/sphx_glr_plot_01_brownian_motion_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    Var W(1) = 1.0174 (theory 1)
    fraction within 2 sd at t = 1: 0.9560 (theory 0.9545)




.. GENERATED FROM PYTHON SOURCE LINES 36-38

Quadratic variation
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 38-42

.. code-block:: Python


    fine = brownian_motion(n_paths=1, n_steps=100000, t_max=1.0, seed=1)
    qv = np.cumsum(np.diff(fine.paths[0]) ** 2)
    print(f"quadratic variation on [0, 1]: {qv[-1]:.4f} (theory 1)")




.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    quadratic variation on [0, 1]: 0.9931 (theory 1)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.049 seconds)


.. _sphx_glr_download_api_gallery_probability_stochastic_processes_plot_01_brownian_motion.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_brownian_motion.ipynb <plot_01_brownian_motion.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_brownian_motion.py <plot_01_brownian_motion.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_brownian_motion.zip <plot_01_brownian_motion.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_