.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/simulation/plot_brownian.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

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

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

.. _sphx_glr_auto_examples_simulation_plot_brownian.py:


Simulation of Brownian motion
=============================

.. GENERATED FROM PYTHON SOURCE LINES 6-16

.. code-block:: Python


    # Author: Steven Golovkine <steven_golovkine@icloud.com>
    # License: MIT

    # Load packages
    import numpy as np

    from FDApy.simulation import Brownian
    from FDApy.visualization import plot








.. GENERATED FROM PYTHON SOURCE LINES 17-18

The package provides a class to simulate different types of Brownian motion using the following classe: :class:`~FDApy.simulation.Brownian`. The type of Brownian motion can be standard, geometric or fractional.

.. GENERATED FROM PYTHON SOURCE LINES 18-31

.. code-block:: Python



    # Set general parameters
    rng = 42
    n_obs = 10
    argvals = np.arange(0, 1.01, 0.01)

    # Set Brownian parameters
    init_point = 1.0
    mu = 1.0
    sigma = 0.5
    hurst = 0.8








.. GENERATED FROM PYTHON SOURCE LINES 32-45

Standard Brownian motion
------------------------

A standard Brownian motion is a stochastic process define as
:math:`\{X_t\}_{t \geq 0}`. The process has the following properties:

* :math:`\{X_t\}_{t \geq 0}` is a Gaussian process.

* For :math:`s, t \geq 0`, :math:`\mathbb{E}(X_t) = 0` and :math:`\mathbb{E}(X_sX_t) = \min(s, t)`.

* The function :math:`t \rightarrow X_t` is continuous with probablity :math:`1`.

To simulate a standard Brownian motion, you can use the following code:

.. GENERATED FROM PYTHON SOURCE LINES 45-51

.. code-block:: Python

    br = Brownian(name="standard", random_state=rng)
    br.new(n_obs=n_obs, argvals=argvals, init_point=init_point)

    _ = plot(br.data)





.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_brownian_001.png
   :alt: plot brownian
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_brownian_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 52-69

Geometric Brownian motion
-------------------------

A geometric Brownian motion is a stochastic process :math:`\{X_t\}_{t \geq 0}`
in which the logarithm of the randomly varying quantity is a Brownian motion
with drift.

The process :math:`\{X_t\}_{t \geq 0}` satisfies the following stochastic
differential equation:

.. math::
      dX_t = \mu X_t dt + \sigma X_t dW_t

where :math:`\{W_t\}_{t \geq 0}` is a Brownian motion, :math:`\mu` is the
percentage drift and :math:`\sigma` is the percentage volatility.

To simulate a geometric Brownian motion, you can use the following code:

.. GENERATED FROM PYTHON SOURCE LINES 69-75

.. code-block:: Python

    br = Brownian(name="geometric", random_state=rng)
    br.new(n_obs=n_obs, argvals=argvals, init_point=init_point, mu=mu, sigma=sigma)

    _ = plot(br.data)





.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_brownian_002.png
   :alt: plot brownian
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_brownian_002.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 76-96

Fractional Brownian motion
--------------------------

A fractional Brownian motion is a stochastic process
:math:`\{X_t\}_{t \geq 0}` that generalize Brownian motion. Let
:math:`H \in (0, 1)` be the Hurst parameter. The process has the following
properties:

* :math:`\{X_t\}_{t \geq 0}` is a Gaussian process.

* For :math:`s, t \geq 0`, :math:`\mathbb{E}(X_t) = 0` and :math:`\mathbb{E}(X_sX_t) = \frac{1}{2}\left(|s|^{2H} + |t|^{2H} - |s - t|^{2H}\right)`.

* The function :math:`t \rightarrow X_t` is continuous with probablity :math:`1`.

The value of :math:`H` defines the process. If :math:`H = 1/2`, :math:`\{X_t\}
_{t \geq 0}` is a Brownian motion. If :math:`H > 1/2`, the increments of
:math:`\{X_t\}_{t \geq 0}` are positively correlated. If :math:`H < 1/2`, the
increments of :math:`\{X_t\}_{t \geq 0}` are negatively correlated.

To simulate a fractional Brownian motion, you can use the following code:

.. GENERATED FROM PYTHON SOURCE LINES 96-100

.. code-block:: Python

    br = Brownian(name="fractional", random_state=rng)
    br.new(n_obs=n_obs, argvals=argvals, hurst=hurst)

    _ = plot(br.data)



.. image-sg:: /auto_examples/simulation/images/sphx_glr_plot_brownian_003.png
   :alt: plot brownian
   :srcset: /auto_examples/simulation/images/sphx_glr_plot_brownian_003.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.279 seconds)


.. _sphx_glr_download_auto_examples_simulation_plot_brownian.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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