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https://github.com/dsagolla/nssvie

mathematics numerical python sde stochastic stochastic-differential-equations volterra-integral-equations

Last synced: 8 days ago
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Host: GitHub
URL: https://github.com/dsagolla/nssvie
Owner: dsagolla
License: gpl-3.0
Created: 2023-11-22T19:26:58.000Z (12 months ago)
Default Branch: main
Last Pushed: 2024-06-11T13:04:28.000Z (5 months ago)
Last Synced: 2024-09-14T10:39:26.272Z (about 2 months ago)
Topics: mathematics, numerical, python, sde, stochastic, stochastic-differential-equations, volterra-integral-equations
Language: Python
Homepage:
Size: 503 KB
Stars: 0
Watchers: 1
Forks: 0
Open Issues: 0
Metadata Files:
- Readme: README.rst

Awesome Lists containing this project

README

        nssvie 

******

|tests| |build| |docs| |pypi| |pyversions| |licence| |linter|

.. |stochastic-volterra-integral-equation| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/stochastic-volterra-integral-equation.svg

.. |X-t| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/X-t.svg

.. |f| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/f.svg

.. |k-1-k-2| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/k-1-k-2.svg

.. |B-t| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/B-t.svg

.. |ito-integral| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/ito-integral.svg

.. |example-1-eq| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/example-1-eq.svg

.. |example-1-f| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main/docs/source/images/example-1-f.svg

.. |example-1-k-1| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/example-1-k-1.svg

.. |example-1-k-2| image:: https://raw.githubusercontent.com/dsagolla/nssvie/main//docs/source/images/example-1-k-2.svg

A python package for computing a numerical solution of stochastic Volterra 

integral equations of the second kind

|stochastic-volterra-integral-equation|

where

+ |X-t| is an unknown process,

+ |f| is a continuous function,

+ |k-1-k-2| are continuous and square integrable functions,

+ |B-t| is the Brownian motion (see `Wiener process `_) and

+ |ito-integral| is the Itô-integral (see `Itô calculus `_)

by a stochastic operational matrix based on block

pulse functions as suggested in `Maleknejad et. al (2012) 

`_ [1]_.

+ `Documentation `_

+ `API Reference `_

+ `Source Code `_

+ `Bug reports `_

``nssvie`` is distributed under the terms of the `GNU GPLv3 `_ license.

Install

-------

Install using either of the following two methods.

1. Install from PyPi

~~~~~~~~~~~~~~~~~~~~

|pypi| |pyversions| |format| 

The ``nssvie`` package is available on `PyPi `_ and can be installed using ``pip``

.. code-block:: bash

    pip install nssvie

2. Install from Source

~~~~~~~~~~~~~~~~~~~~~~

|release| |licence|

Install directly from the source code by

.. code-block:: bash

	git clone https://github.com/dsagolla/nssvie.git

	cd nssvie

	pip install .	

Dependencies

~~~~~~~~~~~~

``nssvie`` uses 

+ `NumPy `_  for many calculations, 

+ `SciPy `_ for computing the block pulse coefficients and

+ `stochastic `_ for sampling the Brownian Motion

Usage 

-----

Consider the following example of a stochastic Volterra integral equation

|example-1-eq|

so 

+ |example-1-f| ,   

+ |example-1-k-1| and   

+ |example-1-k-2|.

.. code-block:: python

	from nssvie import SVIE

	import matplotlib.pyplot as plt

	

	# Define the function and the kernels of the stochastic Volterra 

	# integral equation

	def f(t):

		return 1.0

	

	def k1(s,t):

		return s**2

	

	def k2(s,t):

		return s

	

	# Generate the stochastic Volterra integral equation

	svie = SVIE(

		function_f=f, kernel_2=k1, kernel_1=k2, endpoint=0.5

	)

	# Calculate numerical solution with m=20 intervals  

	approximative_solution = svie.solve_numerical(

		intervals=20

	)

	fig, ax = plt.subplots()

	times = [i * 0.5/20 for i in range(21)]

	ax.step(times, approximative_solution, c='k')

	plt.show()

.. image:: https://raw.githubusercontent.com/dsagolla/nssvie/main/docs/source/images/example.png

	:align: center

The parameters are

+ ``function_f``: the function |f|.

+ ``kernel_1``, ``kernel_2``: the kernels |k-1-k-2|.

+ ``endpoint``: the right hand side of :math:`[0,T)` (default is ``1.0``),

+ ``intervals``: the number of intervals to divide :math:`[0,T)` (default is ``50``)

for the stochastic Volterra integral equation above.

Citation

--------

.. [1] Maleknejad, K., Khodabin, M., & Rostami, M. (2012). Numerical solution of stochastic Volterra integral equations by a stochastic operational matrix based on block pulse functions. Mathematical and computer Modelling, 55(3-4), 791-800. |maleknejad-et-al-2012-doi|    

.. |licence| image:: https://img.shields.io/github/license/dsagolla/nssvie

    :target: https://www.gnu.org/licenses/gpl-3.0.en.html

.. |pypi| image:: https://img.shields.io/pypi/v/nssvie

    :target: https://pypi.org/project/nssvie

.. |release| image:: https://img.shields.io/github/v/release/dsagolla/nssvie?display_name=release&sort=semver

    :target: https://github.com/dsagolla/nssvie/releases

.. |format| image:: https://img.shields.io/pypi/format/nssvie

.. |pyversions| image:: https://img.shields.io/pypi/pyversions/nssvie

    :target: https://www.python.org/

.. |maleknejad-et-al-2012-doi| image:: https://img.shields.io/badge/DOI-10.1016%2Fj.mcm.2011.08.053-blue

    :target: https://doi.org/10.1016/j.mcm.2011.08.053

    :alt: doi: 10.1016/j.mcm.2011.08.053

.. |docs| image:: https://readthedocs.org/projects/nssvie/badge/?version=latest

    :target: https://nssvie.readthedocs.io/en/latest/?badge=latest

.. |build| image:: https://github.com/dsagolla/nssvie/actions/workflows/build.yml/badge.svg

	:target: https://github.com/dsagolla/nssvie/actions/workflows/build.yml

.. |tests| image:: https://github.com/dsagolla/nssvie/actions/workflows/tests.yml/badge.svg

	:target: https://github.com/dsagolla/nssvie/actions/workflows/tests.yml

.. |linter| image:: https://warehouse-camo.ingress.cmh1.psfhosted.org/d6d741fdb0ae96663fc5e9fbfb16b9ee24d52dfd/68747470733a2f2f696d672e736869656c64732e696f2f62616467652f6c696e74696e672d70796c696e742d79656c6c6f77677265656e

	:target: https://github.com/pylint-dev/pylint