nssvie#

A python package for computing a numerical solution of stochastic Volterra integral equations.

GitHub

PyPi

Theory

API Reference

Overview#

A python package for computing a numerical solution of stochastic Volterra integral equations of the second kind

(1)#\[X_t = f(t) + \int\limits_0^t k_1(s,t) X_s \ ds + \int\limits_0^t k_2(s,t) X_s \ dB_s \qquad t \in [0,T),\]

where

\(X_t\) is an unknown process,
\(f \in L^2([0,T))\) is a continuous function,
\(k_1, \ k_2 \in L^2([0,T) \times [0,T))\) are continuous and square integrable functions,
\(B_t\) is the Brownian motion (see Wiener process) and
\(\int_0^t k_2(s,t) X_s dB_s\) 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.

Install#

Install using either of the following two methods.

1. Install from PyPi#

format

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

pip install nssvie

2. Install from Source#

Install directly from the source code by

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

\[X_t = 1 + \int\limits_0^t s^2 X_s \ ds + \int\limits_0^t s X_s \ dB_s \qquad t \in [0,T),\]

so \(f \equiv 1\), \(k_1(s,t) = s^2\) and \(k_2(s,t) = s\) in (1).

from nssvie import StochasticVolterraIntegralEquations

# 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 = StochasticVolterraIntegralEquations(
        f=f, kernel_1=k1, kernel_2=k2, T=0.5
)

# Calculate numerical solution with m=50 intervals
svie_solution = svie.solve_method(m=50, solve_method="bpf")

The parameters are

f: the function \(f\).
kernel_1, kernel_2: the kernels \(k_1\) and \(k_2\).
T: the right hand side of \([0,T)\). Default is 1.0.
m: the number of intervals to divide \([0,T)\). Default is 50.
solve_method: the choosen method based on orthogonal functions. Default is bpf.

for the stochastic Volterra integral equation in (1).

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.