JumpDiff¶
JumpDiff
is a python
library with non-parametric Nadaraya—Watson estimators to extract the parameters of jump-diffusion processes.
With JumpDiff one can extract the parameters of a jump-diffusion process from one-dimensional timeseries, employing both a kernel-density estimation method combined with a set on second-order corrections for a precise retrieval of the parameters for short timeseries.
Installation¶
To install JumpDiff
simply use
pip install JumpDiff
Then on your favourite editor just use
import JumpDiff as jd
The library depends on numpy
, scipy
, and sympy
.
Jump-diffusion processes¶
The theory¶
Jump-diffusion processes1, as the name suggest, are a mixed type of stochastic processes with a diffusive and a jump term. One form of these processes which is mathematically traceable is given by the Stochastic Differential Equation
which has four main elements: a drift term \(a(x,t)\), a diffusion term \(b(x,t)\), linked with a Wiener process \(W(t)\), a jump amplitude term \(\xi(x,t)\), which is given by a Gaussian distribution \(\mathcal{N}(0,\sigma_\xi^2)\) coupled with a jump rate \(\lambda\), which is the rate of the Poissonian jumps \(J(t)\). You can find a good review on this topic in Ref. 2.
Integrating a jump-diffusion process¶
Let us use the functions in JumpDiff
to generate a jump-difussion process, and subsequently retrieve the parameters. This is a good way to understand the usage of the integrator and the non-parametric retrieval of the parameters.
First we need to load our library. We will call it jd
import JumpDiff as jd
Let us thus define a jump-diffusion process and use jdprocess
to integrate it. Do notice here that we need the drift \(a(x,t)\) and diffusion \(b(x,t)\) as functions.
# integration time and time sampling
t_final = 10000
delta_t = 0.001
# A drift function
def a(x):
return -0.5*x
# and a (constant) diffusion term
def b(x):
return 0.75
# Now define a jump amplitude and rate
xi = 2.5
lamb = 1.75
# and simply call the integration function
X = jd.jdprocess(t_final, delta_t, a=a, b=b, xi=xi, lamb=lamb)
This will generate a jump diffusion process X
of length int(10000/0.001)
with the given parameters.
Using JumpDiff
to retrieve the parameters¶
Moments and Kramers─Moyal coefficients¶
Take the timeseries X
and use the function moments
to retrieve the conditional moments of the process.
For now let us focus on the shortest time lag, so we can best approximate the Kramers—Moyal coefficients.
For this case we can simply employ
edges, moments = jd.moments(timeseries = X)
In the array edges
are the limits of our space, and in our array moments
are recorded all 6 powers/order of our conditional moments.
Let us take a look at these before we proceed, to get acquainted with them.
We can plot the first moment with any conventional plotter, so lets use here plotly
from matplotlib
The first moment here (i.e., the first Kramers—Moyal coefficient) is given solely by the drift term that we have selected -0.5*x
And the second moment (i.e., the second Kramers—Moyal coefficient) is a mixture of both the contributions of the diffusive term \(b(x)\) and the jump terms \(\xi\) and \(\lambda\).
You have this stored in moments[2,...]
.
Other functions and options¶
Include in this package is also the Milstein scheme of integration, particularly important when the diffusion term has some spacial x
dependence. moments
can actually calculate the conditional moments for different lags, using the parameter lag
.
In formulae
the set of formulas needed to calculate the second order corrections are given (in sympy
).
Table of Content¶
Literature¶
An extensive review on the subject can be found here.
Funding¶
Helmholtz Association Initiative Energy System 2050 - A Contribution of the Research Field Energy and the grant No. VH-NG-1025 and STORM - Stochastics for Time-Space Risk Models project of the Research Council of Norway (RCN) No. 274410.