Questions tagged [stochastic-calculus]
Questions about stochastic calculus in Mathematica, for example how to use ItoProcess and RandomFunction.
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Is there documentation for "FeynmanKacFormula" under "ItoProcess"? [closed]
Under the command ItoProcess there are "Properties related to ItoProcess" and one is "FeynmanKacFormula". Where can I find documentation on this?...
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Difficulties in solving a system of differential equations with a matrix differential equation
I'm trying to solve a system of differential equations that looks like this:
...
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Applying NDSolveValue on a differential equation
I am trying to solve stochastic Schrodinger equation (Schrodinger equation in the presence of Ornstein Uhlenbeck Process)
$$i\frac{d}{dt}\begin{pmatrix}c_1(t)\\ c_2(t)\end{pmatrix}=H(t)\begin{pmatrix}...
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Product of two Ito processes
I am in the process of learning to use Mathematica for doing stochastic calculus.
I can do Ito's lemma by symbolic manipulation, substitutions, expansions and simplifications.
However, I feel like ...
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Identifying the parameter that flips the solution
I have solved a system of ODEs numerically. As the plot shows, the solution SolJx is positive for the parameters I have chosen. However, if you decrease (more ...
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Evaluating an integral of geometric Brownian motion
I am a Mathematica novice. For an operations research application, I am trying to work with the following wealth process in Mathematica,
$$
d W_t = (\rho W_t + P_t) dt\ ,
$$
where $P_t$ is a stream ...
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How do I simulate an ItoProcess that is not supposed to become negative?
I have a process that is similar to a Cox Ingersoll Ross process. Like a CIR process this process does not become negative. However, when I define the process with ItoProcess and I simulate it with ...
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How can I use RandomFunction to solve an SDE in reverse time?
I am working with a problem of the Ito-stochastic differential form given by
$$\mathrm{d}x=g(x,t)\,\mathrm{d}t+f(x,t)\,\mathrm{d}W,$$
where $W$ is a Wiener process.
I furthermore have to satisfy the ...
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Doing algebra with differential operators
I'm doing some work on stochastic processes, where I use random functions which are defined by their properties over averages, i.e $$\langle f(t) \rangle =0 \\ \langle f(t) f(t')\rangle = \alpha(x,y) ...
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Numerically solving a SDE with Markovian switching?
Consider the reference paper: https://www.sciencedirect.com/science/article/abs/pii/S0005109821004039
How would one go about numerically solving this system?
I tried for the one regime case:
...
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Extension of: Numerically solving a system of SDE's with Levy noise?
A great answer by Alex is to be found here for my original question: Numerically solving a system of SDE's with Levy noise?
Now Let us perturb this system with time delays so the system is:
\begin{...
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Code not getting solved by NMaximize as 1/0 is encountred
I have the following code. This code requires to solve the model with two variables to optimize the function. a1 and a2 are the variables as shown in the code
...
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Why are there problems solving this SDE system?
Following this solution by Alex: Numerically solving a system of SDE's with Levy noise?, I tried Alex's solution for a different model:
...
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Numerically solving a system of SDE's with Levy noise?
Consider this system from the following paper titled: The long-time behaviour of a stochastic SIR epidemic model with distributed delay and multidimensional L´evy jumps
https://arxiv.org/pdf/2003....
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Solving Stochastic Gross-Pitaevskii equation
I am trying to solve the Stochastic Gross-Pitaevskii equation from this paper https://arxiv.org/pdf/1409.0146.pdf. But I have no idea how to solve adding a noise term.
I like to see the wave function ...
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How does MarkovProcessProperties[] calculate the “LimitTransitionMatrix” of a stochastic matrix?
How does Mathematica the function MarkovProcessProperties[] calculate the “LimitTransitionMatrix” of a stochastic matrix using Cesaro limit of a transition matrix? What is it fundamentally doing to ...
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Stochastic process: Understanding Ornstein Uhlenbeck Process
Recently, I have been trying to simulate a random/stochastic variable that follows Gaussian distribution and also has an exponential correlation function $\left\langle X(t)X(s)\right\rangle= e^{-\frac{...
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Solving stochastic equation
I am trying to solve, numerically, a classic stochastic Liouville's equation, namely
\begin{equation}
\frac{dA(t)}{dt} = -B(t)A(t)+ {\cal C}(t)
\end{equation}
with $B(t)=\cos(\omega t)$, and ...
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Solving cubic-quintic stochastic differential equation (Duffing equation)
How do I solve the following cubic-quintic stochastic differential equation (Duffing equation)?
$$
\ddot{x} + \epsilon\gamma\delta \dot{x}-ax+bx^3+cx^5=\epsilon\gamma\delta \cos(\omega t) +\eta[t]
$$
...
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How do you output second derivatives or output the Wiener Process used in ItoProcess?
I have a system of stochastic differential equations and I use ItoProcess to solve them. The following is a snippet in which this is done.
...
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How to reproduce the Lotka-Volterra predator-prey dynamics results? [closed]
In Section 2 of this answer, the stochastic Lotka-Volterra predator-prey dynamics is demonstrated. I have difficulties to reproduce these results. At the end of the mentioned answer, a code is ...
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Simulating/solving a Langevin equation with overdamped dynamics and plotting a phase space plot
I've been reading about the Langevin equation, specifically the case where we are dealing with overdamped dynamics.
I'd like to simulate the dynamics discussed in the second link and effectively ...
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How To Fit Heston Model To Stock Data Stochastic Volatility Model
The stochastic volatility model known as the Heston model can be expressed in Mathematica in the following way
https://reference.wolfram.com/language/example/HestonModel.html
Define the Ito process of ...
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WienerProcess - estimate the expected value after `n` steps
Is there a way to adjust the estimated value of the WienerProcess after n steps?
Eg. could we evaluate the value of
...
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Wrong means given by `RandomFunction` for `StratonovichProcess`
Hello I was trying to compute mean of multiple Stratonovich integral (for $W$ standard Wiener process).
$$
J_{(1,1)} = \int_0^1 \left(\int_0^s 1\,\circ \mathrm{d}W_t\right) \circ \mathrm{d}W_s
$$
...
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Interacting Brownian particles via harmonic repulsive potential
Following this paper (DOI: 10.1103/PhysRevResearch.1.032038) I want to simulate thermally driven particles in a viscous fluid that interact via the harmonic repulsive potential in 2D, namely
$V(r_{ij}...
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I need to calculate the optimal value of a function involving integration
I am working on a capacity sharing problem which involves the function as the picture below.
I want to calculate the derivative of the function with respect to the variable $a_1.$ The profit function ...
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How to find the supremum of a brownian motion?
I imagine this is quite simple but unable to find it, if I simulate a standard brownian motion, $(B_t)_{t \geq 0}$, with
...
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Ito process with white noise
I would like to solve following system of SDEs,
dx_i[t] = f[x_i[t]]*dt + dw_i[t]
where d denotes ...
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System of stochastic differential equations
Is it possible to solve numerically system of coupled non-linear differential equations with noise?
The system looks like
...
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Kloeden–Platen–Schurz algorithm for SDE
RandomFunction's Method setting includes "Kloeden–Platen–Schurz" as a possibility. ...
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Fokker-Planck equation [closed]
Is there any package for simulation of Fokker-Planck equation in Mathematica? Or, in general for stochastic differential equations?
user81393
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Correlated random field generation
I need to generate two correlated gaussian random fields. As far as I know, this question and this other provides the means to generate a single autocorrelated process. However, I am clueless about ...
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Stochastic noise with known propability density function
How can I generate samples for random function, which functional "probability density" is known? When I say "probability density" for random function $\xi(\mathbf{q},t)$, I mean, ...
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How to simulate a non-White Gaussian process? [duplicate]
I am trying to numerically solve a second order stochastic differential equation.
Usually the noise is assumed to be Wiener Process which is white as far as I know.
Is it possible to use a Random ...
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How add noise to a differential equation?
I have a differential equation:
$$\frac{dx}{dt}=\operatorname{sech}(x-1)$$
I want to add noise to it and try to solve it numerically, but it seems that I am programming something wrong, because there ...
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How can I calculate the Allan Variance of a list of Data?
I have a list with over 10.000 elements of data. Now I wanna calculate the Allan Variance of this Measurement. The Allan Variance is defined as following:
$$\sigma_y^2(\tau)=\frac1{2\tau^2}\langle(x_{...
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Mean of an Ito Process
I would like to compute the mean of s[t] which is given by the first equation in these coupled stochastic ordinary differential equation
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Coupled stochastic differential equation [closed]
I am trying to solve the following coupled system of stochastic differential equations:
$\qquad \dot{x}(t)=y(t)\,\eta(t)$
$\qquad \dot{y}(t)=x(t)\,\eta(t)$
$x$ and $y$ are functions to be solved for ...
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Can built-in functions deal with stochastic delay differential equations (SDDE)?
I know that functions like NDSolve can deal with delay differential equations and in the meanwhile, functions like ItoProcess ...
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ItoProcess with matrix of equations
What is the cleanest/simplest way to use ItoProcess to solve the equation
$$i \text{d}\boldsymbol{\psi} = H\cdot\boldsymbol{\psi} \text{d}t + \boldsymbol{\psi}^2\...
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WhenEvent in Stochastic Differential Equation
Is there a way to add events with WhenEvent or similar when using ItoProcess?
Minimal example of my problem would be to change ...
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Why is ItoProcess failing here? (Stochastic Differential Equation) [closed]
Why is this code returning errors and failing to run? If I replace Abs[x[t]]^2 with just x[t] it works perfectly.
...
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Random number in system of equations, solving using NDSolve & WhenEvent
I need to analyse the effect of random forcing on the system of coupled equations. For example, I have shown the equations below.
$$\begin{pmatrix} x1''\\x2''\\x3''\\x4''\\x5'' \end{pmatrix}
+ M_{(5 \...
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How can I can plot a stochastic process? [closed]
How I can plot the following stochastic process using mathmatica
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TransformedProcess: a few questions about it
I have a couple of technical questions that, after searching the internet for hours, I have not been able to find an answer to. Mathematica's online instructions are not even addressing the issue at ...
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Simulating a bivariate Ito process
I would like to simulate a bivariate process where the two components, say $X_1(t) $ and $ X_2(t), $ which are related by the following stochastic differential equations:
...
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Correlated random variables
Assume we have a random variable $X(t)$ that changes as a function of time satisfying a correlation $\left\langle X(0) X(\tau) \right\rangle=e^{-\tau/\tau_c}$. Is Mathematica able to generate random ...
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Having trouble moving program from python to Mathematica
I'm trying to move a Covid-19 model programmed in python to Mathematica and I can't figure out how to translate this segment of the code to Mathematica.
...
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