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Questions tagged [stochastic-calculus]

Questions about stochastic calculus in Mathematica, for example how to use ItoProcess and RandomFunction.

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156 questions
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Solving a stochastic differential equation

How do I solve the following simple stochastic differential equation: $$ m x''[t] + \Gamma x'[t] + k x[t] = \sqrt[]{(2 k_{b} T/\Gamma)} \eta[t] $$ here $\eta[t]$ is Brownian motion, i.e. Wiener process. The equation describes the stochastic motion…
user1188038
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Solve a stochastic equation analytically

I have a function $\vec{F}_i(t)$, which is unknown, but I do know it's mean $\langle \vec{F}_i(t) \rangle = \vec{0}$ and it's variance $\langle \vec{F}_i(t) \cdot \vec{F}_j(t') \rangle = 2 k_B T \gamma \delta_{ij} \delta(t-t')$. I am having a…
asPlankBridge
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Stochastic Schrödinger Equation

I have a stochastic coupled Schrödinger equation to solve. $$i\frac{\mathrm d X_k(t)}{\mathrm dt}=-\left(x_{k+1}(t)+x_{k-1}(t)\right)+V_k x_k(t)+\eta_k t x_k(t)$$ where…
Kazi Ranjibul Islam
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Defining stochastic differential equations and simulating a system of three SDEs

I am trying to work on stochastic differential equations and I have been trying to use Mathematica's built-in function to simulate the system of equations below. When i use the randomfunction to simulate it using the Milstein method. I keep…
Abiy D
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How to solve a stochastic differential equation?

This is a stochastic differential equation, $$ dx(t) = -x(t)dt + e^{(-t)} dw(t)$$ I am not able to determine the next steps to solve this equation.
Vaibhav Garg
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2
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1 answer

Stochastic process, Corelation function, Numerical solution, real data

I am new in Mathematica and stochastic process too. I would like to compute (auto)correlation function from real data. So I decide try/test Mathematica script on OrnsteinUhlenbeck process, before it will be used. The script is proc1 =…
Eduard
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ItoProcess for coupled SDEs

I am trying to create an ItoProcess from the following system of SDEs: $\begin{bmatrix} \mathrm d x\\\mathrm d y\end{bmatrix} = \begin{bmatrix} 0 & 1\\ 0 & \theta\end{bmatrix} \begin{bmatrix} x\\ y\end{bmatrix} + \begin{bmatrix} \mathrm 0…
em70
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How to obtain SliceDistribution or StationaryDistribution for an ItoProcess when it is known to exist?

According to this reference page StationaryDistribution[proc] represents the stationary distribution of the process proc, when it exists. When I define the OrnsteinUhlenbeckProcess by the equivalent ItoProcess, SliceDistribution and…
Tamas
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How to reproduce the Lotka-Volterra predator-prey dynamics results?

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 provided, but when I run it, it gives me errors. Any…
David
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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 all for what I was able to find. Question 1. Is…
Maurice
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Driving a stochastic spiral by a velocity Brownian motion

I want to obtain the position x[t], solution of dx/dt=v and v the Orstein-Ulhenbeck process and use this x[t] in the cartesian coordinate definition of a stochastic spiral x_1(t)=tcos(t+x(t)) , x_2(t)=tsin(t+x(t)). I want to extract x[t] from the…
Nicolas Bian
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Passing the same random values for two stochastic processes

I have two Ito processes as shown below. I can define them separately, and simulate and plot them separately. The thing is, I do not want to apply RandomFunction to proc1 and proc2 separately. Because I need the same exact random Wiener values…
Alex
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How to reformulate differential equation problem as OrnsteinUhlenbeckProcess

I have the following differential equation m*x''[t] + k*x'[t] - randomForce[t] == 0 which describes an oscillating particle with mass m, moving in a medium with friction coefficient k and excited by a random white noise force randomForce. As I…
mrz
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Evaluating an ItoProcesss function

The following is my Stochastic D.E.: proc = ItoProcess[{\[DifferentialD]n[t] == sigma*Sqrt[(2*Um)/(Pi*L)]*\[DifferentialD]w[ t] - (\[DifferentialD]t*(Um*n[t]))/L}, n[t], {n, 1}, t, Distributed[w, WienerProcess[]]] I would like to…
user2457324
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Finding the time at which an Ito process satisfies a constraint

I want to run an Ito stochastic process. I have the following parameters b1b = 0.9; b3b = .8; a1b = 0.1; a3b = 0.2; eps = 0.1; G = (1/eps^2)*b1b ; a1 = (1/eps^2)*a1b; a3 = (1/eps^2)*a3b; xc = Sqrt[a1/a3]; Uc = a1*xc^2/2 - a3*xc^4/4 I want to stop…
Stoc
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