Questions tagged [stochastic-differential-equations]

Stochastic differential equations (SDE) occur where a system described by differential equations is influenced by random noise. Stochastic differential equations are used in finance (interest rate, stock prices, …), biology (population, epidemics, …), physics (particles in fluids, thermal noise, …), and control and signal processing (controller, filtering, …).

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11 views

differential equations with brownian motion

I have an equation like $dX(t)/dt = f(X(t),t)+\int_{0}^t dW_s$. I was wondering if there is a way to solve it (even in the simple case like $f(X(t),t) = g(X(t))h(t)$). Any hint will be much ...
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Eigenfrequency of a Duffing oscillator with variable amplitude

I am cross-posting this question here since I didn't get an answer on physics stackexchange. Say I have a duffing oscillator without a driving force or damping: $$ \ddot x + \alpha x + \beta x^3 = 0 $$...
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Derivation of unique solution to linear stochastic differential equation

I'm reading up on stochastic differential equations. The form of a linear stochastic differential equation is \begin{equation} dX_t = (AX_t + b)dt + \sigma \ dW_t ; \ X_0 = x_0 \end{equation} We can ...
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23 views

Non-Gaussian Ornstein-Uhlenbeck process and AR(1)

I am looking into generalizing the Ornstein-Uhlenbeck process to non-Gaussian noise sources. In the code, I discretized the OU process (Euler-Maryuama discretization) and converted it into an AR(1) ...
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18 views

Second Moments of linear SDEs

Consider and SDE of the form $$ dX = AX\cdot dt + BX\cdot dW_t $$ where $A,B\in\mathbb{R}^2$. How I can calculate the second moments $E[X_1^2]$ and $E[X_2^2]$ of the two variables using Ito's Formula? ...
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Show that a given solution satisfy a Stratonovich stochastic differential equation

I have been attempting to solve this problem for quite some time now and I am still stuck. I hope you guys might be able to help me out! Given any deterministic $T \geq 0$, show that the process $$X_t ...
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Path dependent Markov property

let's consider a function $\Psi\in \mathcal{C}_B(\mathcal{C}[t,T])$ continuous and bounded \begin{align*} \Psi \colon \mathcal{C}[t,T] \longrightarrow [0,+\infty) \end{align*} Then my question is:...
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solving stochastic differential equation - how to start

Can someone please help to solve the following SDE: $$ dX_t=\cos(t)\,dW_t-\tan(t)X_t\,dt, ~~ X_0=0? $$ My idea is to start in this way: $$ X_t = 0 + \int \cos(t)\,dW_t - \int \tan(t)X_t\,dt , $$ but ...
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Feynman-Kac Formula for bounded Cauchy problems

The Feynman-Kac formula is used to give a representation for solutions of partial differential equations. There are multiple versions around of this formula, for example for solutions $u(x,t)$ of $$\...
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23 views

Stationary solution of Fokker-Planck equation for several variables

I am looking for the general stationary solution of the Fokker-Planck equation in two or more dimensions. I know that for the one-dimensional case, given a drift $F(x)$ and position-dependent ...
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Condition to apply the verification theorem

I am solving a optimal control problem, the SDE is this $$ dX(t) = [X(t)(r-\beta(t))+ \theta(t)+k_1e^{\int_t^T (r-\beta(s))ds}]dt + k_2e^{\int_t^T (r-\beta(s))ds}dW(t) $$ with $ X(0)=x_0$, where $\...
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Verification that a function satisfies a PDE

Consider the problem \begin{equation} \begin{cases} \frac{\partial v^{\epsilon}(t,x)}{\partial t}= L^{\epsilon}v^{\epsilon}(t,x)+c(x)v^{\epsilon}(t,x)+g(x); \quad t>0, \: x \in \mathbb{R}^r \\ v^{\...
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Existence of solution for SDE in $L^2(\mathbb{R}^d)$.

Hello, I am looking for an existence result of solution in $L^2(\mathbb{R}^d)$ of the following kind of SDE. $$d X(t) = A(X(t))\,dt+ B(X(t))dW(t)$$, where both $A$ and $B$ are Lipschitz and $X(0)\in L^...
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References for an “approximating stochastic optimal control problems deterministically” argument

Assume you have a stochastic control problem : that is, in brief, we have a sample space $(\Omega,F,\mu)$, and a parameter space $U$ (we can take $U = [0,1]$), such that for each $u \in U$ we have a ...
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Local Continuous Integral Martingale [closed]

Let $dx\left( t\right) =f\left( x\left( t\right) \right) dt+\sigma\left( x\left( t\right) \right) dW\left( t\right) $ be a stochastic differential equation with $x\left( t\right) \in\...
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Ito's chain rule and multidimensional derivation

In the book "Quant Job Interview: Questions and Answers" by M. Joshi in the derivation of the final Black-Scholes formula he makes usage of Ito's chain rule. To get specific, he goes from $$ ...
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Mean Reverting Heston Model?

Is there a name for a variation on the Heston Stochastic Process Model where not only the underlying volatility but the asset price itself is mean-reverting? I'm looking to model long term equity ...
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1answer
49 views

Stochastic calculus: upper bound given Lipschitz drift and diffusion

I am currently stuck on the following problem. Consider the SDE $$\text{d}X_t=\sigma(X_t)\text{d}W_t+\mu(X_t)\text{d}t,$$ with $|\mu(x)|^2+|\sigma(x)|^2\leq A(1+|x|^2)$, where $A$ is a finite constant,...
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the Words 'Diffusion Model'

In ; Stochastic's, SDE, PDE, I have heard the terminology $\textit{"Diffusion Model" or "Diffusion Equation"}$. The heat equation is sometimes also called the Diffusion Equation ( since it represents ...
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1answer
30 views

Is this (stochastic) integral finite?

I am trying to prove a statement and a term comes up for which I need to show that it is finite. Let $(W_s)_{s\geq0}$ be a $\mathbb{P}$-Brownian motion. Is for any $t\geq0$ the following integral ...
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General solution of an ODE from the general solution of a PDE/SDE/SPDE and most general “differential” equation

Is it possible to obtain the general solution of an ODE by solving a PDE, SDE or SPDE? I haven't dived into PDEs, nor SDEs, therefore the question. Another question that arises from my love to ...
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Solution to viscous Hamilton-Jacobi equation can be written as fixed points

Can anyone provide me some source to read more about the fact that, solution to the viscous (https://arxiv.org/abs/2002.06674) $$ -\Delta u + H(x,Du) = \alpha_0\quad \text{in}\;\mathbb{T}^n$$ can be ...
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A kind of coefficient comparison for stochastic integrals

I have the following stochastic process, which is given by $$S_t = s_0 -aq_t, \qquad t \in [0,T], \quad T>0,$$ where $$dq_t=u_tdt+\sigma (1+z_t) W_t, \qquad q_0 \in \mathbb{R},$$ where $(u_t)_{0 \...
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1answer
50 views

How to prove that $X^\varepsilon $ is tight where $dX_t^\varepsilon =b(X_t^\varepsilon )dt+\varepsilon \sigma (X_t^\varepsilon )dW_t$?

Let $b$ and $\sigma $ nice enough so that $$dX_t=b(X_t)dt+\varepsilon \sigma (X_t)dW_t,\quad t\in [0,T]$$ has a unique strong solution. Let $X^\varepsilon =(X_t^\varepsilon )$ the strong solution. How ...
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29 views

How to compute generator of Langevin process?

The (overdamped)Langevin Equation is written as: $$dX_t = -V(X_t)dt + \sqrt{2}dW_t$$ for some function $V(X)$ and $W_t$ is the Brown motion. We can use this SDE to define a continuous Markov chain. A ...
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423 views

Asymptotic behaviour of integral. How should I proceed?

Let us consider the following SDE: $$dY_t=b(Y_t)dt+\sigma(Y_t)dW_t\tag{1}$$ with $b, \sigma: (l, r)\to\mathbb{R}$, $−\infty \leq l < r \leq \infty$ bounded functions on compact intervals of $(l, r)$...
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20 views

Brownian motion - Stratonowich or Ito

I'm coming from physics background, but question seems quite general. I think I understand the definition of both Ito and Stratonowich integral, but I still don't have an intuitive feel for the ...
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26 views

Time trend in stochastic differential equation (SDE)

(just joined, this is my first post), I've been perusing SDE’s, and a simple one is $$\mathrm{d} Y(t) = Y(t)\hspace{0.1cm}\mathrm{d}W(t)$$ where $W$ is stochastic. The solution is $$ Y = \text{exp}[W(...
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1answer
31 views

Back in time Euler–Maruyama method

Given a stochastic differential equation $$ dX(t) = a(X(t), t) \, dt + b(X(t),t) \, dW(t), \qquad (1) $$ we can solve it forward in time with Euler–Maruyama scheme for a finite time step $\Delta t$: $$...
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1answer
21 views

Kinetic Fokker-Planck Equation vs Kramers Equation

Is there any difference between the Kinetic Fokker-Planck Equation and Kramers Equation ? I have seen them both used as a name for the Kolmogorov forward equation describing the time evolution for the ...
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35 views

Predict the number of iterations of a stochastic process that solves a stochastic differential equation

Given initial parameters $T \ge 0 \in \mathbb{R}$, $S, I, R \ge 0 \in \mathbb{N} $, $ N = S+I+R$ the constants $\beta, \gamma$ and a uniform distribution $\mathcal{U}_{[0,1] in \mathbb{R}}$. I wish to ...
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33 views

Ito product rule

Stochastic Differential Equation $dS(t) = rS(t)(α − S(t))dt + σS(t)dW(t), S(0) = x.$ where W(t) is a standard Brownian motion and r, α, σ are positive constants. I have no idea why this question need ...
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About the filtering equation (Kushner-Stratonovich)

Let $$dX_t = \mu(X_t,t) dt + \sigma dB_t$$ $$dY_t = h(X_t,Y_t,t) dt + \eta dW_t$$ where $B$ and $W$ are independent standard Brownian motions, $\eta, \sigma$ are positive real numbers. What equation ...
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82 views

Is there some analytical result for Wright-Fisher SDE?

Are there already known analytical results (e.g. Laplace transform) about the Wright-Fisher SDE $$dY=(A-(A+B)Y)dt+C\sqrt{Y(1-Y)}dW$$ with $A$, $B$ and $C$ parameters? If so, could you please quote ...
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Difficulty in understanding Feller test for explosions for SDE. Any other source?

I was focusing on Feller test for explosions for a SDE like this $$dX_t=\mu(X_t)\cdot dt + \sigma^2(X_t)\cdot dW_t$$ Particularly, I was focusing on Karatzas, Shreve and attention is on exit time of ...
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1answer
30 views

Integrating $dX_t=-\beta X_t dt + dW_t$

I came across this question in the book "One Thousand Exercises in Probability" by Grimmett and Stirzaker: it is a question number 5 in section 13.8. My natural approach would be to apply ...
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Notation in SDEs : angle brackets

Can someone explain me what this notation $\langle X \rangle_t$ means and why it is used in the context of SDEs ? For example reading a book I have found this two example: Given $ \frac{dS_t}{S_t}=\...
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55 views

Integral of $W_t$ with respect to $dW_t$ by first principles

As an example, I am able to integrate $\int_{h=0}^t h dW_h$ from first principles as follows: $$ \int_{h=0}^t h dW_h = \lim_{n\to\infty}\sum_{j=0}^{n-1}h_j(W_{h_{j+1}}-W_{h_j})=\\= \lim_{n\to\infty}\...
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1answer
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How to solve $ \frac{dS(t)}{S(t-)}=\mu dt+\sigma dW(t)+d\left( \sum_{i=1}^{N(t)}\left( V_{i}-1\right) \right) $

Hey I found following Ito formula for jump diffusion process. Let $$X_{t}=X_{0}+\int_{0}^{t}b_{s}ds+\int_{0}^{t}\sigma_{s}dW_{s}+\sum_{i=1}% ^{N_{t}}\Delta X_{i},$$ where $N_t$ is Poisson process and $...
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1answer
33 views

From S.D.E to Fokker-Plank-Smoluchowski equation

Let me use as reference these lecture notes : http://wwwf.imperial.ac.uk/~pavl/lecture_notesM4A42.pdf In section 5.4 (page 59) here the author argues how the sample paths for a diffusion (from which ...
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23 views

Some Growth Conditions and Locally Lipschitz

A function on $\mathbb{R}^d$ is Locally Lipschitz if when restricted to a compact subset it is Lipshitz see : https://en.wikipedia.org/wiki/Lipschitz_continuity . I have been told that if a function $...
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1answer
50 views

The importance of approximating $\mathbb{E}^x(f(B_t)) \approx f(x)+ \frac{t}{2} f''(x) $

In order to show that the infinitesimal generator of the Brownian motion is $\frac{1}{2}\Delta$, in this answer, first he writes equation $$ \frac{d}{dt} P_t f(x) = A P_tf(x), \tag{1} $$ then he ...
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1answer
43 views

SDE of a standard Brownian motion - Langevin equation

Assume we have a standard Brownian motion $W_t$ as the solution to the following SDE $dX_t=\mu dt+\varepsilon dW_t$ Which kind of SDE it is? Ito process? For which kind of processes we can say that ...
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25 views

Is optimal stopping problem a special case of stochastic optimal control problem?

Let's say all the processes are Markov. Is optimal stopping problem a special case of stochastic optimal control problem, in the sense that, to choose an admissible stopping time in an optimal ...
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2answers
73 views

Numerical Solution of SDEs

I want to find a sample path of the following stochastic process: $$dx(t)=f(x(t))dt+g(x(t))dB(t)$$ where $B$ is the Brownian motion. Let $x_0$ be an initial condition. Can I discrete the process as ...
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16 views

Determining expectation of a coloured noise process

When you have the following colored noise process: $dM_t=-aM_t+adW_t$ (with $W_t$ a wiener process) How do you compute the following values? $E[M_t]=0$ $E[M_tM_s]=\frac{a}{2}$exp(-a|t-s|) The hint was ...
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78 views

Infinitesimal generator - Is it obtained from a stochastic process or It can help constructing the process

We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere(or manifolds) ...
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92 views

Convergence of the random walk to the Brownian motion on $S^2$

We know (see here ) that the random walk generated in $R^1$ can converge in distribution to the standard Brownian motion $B_t$ in $R^1$. Could anybody provide a rigorous mathematical proof, how a ...
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92 views

Sample paths of a diffusion are almost surely non-constant?

Consider a scalar diffusion $X=(X_t)_{t\geq 0}$ given by $\mathrm{d}X_t = b(t,X_t)\mathrm{d}t + \sigma(t,X_t)\mathrm{d}B_t, \quad X_0 = \xi$ for sufficiently regular coefficients $b$ and $\sigma$ and ...

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