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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Regularity of the initial condition for a stochastic differential equation

We consider a continuous stochastic stochastic process $X_t$ with the following dynamic on $[0,T]$ : $$ dX_{t}^{x} = rX_{t}^{x}dt + \sigma X_{t}^{x}dB_t $$ Where $X_{0}^{x}=x$ is the initial condition,...
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Finite dimensional distribution of a stochastic process Lipschitz on every relatively compact set

Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE \begin{equation} dX_t = \mu(X_t)dt + \sigma(X_t)dW_t\,. \end{equation} Let $f(x,t|x_0,0)$ denote its transition density function. ...
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How to prove unique solvability of an SDE?

I have a stochastic differential equation of the type: $$ dX(t) = \mu(t) X(t)dt + \sigma(t) X(t) dW(t) \tag{1} $$ However, my $\mu(t)$ is a complicated function of $t$ as well as $W(t)$, somewhat like:...
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Existence and uniqueness of SDE solution in $L^p(\Omega\times [0,T])$, for $p\geq 2$

I have been studying stochastic differential equations (SDE) and came across the following questions. Let $T \in (0,\infty)$. Let also $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be any filtered ...
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partial derivative of geometric brownian motion wrt time? $S_t = S_0 e^{\mu t - \frac{1}{2}\sigma^2t + \sigma W_t}$ [closed]

If geometric brownian motion is given by: $$S_t = S_0 e^{\mu t - \frac{1}{2}\sigma^2t + \sigma W_t}$$ Then what would $\frac{\partial S_t}{\partial t}$ be?
THAT'S MY QUANT MY QUANTITATIV's user avatar
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Differentiability and continuity of the value function of optimal stopping problem

This question arise from Lemma 4.14 of Kwon, H. D., & Palczewski, J. (2022). Exit game with private information. arXiv preprint arXiv:2210.01610. Let us consider the following optimal stopping ...
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ornstein-uhlenbeck process with normal initial value stochastic filtration

Let's take an input-output stochastic system with a given noise. First of all we should define the noise process so we will have: $$d x_t=-\theta x_t d t+\sigma d W_t$$ where the $d W_t$ is a standard ...
kasra's user avatar
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How to add other constraints on boundary to enforce the pde not to be identically 0 in the domain

I need to solve a stationary density distribution of a Fokker Planck equation $-\nabla \cdot(\mu \nabla V) + \frac{1}{2}\sigma^2 \Delta\mu = 0 $ where $\mu$ is the stationary density of the SDE: $dX = ...
Eugene's user avatar
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How we define the Lie symmetry of a stochastic differential equation?

In the literature of symmetry groups, Sophus Lie define the symmetry of a pde or an ode by a vector field defined in the tangent space of the submanifold (defined by solutions of the pde or ode) and ...
Anas Cobain's user avatar
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Minimal set of condition for the Feynmann-Kac formula to hold

The Feynmann Kac formula tell us that the solution of the PDE $$ \frac{\partial u}{\partial t}(x,t) + \mu(x,t) \frac{\partial u}{\partial x}(x,t) + \tfrac{1}{2} \sigma^2(x,t) \frac{\partial^2 u}{\...
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Ornstein Uhlenbeck Stationary distribution and the Langevin equation

The Orstein-Uhlenbeck SDE $$dx = \theta (\mu-x)dt + \sqrt{2}\sigma dW_t$$ Has stationary measure given by the normal distribution $\mathcal{N}(\mu,\frac{\sigma^2}{\theta})$. The Langevin equation \...
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Ito's lemma and Stratonovich calculus

I tried converting Ito's lemma to Stratonovich form, but I got inconsistent results. Consider a 1-D SDE in both Ito and Stratonovich sense: $$ dX_t = \mu(X_t) dt + \sigma(X_t) dW_t = \bar{\mu}(X_t) dt ...
Fred's user avatar
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Marginal densities of same SDE with different initial distributions

Consider the Ornstein-Uhlenbeck SDE: $$dX_t=-\gamma_t X_t dt + \sigma_t dB_t$$ where $\gamma_t$ and $\sigma_t$ are time-varying constants independent of $X_t$. Now assume I have two versions of the ...
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Boundary condition of a PDE

I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "...
Valentin's user avatar
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If two Stratonovich SDEs are equal in distribution, do they have the same drifts?

General problem: Let $X$ and $Y$ be processes taking values in $\mathbb{R}^n$ which solve the Stratonovich SDEs $$\partial X_t = \sigma(X_t) \partial W_t$$ $$\partial Y_t = \xi(Y_t) \partial B_t,$$ ...
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Find the solution of SDE $dY_t=rdt+\alpha Y_tdB_t$

Problem: find the explicit solution of the following SDE $$dY_t=r dt+\alpha Y_t dB_t$$ with initial condition $Y_0=y_0$ and $B_t$ is a Brownian motion. Hint: consider $Z_t=\exp(-\alpha B_t+\frac{1}{2}\...
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Question about Ito's lemma

Suppose you have a continuous-time stochastic process $\textbf{X}_t$, that takes values in $\mathbb{R}^d$. Now assume that we define a second stochastic process $\textbf{Y}_t$, also in $\mathbb{R}^d$ ...
Asasuser's user avatar
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Doob-Meyer decomposition for inequalities

Consider a family of probability measures $\mathcal{P}$ such that the processes \begin{equation} X_t:=ess\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^{\mathbb{P}}[ B|\mathcal{F}_t], \end{equation} is a $\...
Don P.'s user avatar
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Infinitesimal generator for n-dimensional Geometric Brownian Motion

I'm aware that the generator for one-dimensional geomtric BM is given by: $$\mathrm{A}f(x) = \mu xf'(x)+\frac{1}{2}\sigma^2x^2f''(x),$$ but I am trying to find the general infinitesimal generator for ...
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Is usually a weak solution to a SDE (with coefficients independent of $\Omega$) adapted to the augmented natural filtration generated by the noise?

Consider an SDE $$dX_t=b(X_t)dt+\sigma(X_t)dW_t \quad X_0=x \in \mathbb R^n.$$ with $b \colon \mathbb R^n \to \mathbb R^n, \sigma \colon \mathbb R^n \to \mathbb R^{n \times q},$ $W$ is an $R^q$ valued ...
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50 views

Conditional expectation of a Ito process

Consider a standard Brownian motion $(\Omega,\mathcal{F}_t,\mathcal{F}, W_t,\mathbb{P})$. Let $dX_t = b(X_t)dt + \sigma(X_t)dW_t$ be a SDE. Assume the existence of a unique strong solution. Let $X_t(x,...
Emanuele Angilè's user avatar
2 votes
0 answers
172 views

How to solve the following inverse-time non-linear vector SDE?

In our recent studies on the diffusion-based generative models, we need to solve an inverse-time process of the diffusion model. Specifically, the inverse-time process of interest can be formulated as ...
Anton W's user avatar
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3 votes
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Estimating jump diffusion parameters from first passage time data?

I know that there is a literature on approximating the first passage time distribution of jump diffusion processes. I know there is also a literature on estimating parameters of jump diffusion ...
ThinkConnect's user avatar
2 votes
1 answer
63 views

What is the correct of the Kolomogorov backwards equation corrosponding to an Ito Stochastic Differential Equation

I am trying to solve an backwards integrated Ito Stochastic Differential equation of the form $$ dx = a dt + bdW \tag{1}$$ where dW is the random variable associated with a Wiener process. However I ...
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Product Property for Lévy Processes

The product property for random variables (RVs) [1] (Page 20) provides a way to transform symmetric $\alpha$-stable ($S\alpha S$) random variables to ones with different indices. In particular, let $X$...
Tolga Birdal's user avatar
1 vote
1 answer
101 views

nonlinear stochastic differential equation

I have problem with solving this nonlinear SDE $$dX_t=X_t(1+X^{2}_t)dt + (1+X^{2}_t)dW_t$$ So far I've tried using Ito's formula but without any success. Could you give me an advice on how to start ...
bobby shmurda's user avatar
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34 views

How would I derive the Heun scheme for stochastic differential equations?

I know that the Heun scheme takes the following form: $X^*_{n+1} = X_n + f(X_n,n)\Delta t + g(X_n, n)\Delta W_n$ $X_{n+1} = X_n + \frac{1}{2}\left( f(X_n,n) + f(X^*_{n+1},n+1) \right)\Delta t + \frac{...
abcdefg123's user avatar
2 votes
0 answers
77 views

Brownian motion on product manifolds with the product metric

I recently stumbled upon a proof while reading a paper in the field of my study. It says: \begin{gather*} For\ any\ t >0,\ let\ B_{SE( 3)}^{( t)} :=\left[ B_{SO( 3)}^{( t)} ,\ B_{R^{3}}^{( t)}\...
Lee Youngjun's user avatar
2 votes
0 answers
52 views

How to estimate the difference between two Ito diffusions?

Suppose $b : \mathbb{R}^d\to \mathbb{R}^d$, $\sigma: \mathbb{R}^d \to \mathbb{R}^{d\times d}$ are measurable functions and satisfy $$ 2\langle x-y, b(x)-b(y)\rangle + \|\sigma(x) - \sigma(y)\|^2 \le K|...
epsilon's user avatar
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2 votes
1 answer
44 views

Re the Fokker-Planck equation: How to reconcile boundary conditions for reflecting barriers with arbitrary initial conditions?

I'm learning about boundary conditions for the Fokker-Planck equation and am misunderstanding something fundamental. In one dimension, the Fokker-Planck equation is $$\partial_t f(x,t) = - \partial_x ...
Stig's user avatar
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1 vote
1 answer
53 views

Solving a two-dimensional stochastic differential equation [duplicate]

I'm attempting to solve the following two-dimensional stochastic differential equation: $$ dX_1(t) = \big(\alpha X_2(t) - \theta_1 X_1(t)\big)dt + \sigma_1 dW_1(t), $$ $$ dX_2(t) = - \theta_2 X_2(t)dt ...
WilliamMorris's user avatar
5 votes
0 answers
110 views

Ornstein Uhlenbeck velocity with displacement-based drift

Apologies, I'm not very good at stochastic calculus so I'll ask the below as best I can..! I have a particle I am trying to model which bounces around the origin, but which has a momentum component. I'...
SRB121's user avatar
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2 votes
1 answer
41 views

Stratonovich SDE and main hint for Ito's lemma

I've stuck on a SDE problem. Namely, I've got to solve equation $$ \text{d}X_t = e^{-X_t}\circ\text{d}B_t. $$ So in order to apply Ito's lemma, I transformed this SDE into form $$ \text{d}X_t = -\frac{...
Maciej Ostapiuk's user avatar
8 votes
0 answers
209 views

Why the renormalization constant of the regularized $2D$ noise diverges as a logarithm

Motivation and overview I'm trying to understand the theory of regularity structures and in particular, following this paper I'm looking to the $\Phi^4_d$ model on the $d$-dimensional torus $T^d$ i....
Marco's user avatar
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2 votes
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Generator of the Brownian motion constructed using a time change

In example 8.5.8 of Oksendal text book, the Brownian motion is constructed on the unit sphere by defining time change $Z_t(\omega) = Y_{a(t,\omega)}(\omega)$ and $$ \alpha_t=\beta_t^{-1}\,,\quad \...
Kevin's user avatar
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2 votes
0 answers
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Generate random numbers using SDE of the Legendre process [duplicate]

Assume I know the SDE of a stochastic process as: $$ d\theta(t) = \frac{\sigma^2 cot(\theta(t))}{2 }dt + \sigma dB(t). $$ What are the steps to generate random angles (random walks) corresponding to ...
K252's user avatar
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2 votes
1 answer
70 views

Expected Value of Ito process

I am trying to find the 3rd moment of the a process $X_{t}$ that satisfies the following differential equation: $$dX_{t} = (AX_{t} + a)dt + (BX_{t} + b)dWt$$ where A,a,B,b are constants. I came across ...
Ethan Davitt's user avatar
5 votes
2 answers
128 views

A SDE that "forgets" its input

Let $W$ be the Wiener process. Consider the coupled SDE $$ dX_t = - X_t dt + \sqrt{2} dW_t$$ $$ d\bar W_t = -\sqrt{2} X_t dt + dW_t$$ with initial conditions $\bar W_0 = 0$ and $X_0 \sim N(0, 1)$. The ...
Mark's user avatar
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1 vote
1 answer
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Validity of infinitesimal manipulations in Ito SDEs

Suppose I have the Ito SDE $$dX_t = -\theta X_t dt + \sigma dW_t$$ with some initial probability distribution $X_0 \sim p_{X_0}$. Then the process $X$ evolves as an Ornstein Uhlenbeck process. Since $...
Mark's user avatar
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2 votes
1 answer
86 views

Continuous dependence of Cox-Ingersoll-Ross (CIR) process on its initial value and model parameters

Suppose we are considering a Cox-Ingersoll-Ross (CIR) process $$\mathrm{d}\nu_t=\kappa\left(\theta-\nu_t\right)\mathrm{d}t+\sigma\sqrt{\nu_t}\mathrm{d}W_t,$$ with deterministic initial value $\nu_0>...
Shiningale's user avatar
1 vote
0 answers
42 views

Solving an SDE - tips and tricks [duplicate]

Consider the stochastic differential equation $$ dX_t = \ln(1 + X^2_t)\space dt + X^+_tdB_t $$ with $X_0 = a$ and $x^+ = \max({x, 0})$ is the positive part of real number $x$, and $a$ is a real ...
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2 votes
1 answer
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On a stochastic differential equation

Consider the SDE $dX_t = \sqrt{1 -X_t^2}\;1_{|X_t| \leq 1}\;dW_t$ with $X_0 =0$. Lemma 6 of "Noise stability on the Boolean hypercube via a renormalized Brownian motion (Eldan, Mikulincer, ...
Mathews Boban's user avatar
2 votes
0 answers
56 views

How can Brownian motion be a function of space only?

Typically, we see the notation $W(t)$ which represents an $\mathbb{R}^n$-dimensional Brownian motion (BM). What about $W(x)$? I've seen this in some papers where it is considered a space-dependent BM. ...
900edges's user avatar
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2 votes
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Spatial discretization of Brownian motion/Wiener process

Consider a spatio-temporal Brownian motion/Wiener process $dW(x,t)$ for $x\in [0,1]$ and $t\geq 0$. I want to come up with an "$n$-sized" discretization of the form $dW_n(x_i,t)$ for $i=1,\...
900edges's user avatar
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1 vote
1 answer
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Exploring Regularity of Solutions in Stochastic Differential Equations

On our platform, we delve into the analysis of stochastic differential equations (SDEs) with a focus on understanding the regularity of solutions. Allow me to introduce a scenario involving a $q$-...
Oussama's user avatar
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4 votes
2 answers
238 views

Converting an Ito integral into a Backward Ito Integral

In "Reverse Time Diffusion Equation Models", Brian D.O. Anderson begins with a multidimensional Ito SDE $$ dx_t = f(x_t,t) dt + g(x_t, t) dw_t,$$ with some initial condition $p(x_0, t_0)$. ...
Mark's user avatar
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3 votes
0 answers
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Understanding a proof of the covariance matrix of a random process in SDE

Considering the following SDE with multiplicative as well as additive stochastic uncertainty, $$ \dot{x}=A x+\sum_{\ell=1}^m \sigma_{\ell} B_{\ell} x \xi_{\ell}+H \eta $$ where $x \in \mathbb{R}^n, H \...
yuqi Xu's user avatar
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What is the meaning of $dX(t)$ when $X(t)$ is a random process?

In the book about Stochastic Differential Equations, the term $dX(t)$ appears everywhere and I have a question about the meaning of it. Because for every $t\in [0,T]$, $X(t)$ is a random variable ...
Gang men's user avatar
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6 votes
1 answer
85 views

Why does the Yamada-Watanabe Uniqueness Theorem for SDEs not hold in the multidimensional case?

Consider the stochastic differential equation $$ dX_t = b(t,X_t)\,dt + \sigma(t,X_t) \, dB_t, \quad t \geq 0. $$ where $b:[0, \infty) \times \mathbb{R}^n \to \mathbb{R}^n$, $\sigma: [0,\infty) \times \...
Alex's user avatar
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-1 votes
1 answer
98 views

How to simulate from $dY_i=Y_i( \mu dt$ $+ \sigma_{(2)}( \alpha dB^{(1)}_i + \sqrt{1- \alpha ^2} dB^{(2)}_i))$ for $\mu, \sigma>0, \alpha \in [-1,1]$?

I am studying numerical methods from the textbook Monte Carlo Methods in Financial Engineering by Paul Glasserman and have encountered the following exercise: I want to simulate from the stochastic ...
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