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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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Discretization formula for system of differential equations. "Solution to one of these is the initial condition of the other". In which sense?

Consider the following stochastic differential equation \begin{equation} dy=\left(A-\left(A+B\right)y\right)dt+C\sqrt{y\left(1-y\right)}dW\tag{1} \end{equation} where $A$, $B$ and $C$ are parameters ...
Strictly_increasing's user avatar
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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....
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Forward vs backward formulation in Feynman-Kac

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a nice filtered probability space with an $m$-dimensional standard Brownian motion $W$. Fix a time horizon $T>0$. Let $\mu \colon [0,T] \times \mathbb{R}^d \...
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Langevin equation and convergence to stationary solutions. Free energy. SDE. FPE.

Let $f\geq 0$ be Lipschtiz. The overdamped Langevin equation \begin{equation}\label{eq overdamped Langevin SDE} dX=-\nabla f(X)dt+\sqrt{2} dW_t \end{equation} with Kolmogorov forward equation \...
orange is the new f's user avatar
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Reference request for a Riemannian Fokker-Planck equation

I am looking for any reference which states, and proves, a Fokker-Planck equation for Riemannian manifolds. In particular, if $\mathrm{d}X_t=\mu(X_t)~\mathrm{d}t + \sigma(X_t)~\mathrm{d}B_t$ is a ...
5d41402abc4's user avatar
8 votes
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Recast the scalar SPDE $du_t(Φ_t(x))=f_t(Φ_t(x))dt+∇ u_t(Φ_t(x))⋅ξ_t(Φ_t(x))dW_t$ into a SDE in an infinite dimensional function space.

Let$^1$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space $U$ be a separable Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric operator on $U$ with finite trace $(W_t)_{t\ge0}...
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7 votes
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How are Markov Kernels Related to SDEs

(Disclaimer: I've been working with SDEs for some years now but have not worked with general Markov processes before... so I'm trying to reconcile some ideas with this post.) I recently read the ...
Joe_Affine's user avatar
7 votes
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752 views

proving equalities in stochastic calculus

I am struggling with this question: FIRST PART (almost done, but stuck somewhere): Let $Z $~$ N(0,1)$ be a standard normal random variable, and define a function $F$ by the formula \begin{equation} ...
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Will solutions of SDE with different initializations intersect at some point?

Inspired by the result for ODE here, which shows that solutions to the same ODE with different initializations do not intersect, I am wondering if similar results also hold for SDE? Consider the SDE $$...
learner with 's user avatar
6 votes
1 answer
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deriving covariance of SDE from fokker-planck

In the book 1 the covariance of an SDE is derived. I am not sure about a particular step. Let me describe it in a TLDR version, then in a longer version. We have an SDE $$dx = f(x,t) dt + L(x,t) d\...
black's user avatar
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Langevin dynamics

From this paper, if the deterministic dynamics of $x_t$ is $dx_t=v_tdt$ where $v_t=\nabla\log\pi-\nabla\log\mu_t$ with $\mu_t$ denotes the law of $x_t$ and $\pi$ is a distribution depending on $x_t$, ...
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Intuition for expression of most likely trajectory of an SDE

Consider a stochastic differential equation evolving on $\mathbb R$ \begin{equation} dx_t = f(x_t)dt + c dw_t ,\quad x_0 = y \in \mathbb R \end{equation} where $f: \mathbb R \to \mathbb R, c \in \...
Lance's user avatar
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Convergence of a stochastic integral to a normal random variable

Let $X_t$ be the Ornstein-Uhlenbeck process defined by: $$ X_t = X_0 \, e^{-t} + \int_0^t e^{-(t-s)} dW_s. $$ Is it possible to show using elementary tools, in particular without using the central ...
Roberto Rastapopoulos's user avatar
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Question about a Bessel process

Are there any explicit path-wise solutions for a 3 dimensional Bessel process? E.g. the Ito SDE: $$dX_t= \frac{dt}{X_t} + dW_t, \ \ X_0 =x >0 $$ where $W_t$ is a standard Wiener process.
user48672's user avatar
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How to solve the SDE $\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t)$?

Recently, I have read Khasminskii's book (Stochastic Stability of Differential Equations). In Section 3.4, the author claims that the problem $$\mathrm{d}X(t)=-X^{3}(t)\mathrm{d}t+\mathrm{d}B(t),X(0)=...
R-CH2OH's user avatar
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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'...
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What is known about pairs of "physical" SDEs and "statistical" SDEs?

Background: Recall that a Langevin motion on a Riemannian manifold $(M, g)$ in $\mathbb{R}^D$ can be written down as the solution to the SDE in a local chart $U\subset \mathbb{R}^d$ (open) $$dY_t = [\...
Nap D. Lover's user avatar
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Girsanov from stochastic starting points.

Consider a canonical Wiener space $(\Omega,\mathscr{F},\mathbb{P})$. Consider two SDEs: $$dX_t = \alpha(X_t,t)dt+dW_t, \ \ \text{Law}(X_0)=\mu$$ and $$dY_t = \beta(Y_t,t)dt+dW_t, \ \ \text{Law}(Y_0)=\...
Rabbithawke's user avatar
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Form of invariant measures for SDsE on the toroidal domain $[0,1)^d$

Consider the SDE $${\rm d}X_t=b(X_t){\rm d}t+\sigma(X_t){\rm d}W_t\tag1$$ with Lipschitz continous $b:\mathbb R^d\to\mathbb R^d,\sigma:\mathbb R^{d\times d}\to\mathbb R$ and a $d$-dimensional Brownian ...
0xbadf00d's user avatar
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5 votes
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Ito's formula and Feynman Kac formula for path-dependent SDEs

In Rogers and Williams Volume 2, Chapter V Section 2 Subsection 8, we have introduced there the notion of a general form of SDEs where the coefficients are taken to be previsible path functionals: $\...
Nap D. Lover's user avatar
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238 views

Diffusion process and jump process

I am reading diffusion process from a textbook and noticed that the author claims that condition (1) implies the stochastic process $X_{t}$ cannot have instantaneous jumps. So I wonder does "...
Talking Puppet's user avatar
5 votes
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165 views

Radon-Nikodym derivative of pushforward measures and Girsanov theorem

Let $\mu$ and $\nu$ be two measures on a measure space $(\Omega, \Sigma)$, and $\mu$ is absolute continuous w.r.t. $\nu$. Also let $X\colon \Omega \to H$ be a measurable functions mapping to another ...
null's user avatar
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If a diffusion is Gaussian, what does it imply to its drift and volatility?

Let $(Y_t)$ be a stochastic process solution to the SDE $$dY_t= \lambda(Y_t,t) dt + \sigma(Y_t,t) dB_t. $$ If we know that $(Y_t)$ is a Gaussian process, what does it inform us on the drift $\lambda$ ...
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Estimate to arrive at $\mathbb{E}[(|X^{i}_t|+\frac{1}{N}\sum_{j=1}^N|X^j_t|)^q\mid X_0^{i}=x]\leq C x^q$

Setup We have a system of $N$ diffusion processes described by $$X_t^{i}=X_0^{i}+\int_0^t \mu(s,X_s)ds+\int_0^t\sigma(s,X_s)dW^{i}_s,$$ with the Brownian motions $W^{i}$ as well as the drift $\mu$ and ...
Leoncino's user avatar
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0 answers
358 views

Deriving Stochastic Differential Equations from Autocorrelation

Suppose I know my stationary stochastic process has the following autocorrelation function: $$ R(\tau) = \sigma^{2} e^{-\alpha |\tau|} \cos(\omega \tau) $$ How can I derive the stochastic ...
the_src_dude's user avatar
5 votes
0 answers
602 views

How can I numerically compute a stochastic integral?

I am trying simulate a to solve a 2-dimensional stochastic process and $Y_t^1$ is a mean-reverting square root process which I simulated on a time grid using its known conditional distribution. I ...
user3503589's user avatar
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208 views

Are SDE's really "differential"?

An SDE of the form $$dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dB_t$$ is really short-hand notation for an equation involving Ito integrals: $$X_t = X_0 + \int_0^t \mu(X_s,s)\,ds + \int_0^t \sigma(X_s,s)\,...
Stefan Perko's user avatar
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0 answers
705 views

Brownian motion on sphere proof?

proving the brownian motion on the sphere equation the stratonovich form differential equation $$\partial X=n(X)\times \partial B$$ the equation in ito's form becomes $$dX=n(X)\times dB+H(X)n(X)...
Nebo Alex's user avatar
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costruction of brownian motion on sphere?

i am trying to construct a brownian motion on the sphere using the method given in Price and williams paper.$\partial$ represents the SDE of stratonovich type which is converted to ito form in last ...
Nebo Alex's user avatar
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5 votes
1 answer
438 views

Continuous dependence on an initial condition (SDE)

Let's say I have a (one-dimensional) diffusion process $$dX=\mu(X_t)dt+\sigma(X_t)dW.$$ Assume we have fixed $\epsilon > 0$ and $t >0$ Under what conditions is $\mathbb{P}^x(X_t < \epsilon)$...
Trademark's user avatar
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0 answers
94 views

Feynman-Kac theorem of the weak solution of parabolic PDEs

Is there any reference on the Feynman-Kac theorem of the weak solution of parabolic PDEs? So far I can only find the one for classical solution.
mnmn1993's user avatar
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1 answer
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Estimate the norm of the (stochastic) heat equation with time-dependent diffusion coefficient

I'm considering the following (stochastic) PDE: $${\rm d}U_t=\kappa(t)\Delta U_t{\rm d}t+\sigma W_t\tag1$$ on $[0,1)^2$ with Neumann boundary conditions, where $\kappa:[0,T]\to(0,\infty)$ is linear ...
0xbadf00d's user avatar
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4 votes
0 answers
79 views

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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4 votes
1 answer
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When can the order of a Ito integral and a.s. limit be interchanged

When can the order of a Ito integral and a.s. limit be interchanged i.e. assume you got $Y_t^n\rightarrow Y_t$. When can we say $$\int\limits_0^t Y_t^n dW_s \rightarrow \int\limits_0^t Y_t dW_s$$ I ...
MackeyTopology's user avatar
4 votes
0 answers
136 views

Transform multiplicative noise to additive noise with singular matrices

I have a stochastic differential equation with multiplicative noise $\alpha(t)$ \begin{equation} \dot{\textbf{X}}=\textbf{A}\textbf{X}+\alpha(t)\textbf{B}\textbf{X}-\alpha^*(t)\textbf{B}^T\textbf{X}...
J.Agusti's user avatar
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4 votes
0 answers
129 views

Conditioning on initial condition in stochastic differential equation

Consider the following stochastic differential equation $$ \mathrm d X_t = b(X_t) \, \mathrm d t + \mathrm d W_t, \qquad X_0 = \xi, \qquad \xi \sim \mu $$ where $(W_t)_{t \geq 0}$ is a standard ...
Roberto Rastapopoulos's user avatar
4 votes
0 answers
41 views

Why does Brownian motion cluster around singularities of the potential?

Suppose I give you the potential plotted on the left (with toroidal boundaries). On the right, I've plotted the associated Gibbs measure, which is how I'd naively expect a Brownian particle to spend ...
Jesse's user avatar
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4 votes
0 answers
86 views

Is it possible to find the distribution of this nonlinear SDE?

Consider the nonlinear SDE for $(X_t)_{t\geq 0}$ \begin{equation} \mathop{dX_t}=X_t\left(\mu\mathop{dt}+\sqrt{v_0+\omega\left(\phi(t)-\ln X_t\right)^2}\mathop{dW_t}\right), \end{equation} where $\phi:...
UNOwen's user avatar
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0 answers
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Calculating expectation of product of two Cox-Ingersoll-Ross (CIR) processes

I would like to calculate expectation $\mathbb{E}\left[\nu_t\tilde{\nu}_t\right]$ of product of two Cox-Ingersoll-Ross processes $\nu_t$ and $\tilde{\nu}_t$. They are described by the following SDEs: $...
Shiningale's user avatar
4 votes
0 answers
172 views

Deriving PDE from stochastic representation formula

This is a question from Exercise 5.10 of Arbitrage Theory in Continuous Time (2009) by Tomas Bjork (rest in peace). The problem states that, Consider the following boundary value problem in the domain ...
Jerry's user avatar
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0 answers
381 views

Converting SDE to ODE via Dynkin's formula

I'm reading on Dynkin's formula. Given a stationary diffusion process $X(t)$ in the form $$dX(t) = \mu(X(t))dt +\sigma(X(t))dW$$ , as a generalization of the 2nd fundamental theorem of Calculus $F(b)-...
athos's user avatar
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4 votes
0 answers
232 views

Forward Fokker-Planck equation for stochastic differential equation with colored noise

It comes from an exam I took years ago, but until today, I still don't know the answer. Given a Langevin equation $$ \dot x = -kx + \sqrt{\Gamma}\xi, $$ where $\xi(t)$ is a colored noise satisfying $\...
Sato's user avatar
  • 185
4 votes
0 answers
91 views

Efficient numerical solution of brownian motion SDE with non-smooth diffusivity

I'm trying to numerically solve the 1-dimensional stochastic differential equation for brownian motion (Fokker-Planck) with rapidly varying diffusivity $K$, $$ \textrm{(Ito)}\quad dX_t = \frac{\...
akvilas's user avatar
  • 163
4 votes
0 answers
322 views

Ito's Lemma for Markov Chain

I'm sorry if this question may have been asked before but given the dearth of references, I doubt it. Now, here's my problem. I have this continuous function $f(t,w,x)$ which is dependent on time t, ...
Benihime Atarame's user avatar
4 votes
0 answers
579 views

Kolmogorov Backward and Forward Equations: Why are there different derivations for the forward and backward dynamics?

Dear knowledgeable people of math.stackexchange :) , In Stochastic Analysis and Diffusion Processes by Kallianpur on pages 218 to 221 the derivations for the forward and backward Kolmogorov equations ...
LudiWin's user avatar
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0 answers
76 views

Solving/Rewriting SDEs in Non-Matrix Lie Groups

I'm working on trying to solve a state estimation problem in a non-matrix Lie group. I have found some good resources for state estimation in certain matrix Lie groups. For instance, in this paper ...
Mnifldz's user avatar
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4 votes
0 answers
285 views

Independence of solution to SDE and initial condition

Let $(\Omega,F,(F_t)_{t\geq 0},P)$ be a filtered probability space with the standard condition. Let $W_t$ be the $F_t$-Wiener process, and let $(X_t)_{t\geq 0}\subset {\mathbb{R}}$ be the (strong) ...
kisten's user avatar
  • 314
4 votes
1 answer
35 views

Show that $P(B)>0$ where $B = \{\tau < T \text{ and } |X_t| \leq \theta -1 \text{ for all } 0 \leq t \leq \tau\}$ and $X$ is the sol of an Ito SDE

This question comes from a step in a proof in Mao's book on SDEs (page 120), where the author states that the below is true but he doesn't justify it. Question: Consider the SDE on $t \geq 0$ $$dX_t = ...
UBM's user avatar
  • 1,861
4 votes
1 answer
240 views

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 ...
W. Volante's user avatar
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4 votes
0 answers
695 views

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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