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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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Why do trajectories differ when introducing a second Brownian motion in SDE Simulation? [closed]

I am working on simulating a SDE using both the direct approach and a transformed variable approach. My SDE is: \begin{equation} dX_t = \kappa X_t \, dt + \sigma \sqrt{X_t} \, dB_{1,t} - \gamma \sqrt{...
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Stochastic differential equations with only time integrals

I want to reason about the stochastic differential equation $$ dX_t = A_t X_t dt $$ Where $A_t$ is a matrix valued stochastic process, and hence $X_t$ is a vector valued stochastic process. Are there ...
rufus_lawrence's user avatar
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15 views

estimation of different $L^p$ norms [closed]

I am wondering if it is possible to find a constant $C=C(p,T)$ such that $\mathbb E[\int_0^T|Y_t|^p\mathrm{d} t]\le C(p,T) \mathbb E[\sup_{t\in [0,T]}|Y_t|^2],$ where $p>1$, $T$ some finite time ...
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1 vote
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65 views

SDEs and irreversibility in statistical mechanics

In statistical mechanics, the equations governing particle behavior are typically deterministic ordinary differential equations (ODEs). However, in real-life particle systems, these systems are never ...
Zhang Yuhan's user avatar
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33 views

Whether we can get energy estimate $d\|X\|^2+2\|X\|_{H^1}^2 dt=2(X, B dW) +{\rm Tr}(BB^*)dt$ of a weak solution of stochastic differential equations? [closed]

Set an probability space $(\Omega,{\mathcal F},P)$ with filtration ${\mathcal F}_{t,t \ge 0}$, and seperatable Hibert space $H$ and $U$ a self adjoint, sectoral,densely defined operator A on H with ...
shanlilinghuo's user avatar
2 votes
0 answers
57 views

Solution of a square root SDE

In my research I have a family of SDEs that frequently pop up and I can't find an analytical solution for them, nor can I sample from them exactly. For $a, b \in \mathbb{R}$ the SDE is \begin{equation}...
perojov's user avatar
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-1 votes
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covariance function of sub-VP SDE

For Score-Based Generative Modeling through Stochastic Differential Equations , could anyone help to derive equation (28) which is the covariance function of sub-VP SDE ? Note: I managed to ...
super_mario's user avatar
-1 votes
0 answers
13 views

Reverse-time Weiner term for Diffusion [closed]

How to derive the reverse-time Weiner term using the following 4 pictures ? Note: They are extracted from Time Reversal of Diffusions and Smoothing of a diffusion process conditionned at final time ...
super_mario's user avatar
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12 views

Is there a way to measure interdependence of steps in diffusion/reverse diffusion process? [closed]

I know that that diffusion process is Markovian in theory, in that the consecutive steps do not depend on each other. Is there a way to draw some predictions (with measurable uncertainty) for the next ...
Yeshwanth Venkatesha's user avatar
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Distributional equality between SDEs

Let $B(t)$ be a Wiener process. We know that $X^{(1)}(t) = B(t)$ and $X^{(2)}(t) = -B(t)$ both weak solutions to the SDE \begin{equation} dX(t) = dB(t), \quad X(0) = 0. \end{equation} Now let $B(t)$ ...
perojov's user avatar
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How can an Inhomogeneous Poisson Process as a model for probability have a jump size of 1?

I have the probability of an event, happening at time $t$, $\operatorname{v}\left(t\right)$ being modelled by the following equation: $$ \tau\, \frac{{\rm d}\operatorname{v}\left(t\right)}{{\rm d}t} = ...
James Stirling's user avatar
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34 views

Can a random field of Ito diffusions spend positive time in a Lebesgue measure 0 set?

Suppose I have a family of Itô diffusions governed by the following SDEs: $$dX_t(x) = b(t,x, X_t(x))dt + \sigma(t,x,X_t(x))dW_t $$ with $X_0(x) = h(x)$. Suppose $x \in \mathbb{R}^d$, $X_t(x)$ is $\...
qp212223's user avatar
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2 votes
0 answers
95 views

Brownian Motion / heat flow generated by Hodge Laplacian

Let $\square_M = - (dd^* + d^*d)$ be the Hodge Laplacian on the differential forms $\Omega(M)$ (or if you wish, on a fixed $\Omega^k(M)$). What is the stochastic process generated by this operator? ...
Alex's user avatar
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Expectation of bessel process conditioned on starting point

I want to calculate the expectation of the Bessel process for $n=3$ $$\begin{align} dX_t = dW_t + \frac{1}{X_t} dt \end{align} $$ given that we start at initial point $X_0=1$. My attempt was the ...
black's user avatar
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1 vote
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Converging PDEs as hydrodynamic limits of converging particle processes

Suppose that some interacting particle process $\{X^i_t\}_{i=1}^n$ has a hydrodynamic limit which is a PDE $$\rho_t = L \rho + f\quad \quad \quad (1) $$ where $L$ and $f$ are possibly nonlinear. By ...
900edges's user avatar
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43 views

SDE's and Frechet derivatives

I'm learning the basics of SDE's, and it's usually stated that an SDE like $$dX_t = \mu(X_t, t)dt + \sigma(X_t,t)dW_t \tag{1}$$ is an abbreviation for the stochastic integral equation $$X_{t+h} - X_t =...
user541020's user avatar
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Two SDEs that share a Brownian motion

I have the system \begin{align} dX_t & = \beta(\alpha - X_t)dt + Y_t dt + dB^1_t + dB^2_t \newline dY_t & = \beta(\alpha - Y_t)dt + X_t dt + dB^2_t + dB^3_t \end{align} Now, I am creating a ...
Pero's user avatar
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Why is the autocorrelation of an uncorrelated random noise process the dirac delta distribution?

I am reading Stochastic Methods by Gardiner and in the beginning of chapter 4 he motivates the rigorous interpretation of a Stochastic Differential equation by describing the properties of a "...
Mashe Burnedead's user avatar
1 vote
1 answer
83 views

Covariance for two correlated Ornstein Uhlenbeck processes [closed]

Given two Ornstein Uhlenbeck stochastic differential equations: $$ dX^1_t = \theta_1(X_t^1 - \mu_1)dt + \sigma_1 dW^1_t $$ $$ dX^2_t = \theta_2(X_t^2 - \mu_2)dt + \sigma_2 dW^2_t $$ Where $W^1_t$ and $...
spie227's user avatar
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5 votes
1 answer
63 views

Showing bounds of Stochastic Process

Suppose that we have the SDE: $$ dZ_t = 2Z_t(1-Z_t)dt + 4Z_t(1-Z_t)dB_t $$ With $Z_0 = \frac{1}{3}$. How can I show that $0 \leq Z_t \leq 1$. I have tried solving the 'alalogous' differential ...
Lehmann's user avatar
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3 votes
0 answers
105 views

Understanding the reverse SDE

When reading papers/blogs about diffusion models, it is stated that if $X_t$ is the unique solution to an SDE $$dX_t=f dt+g dB_t,$$ where $B_t$ is the standard Brownian motion (starting from $B_0=0$), ...
learner with 's user avatar
2 votes
1 answer
78 views

Find a PDE for $f$ satisfying $f(t,Y_t) = \exp(- \frac{\gamma^2}{2} t + \gamma W_t) E[\exp(\frac{\gamma^2}{2} T - \gamma W_T) F(Y_T) | \mathcal{F}_t]$

I am studying a course on Stochastic Calculus for Finance and am struggling with the following question: Given $dY_t = b(t,Y_t) \, dt + \sigma(t, Y_t) \, dW_t$ where $\gamma \neq 0$, and $$f(t,Y_t) = ...
FD_bfa's user avatar
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1 vote
0 answers
30 views

For $A_t= B_t^l Y_t \mathbb{E}(\frac{A_T}{B_T^l Y_T} \mid \mathcal{F}_t)$, find the values of $l$ to replicate $A_t$ by a self financing portfolio $X$

Background: In attempting to resolve the below problem, I have arrived at an answer that appears to counter intuition (and therefore, I suspect that it is wrong). I would appreciate assistance in ...
FD_bfa's user avatar
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2 votes
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30 views

Expected value of mean reverting spot price - triple numerical integral

I have been reading this paper: https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=7d38b03cfc62a15bdfd755c793d4e70a821725cc and having trouble trying to implement the expected ...
loprocto's user avatar
2 votes
0 answers
75 views

Solving a Fokker-Planck equation with discontinuous drift coefficients

Recently I'm trying to solve a Fokker-Planck equation corresponding to a piecewise-linear diffusion process with discontinuous coefficients. Namely, I'm dealing with the equation \begin{equation}\...
painday's user avatar
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1 vote
0 answers
44 views

Understanding Uniqueness of SDEs

I am a little confused about SDEs and their unique solutions. Let's say I write an SDE $dX_t = A_t dt + B_t dW_t$. Is it not the case that I can write the solution $X_t = X_0 + \int_0^t A_s ds + \...
I_cosine_this's user avatar
1 vote
0 answers
26 views

Is there a continuous time analog of linear regression for SDEs?

The ordinary linear regression $$Y_n = \alpha+\beta X_n +\epsilon_n$$ has a closed form solution for $\beta$ $$\beta = \frac{\operatorname{Cov}(X, Y)}{\sigma_X^2}.$$ Question: Is there a continuous ...
Nap D. Lover's user avatar
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1 vote
0 answers
32 views

Geometric Levy motion

Which Levy processes $(L_t)$ admit a geometric motion, i.e., an almost sure global solution of $$ dX_t = X_tdL_t\\ X_0=1 $$ (for some sensible but unspecified notion of "solution")? If the ...
Bananach's user avatar
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1 vote
2 answers
131 views

Prove that $(t+1) X_{\frac{t}{t+1}}$ is a Brownian Motion using Levy's characterisation where $X$ is a Brownian Bridge

The following result is well documented and is a result of the Stochastic Processes course I am following. Below the result, I present the standard proof presented in my course (Method 1) and my ...
FD_bfa's user avatar
  • 4,331
2 votes
1 answer
64 views

Finding $\mathbb{E}(X_t^2)$ for a SDE

I have the following SDE: $$dX_t=aX_tdt+bdW_t$$ I wanted to find $dX^2_t$ and from that I can get $d\mathbb{E}(X^2_t)$, from which i got: $$dX^2_t=(2aX^2_t+b^2)dt+2bX_tdW_t$$ And taking expectation (...
burneracc acc's user avatar
1 vote
1 answer
50 views

On the integrand of the infinite dimensional stochastic integral

Let $W$ be a $Q$-Wiener process taking values in a Hilbert space $U$, $U_0:=Q^{\frac{1}{2}}U$ be the reproducing kernel Hilbert space of $W$. In [Da prato and Zabczyk,2014], it states that the ...
George's user avatar
  • 115
3 votes
1 answer
58 views

Coupled non-linear stochastic differential equation

Given two stochastic variables $X, Z$ with the same Wiener increment $dW$, I would like to solve the following set of Itô equations: \begin{align} &dX=-\gamma X dt + \frac{g}{1+\kappa Z^2} dW\\...
J.Agusti's user avatar
  • 155
2 votes
0 answers
42 views

Linearizing/Approximating Nonlinear Stochastic Differential Equations

I'm working with a multivariate stochastic differential equation (SDE) of the form \begin{equation} \mathrm{d} \boldsymbol{x} = \boldsymbol{f}(\boldsymbol{x},t) \mathrm{d}t + \boldsymbol{G}(\...
kjc93's user avatar
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0 votes
1 answer
36 views

System of Stratonovitch SDEs $dX = \sigma X \circ dW$ to a system of Ito SDEs

I'm aware of several related stack questions, but my case is a bit different because I assume that the system of SDEs is multiplied by 1-dimensional increment $dW$. Suppose $\sigma$ is an $n \times n$ ...
MonteNero's user avatar
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0 votes
0 answers
44 views

On the definition of Gelfand triple

Gelfand triple is a useful concept in functional analysis and stochastic PDEs. In chapter 4 of [Rockner and Liu, 2015], it states that: Let $H$ be a Hilbert space, and $V$ be a reflexive Banach space ...
George's user avatar
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0 votes
1 answer
49 views

Brownian bridge satisfied SDE

I am trying to solve the following problem: given, as usual, a Brownian motion B and Brownian bridge $Y_t = a(1-t)+bt+(1-t)\int\limits_0^t\frac{\mathrm dB_s}{1-s}$, prove that it satisfies the SDE $...
Noli's user avatar
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6 votes
0 answers
52 views

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
2 votes
0 answers
27 views

Connection between SDE and Doob-Meyer decomposition

Suppose that the stochastic process $(X_t)$ is the solution of the following SDE: $$dX_t = f_t dt + h_tdW_t,$$ (where $W_t$ is the stanrdard Wiener process) so that with probability $1$, it equals to ...
Mingzhou Liu's user avatar
1 vote
0 answers
29 views

Writing $\frac{dM}{dt}(t)$ for $M(t)$ being a stochastic process.

In the paper, the equation 2.35 is an implicit form of a stochastic differential equation. Here is the excerpt from the paper: The implicit form of a stochastic differential equation is given as $$\...
MonteNero's user avatar
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5 votes
0 answers
84 views

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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4 votes
0 answers
99 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
  • 395
7 votes
1 answer
736 views

Are differentials on their own in stochastic calculus just an abuse of notation?

In stochastic calculus, it is often standard to write a DE in differential form, such as $\mathrm dY = H \, \mathrm dX$ for the stochastic integral $$\displaystyle\int\limits_0^t H \, \mathrm d X := \...
Markus Klyver's user avatar
1 vote
0 answers
37 views

ARCH-Vasicek model closed-form solution

I understand how we can obtain the solution of Vasicek model $dr_t=\alpha(\mu-r_t)dt+\sigma dW_t$: $$ r_t=r_0e^{-\alpha t}+\mu(1-e^{-\alpha t})+\sigma\int_0^te^{-\alpha(t-s)dW_{s}} $$ This easily ...
KiNest's user avatar
  • 11
1 vote
0 answers
27 views

Mean, Variance and Correlation Function of a quadratic SDE

I am struggling with the following nonlinear SDE: $ ds=dt(-\Omega s^2(t)+\alpha s(t)+\beta) + d\xi(t)(\gamma (1-s(t))) $ $ d\xi = dt(-\frac{1}{\tau} \xi(t)) + \sigma dW(t) $ Where $\alpha$, $\Omega$, ...
duodenum's user avatar
3 votes
0 answers
46 views

Canonical representation for SDEs / diffusion semimartingales on manifolds

In $\mathbb{R}^d$, there is a "canonical" expression for diffusion semimartingales, given by the SDE $d X_t = b(X_t) dt + \sigma(X_t) dB_t$ or generator $L = b \cdot \nabla + \frac{1}{2} \...
Alex's user avatar
  • 637
2 votes
0 answers
33 views

Initial Condition Dependence of an SDE

Now consider the following SDE system: $$\begin{aligned} \mathrm{d}r_t&=-\kappa(r_t-\bar{r})\mathrm{d}t+\sigma\sqrt{v_t}\mathrm{d}B_{rt}\\ \mathrm{d}v_t&=-K(v_t-\bar{v})\mathrm{d}t+\sigma_v\...
Ben's user avatar
  • 21
3 votes
1 answer
110 views

How to prove an adapted Feynman Kac Formula for $v_t + \frac{1}2 \sigma ^2 (t,y) v_{yy} + b(t,y) v_y - \delta(t,y) v + h(y) = 0$ using SDE techniques?

I am considering a generalisation of the Feynman-Kac (FK) Theorem. The traditional FK Theorem states the following: Fix a filtered probability space $(\Omega, \mathscr{F}, (\mathscr{F}_t), \mathbb{P}^...
FD_bfa's user avatar
  • 4,331
0 votes
1 answer
80 views

Solve SDE $dX_t = X_tW_tdt + dW_t,$ [closed]

I was working on question 7.5d in Applied Stochastic Analysis by by Eric Vanden-Eijnden, Tiejun Li, and Weinan E, which is given as $$dX_t = X_tW_tdt + dW_t, X_{t|t=0} = X_0, t\in \mathbb{R}_+$$ ...
HHHHHHHH's user avatar
0 votes
0 answers
32 views

Math requirements for understanding the theory of diffusion probabilistic model

I'm currently computer science major student, with concentration on Artificial Intelligence. Recently, I found that the topic of diffusion model interests me a lot and I started to learn its ...
martin-sy's user avatar
1 vote
0 answers
41 views

Approximating the Gaussian White Noise stochastic process for numerical applications

The problem I have at hand is a rather amateurish one. For a modeling task, I have worked out the stochastic differential equation the random process of interest should obey, and the gaussian white ...
Waylander's user avatar

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