# 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: 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 ...
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### 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 ...
1 vote
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### 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 ...
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### 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 ...
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 ...
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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 ...
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### 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 ...
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### 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 ...
32 views

### 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 $$dX(t) = dB(t), \quad X(0) = 0.$$ Now let $B(t)$ ...
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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 ...
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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 "...
1 vote
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### 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 $$dX_t = X_tW_tdt + dW_t, X_{t|t=0} = X_0, t\in \mathbb{R}_+$$ ...