# Questions tagged [sde]

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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### Is this stochastic Picard iterator well-defined?

Preliminaries Let $x_0 \in \mathbb{R}^d$. Let $T \in (0, \infty)$. Let $$\sigma: \mathbb{R}^d \rightarrow \mathbb{R}^{d \times d}$$ and $$\mu: \mathbb{R}^d \rightarrow \mathbb{R}^{d}$$ be affine ...
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### lower bound for advection diffusion equation on the Torus

The motivation for this question comes from the Kolmogorov equation for basic SDE. Consider the PDE $$\partial_t f + u \cdot \nabla f = \Delta f.$$ posed on $\mathbb{T}^2$, where the drift $u$ is ...
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### Solving second order nonhomogeneous ODE where the RHS is a random process

Context: I'm trying to characterize the metastability behavior of a digital latch. I'm modeling two cross-coupled inverters as RC circuits with negative gain. One of the inverters has a source of ...
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### Show that the larger $c$ is the faster ${\rm d}U_t^c=\frac c2h'(U_t^c){\rm d}t+\sqrt c{\rm d}W_t$ converges to its stationary distribution

Given two Markov chains $\left(X^{(1)}_n\right)_{n\in\mathbb N_0}$ and $\left(X^{(2)}_n\right)_{n\in\mathbb N_0}$ with transition kernel $\kappa_1$ and $\kappa_2$, respectively, and a common ...
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### What is the connection between regularity structure and rough path theory?

This refers to the page https://en.wikipedia.org/wiki/Rough_path where it mentions about rough path and regularity structure as explains in the page:https://en.wikipedia.org/wiki/Regularity_structure ...
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### Exact (!) relation between Martingale Problem, SDEs and Markov processes

I am currently trying to understand the big picture/connections of Martingale Problem, Fokker-Planck-equations (although, until now, I have mostly kept these out of my considerations), SDEs and Markov ...
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### SDEs: Find the value of $\Bbb E[V(t_1)V(t_2)]$

Question: Consider the following SDE: $$V(t+\Delta t) = V(t)-V(t)\Delta t + \sqrt{\Delta t}\xi \qquad , \qquad V(0) = v_0$$ where $\xi \sim \mathcal N(0,1)$ (standard normal distribution) is ...
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### Expectation of a function of Ito diffusion

Given an Ito Diffusion i.e.: $$dX(t) = \mu dt + \sigma dW(t)$$ and a function $$k(x) = \lambda x^2$$ and I want to find the expected value $E[k(X(t)]$ of the function - the only way I know ...
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### Does solution exit for the stochastic differential equation (SDE) $d X_t = -\frac{1}{2} e^{-2 X_t} dt + e^{-X_t} dW_t$

Let $W_t$ be a standard Brownian Motion. Solve the stochastic differential equation: $$d X_t = -\frac{1}{2} e^{-2 X_t} dt + e^{-X_t} dW_t$$ Does the solution exist for all times t?
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