# 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 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 ...
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### Are these conditions sufficient for the process to be continuous?

The text book I’m working on considers the following SDE: $dX(t) = \mu(t)dt + \sigma(t)dW(t)$, where $\mu$ is defined to be a càdlàg predictable and finite variation process, while $\sigma$ is a ...
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### Pushforward of Gaussian Measure by Solution Operator to SDE

Consider an n-dimensional Ito process $$X_t^x = x + \int_0^t\, \alpha(s)ds + \int_0^t\,\beta(s)\,dW(s)$$ driven by an n-dimensional Brownian motion; where $\alpha,\beta$ are Lipschitz functions with ...
1 vote
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### Verify Langevin equation

Consider the Langevin equation: $$dX_t = -bX_tdt +adB_t, ~~~ X_0 = x_0,$$ where $a,b>0$. We know that the solution is $$X_t = e^{-bt}x_0 + ae^{-bt} \int_0^t e^{bs} dB_s.$$ Now I want to verify the ...
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### Calculating the unconditional expectation of an Ito process

Suppose that: $$dX_t =\mu(t,X_t)dt+\sigma(t,X_t)dW_t,$$ where $X_t$ is a vector valued stochastic process, $W_t$ is a vector of Brownian motions, $\mu$ is a vector valued function and $\sigma$ is a ...
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### Reference for Mckean-Vlasov stochastic differential equations

Could someone please provide me with some reference books on nonlinear stochastic differential equations in the sense of McKean-Vlasov, the propagation of chaos in corresponding systems of particles, ...
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### Is this stochastic equation numerically solvable?

My knowledge of Stochastic ODEs is poor at best, but I have a problem I need to solve. I've boiled it down to a more straightforward scaler form. $dX_t = e^{-X_t}X_t+dW_t e^{-X_t}$ I was wondering if ...
1 vote
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### Optimal filtering with changing variance

I have seen on some books (e.g. Lipster and Shiryayev (1977)) some concepts from optimal filtering. One idea is that, given a hidden state $\theta$ (say time invariant and taking finitely many values) ...
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### The Riemannian metric induced by an SDE

Following this paper, a diffusion process in $\mathbb{R}^d$ $$dX_t = f(X_t) dt + \sigma(X_t) dW_t ,$$ with $\sigma(x) \in \mathbb{R}^{d \times m}$ and $m$ dimensional Brownian motion can be considered ...
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### Is there an inverse Lamperti transformation for diffusions?

The Lamperti transformation is commonly used to transform SDEs with state dependent coefficients into SDEs with constant diffusion. For multidimensional processes there are some conditions on the ...
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