All Questions

Consider the following equation for $X(t)$: $$X(t)=e^{-bt}X(0)+\sigma\int_{0}^{b}e^{-b(t-s)}dW(t) \, ,$$ where $0 < b, \sigma\in\mathbb{R} $, $X(0)$ is the initial distribution of $X(t)$, ...

Let $X_t$ be a squared Bessel process satisfying the SDE: $$ dX_t=\left(1-\frac{\beta}{(1-\beta)(1-\rho^2)} \right) dt +2\sqrt{X_t}dW^{(1)}_t $$ and $v_t=v_0e^{-\alpha^2 t/2+\alpha W^{(2)}_t}$ be a ...

Let $S_t$ be the Geometric Brownian Motion, we know that $$dS_t=rS_tdt+\sigma S_tdW_t, t\in [0,T], S_0>0, r>0,\sigma>0$$ and the distribution of $S_t$ is known explicitly. Please see the ...

I am studying the paper "Score-Based Generative Modeling through Stochastic Differential Equations" (arXiv:2011.13456) by Yang et al. The authors use the following loss function (Equation 7 ...

Suppose that I'm given a sample from time-series $(x_n)_{n=1}^N$ and want to decide if it comes from an OU process or not. Is there a (rigorous) test I can use? So far, everything I've seen is hand-...

Suppose there are two given SDE in $\mathbb{R}^d$: $$ \begin{align} \left\{ \begin{aligned} dX_t&=\begin{bmatrix}-\nabla V(X_t)+2\beta^{-1}v_F^\theta(X_t)\end{bmatrix}dt+\sqrt{2\beta^{-1}}dW_t,&...