# Questions tagged [stochastic-processes]

A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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### Does this Condition on Exit Times imply $X_t$ is a Local Supermartingale?

Let $(X_t)_{t\geq 0}$ be a continuous (or càdlàg), real-valued process, and define stopping times $$\tau_{s,a,b}=\inf~ [s,\infty)\cap\{t:X_t\notin (a,b)\}.$$ We can interpret $\tau_{s,a,b}$ as the ...
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### Why is a predictable stochastic process called *predictable*?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I$ be an index set $\mathbb F=(\mathcal F)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $X=(X_t)_{t\in I}$ be a stochastic ...
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### Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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### Has Math of Finance education become unnecessarily inaccessible?

EDIT: Post shortened, following the suggestions of @Noah Schweber (Thank you!) EDIT#2: I have spent several hours trying to properly form these questions, so instead of voting this down, please ...
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### Infinitesimal generator of time-dependent Markov diffusion

If one talks about homogeneous Markov diffusion $$\mathrm dX_t = \mu(X_t)\mathrm dt+\sigma(X_t)\mathrm dw_t$$ with $\mu,\sigma$ sufficiently differentiable and of appropriate dimensions, there is ...
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### Understanding the Poincaré cone condition and its intuition (in probabilistic solutions to PDEs)

So, context. Theorem. Let $D\subseteq\mathbb R^d$ be bounded and satisfy the Poincaré cone condition and let $\varphi:\partial D\to\mathbb R$ be $C^0$ boundary conditions. Let $(B_t)_{t\geq0}$ be a ...
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### What is the Girsanov density as a functional on the canonical path space $C[0,1]$?

I'll formulate the question via an example. On $( C[0,1], \mathcal{C} )$, where $C[0,1]$ is the set of continuous functions on $[0,1]$ and $\mathcal{C}$ the Borel $\sigma$-algebra given by uniform ...
I encountered the following problem. $\{x_t\}$: Markov chain in discrete time; $\Omega$: a finite state space s.t. $|\Omega|=n<\infty$; $\tau_w\equiv\min\{t\ge 0\,|\,x_t=w\}$, $w\in\Omega$ (first ...
It is a classical result that if $X$ is a process with values in $\mathbb{R}^d$ and \begin{align*} M_t^i = X_t^i - \int_0^t b_i(X_s) ds \\ M_t^i M_t^j - \int_0^t a_{ij} (X_s) ds \end{align*} are both (...