# Tagged Questions

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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### How to Prove the Stochastic Fubini Theorem? (Exercise 2.19 in Chapter IV of Revuz and Yor)

Here is the theorem statement: Let $B$ and $C$ be two independent standard Brownian motions. If $\phi$ is square integrable on the unit square ($\phi \in L^2([0,1]^2)$ ), by suitable filtrations, ...
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### A strange-looking Feynman-Kac formula

I have a PDE which involves the predictable finite-variation parts of some semi-martingales and a quadratic-covariation process and I tried to derive a Feynman-Kac style expectation from the PDE. ...
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### Equivalent definitions of Poisson process

Define a Poisson process with parameter $\lambda$ is a counting process $(N(t))_{t\ge 0}$ such that: (i) $N(0)=0$; (ii) It has independent increment property; (iii) $N(t+h)-N(t)$ has Poisson ...
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### Is there an analytic solution for this Fokker-Planck equation?

The Fokker-Planck equation for a probability distribution $P(\theta,t)$: \begin{align} \frac{\partial P(\theta,t)}{\partial t}=-\frac{\partial}{\partial\theta}\Big[[\sin(k\theta)+f]P(\theta,t)-D\frac{\...
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### Why row vectors in stochastic processes?

It seems reasonable to state that column vectors $\mathbf{x}$ are the most frequently seen standard notation, often using $\mathbf{x}^\intercal$ to denote a row vector (transposed column vector). ...
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### Modified Doob's $L^1$ inequality

Let $X_n$ be a non-negative submartingale. Show that for all $\lambda >0$ $$P(\sup_{k\leq n} X_n \geq 2\lambda) \leq \frac{1}{\lambda} \int_{X_n \geq \lambda} X_n dP$$ In Doob's weak $L^1$ ...
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### When can we interchange the derivative with an expectation?

Let $(X_t)$ be a stochastic process, and define a new stochastic process by $Y_t = \int_0^t f(X_s) ds$. Is it true in general that $\frac{d} {dt} \mathbb{E}(Y_t) = \mathbb{E}(f(X_t))$? If not, ...
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### Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
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### Is the integral of an Ito process still an Ito process?

Assume $r(t)$ is an Ito diffusion: $$dr_t = \mu_tdt + \sigma_tdW_t$$ Then, define the following process: $$X_t = \int_0^tr(s)ds$$ Is $X_t$ still an Ito diffusion?
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### Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
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### Stochastic domination

Suppose we have two probability measures on a space $X$, $\mu$ and $\nu$, such that $\nu$ stochastically dominates $\mu$, i.e.there exist a coupling of $\mu$ and $\nu$ on the product space $X \times X$...
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### For a simple random walk $S_n$ and for a stopping time $\tau$, what is the intuitive interpretation of $P(\tau < \infty) = 1$?

Suppose we have a simple random walk $S_n$ and we define a stopping time to be $\tau = min\{n: S_n = A \ \text{or} \ S_n = -B\}$. That is, we stop the first time we hit $A$ or $-B$. With this, I have ...
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### Characterization of point process, given the number of points

For a point process with independent and identically distributed (i.i.d) inter-renewals, with distribution $p(x)$, we observed $N$ points on $[0,T]$. What is the joint probability distribution ...
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### Methods of SDE calibration

There is somewhere summary of methods that can be used to estimate parameters of SDE? I currently using MLE and regression due to linear dependence between samples. I searching for something ...
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### Clarification on the definition of progressively measurable processes

I am finding difficulties in becoming familiar with the definition of progressively measurable processes. Definition A stochastic process $X$ is called progressively measurable with respect to the ...
Let $(X_t)_{t\in\mathbb{R}}$ be a Levy Process, i.e. $X_0 = 0$ a.s., $X$ has independent and stationary increments, and almost all paths $t\mapsto X_t(\omega)$ are right continuous with left hand ...