A stochastic process is a collection of random variables usually indexed by a totally ordered set.

**4**

votes

**2**answers

239 views

### Is $B(t-1)$ an Ito process?

Let $I(\cdot)$ be an indicator, and $B_{t}$ be an 1-dim standard Brownian motion in a nice filtered probability space
$(\Omega, \mathcal{F}, P, \mathcal{F}_{t})$. We consider a random process
$$Y_{t} ...

**0**

votes

**1**answer

104 views

### probability in galton watson processes [closed]

I am trying to study the Elementary new proofs of classical limit theorems for Galton Watson processes written by Jochen Geiger.
I don't understand what Z_(n,i) stand for.
And in the proof of Theorem ...

**1**

vote

**2**answers

163 views

### Principal bundles and Subriemannian Geometry

In sub-Riemannian geometry, one considers manifolds $P$ equipped with a subbundle $\mathcal{H}$ of $TP$, the horizontal distribution. One then has a Riemannian metric only on this distribution ...

**0**

votes

**0**answers

48 views

### Strong Markov Property of the joint process $(B_t,L_t)_{t\ge 0}$

Let $B=(B_t)_{t\ge 0}$ be a Brownian motion and $L=(L_t)_{t\ge 0}$ be its local time in zero. Given two strictly increasing functions $\phi_1$, $\phi_2: \mathbb R_+\to\mathbb R$ such that ...

**7**

votes

**1**answer

295 views

### Central limit theorem for biased random walk

Define random variables $X_n$ by $X_0 = 0$ and
\begin{equation*}X_n - X_{n-1} = \begin{cases}
1 & \text{with probability } g(X_{n-1}) \\
0 & \text{with probability } 1-g(X_{n-1})
\end{cases}
...

**2**

votes

**1**answer

257 views

### Stochastic interpretation of heat kernel on fiber bundle

I'm looking for a stochastic interpretation of the heat equation for vector valued function.
The classical set up is the following :
If $(M,g)$ is a riemannian manifold then we could consider the ...

**1**

vote

**0**answers

141 views

### Joint law of a standard Brownian motion and its local time at a nonzero level

Let $B_t$ be the standard Brownian motion and $L_t^a$ be the local time at level $a$. It is known that the joint-density of $(L_t^0,B_t)$ is
$$
P\left(B_t\in d y, L_t^0\in d v\right) = ...

**0**

votes

**0**answers

45 views

### Recursive parameter estimation for partially observed Ito SDEs

I'm trying to get my head around online (recursive) maximum-likelihood parameter estimation in the language of stochastic processes and in the context of stochastic filtering, i.e. where we have a ...

**1**

vote

**0**answers

105 views

### Horizontal vs Vertical sides Exit from a Rectangle for simple symmetric Random Walk on $\textbf{Z}^{2}$

Consider simple symmetric random walk, $X_{n} = (X_{n}^{(1)}, X_{n}^{(2)})$ with $X_0= (0,0)$, on the 2 dimensional integer lattice, $\textbf{Z}^{2}$.
Let $T_{M}, T_{N}$ be the smallest $n$ such ...

**6**

votes

**1**answer

132 views

### Does a Gaussian process shrink under a contraction map

Let $T \subset \mathbb R^n$, and assume it's a finite set if that helps. Consider the symmetric Gaussian process $(X_t)_{t\in T}$ defined by $X_t = \langle G, t\rangle$, where $G$ is a standard ...

**2**

votes

**1**answer

152 views

### Stochastic differential equation associated with an optimal control problem

We know how to find the stochastic differential equation (Hamilton-Jacobi-Bellman equation, HJB) of the control problem where a process $X_t$ is controlled up until it is stopped at a stopping time ...

**1**

vote

**0**answers

43 views

### Question about the characteristics of semimartingales

Let $D=D([0,1,R)$ be the space of cadlag (right-continuous with left limits) functions defined on [0,1] and $X:=(X_t)_{t\in [0,1]}$ be the canonical process on $D$, i.e. $X_t(x)=x(t)$ for all $x\in ...

**5**

votes

**0**answers

246 views

### Quadratic variation and predictable quadratic variation for martingales

Let $(M_{t})_{0\le t\le 1}$ be a continuous martingale with respect to the filtration $(\mathcal{F}_{t})_{0\le t\le 1}$. Assume that $E M_1^2<\infty$.
Fix $N$ and consider now a discrete version ...

**4**

votes

**1**answer

183 views

### Area enclosed by Brownian motion (without winding number)

The question Average Value of Area Closed by Brownian Motion turned out to be about the Lévy area process, which measures "signed area with multiplicity" enclosed by Brownian motion (e.g. each ...

**1**

vote

**2**answers

66 views

### SDEs: Bounding the variance of a solution

I've been thinking about something that would seem intuitive, but I haven't really been able to dig a direct answer to. This is a rough draft of it.
Let
$$X_t = \mu_{X,t} \mathrm{d}t + \sigma_{X,t} ...

**2**

votes

**1**answer

134 views

### Criterion for weak convergence of probability measures on S' or D'

Let $X_n$ in $S'$ and $\mu_n$, $\mu$ in $M(S')$. $S'$ is the space of tempered distributions. I'm looking for a reference that says if $< f, X_n >$ converges in distribution to $< f,X>$ ...

**30**

votes

**6**answers

2k views

### Deep Learning / Deep neural nets for mathematician

I am interested in finding out the math ideas behind the technologies that are under the umbrella of "Deep Learning" or "Deep neural nets".
Most of the papers/books that are often quoted in ...

**9**

votes

**1**answer

340 views

### a question on 0-1 valued stochastic process

Here's a question on probability theory from a layman (I'm a game theorist). It is very likely that the question will be a straightforward matter for someone who is a probability theorist. I guess I'm ...

**0**

votes

**0**answers

27 views

### a judicious choice of a parametrix expansion for a Kolmogorov equation?

Consider two SDEs:
Let $b, \sigma$ be two smooth lipshitz functions on $\mathbb{R}$ and consider $\left \{ X(s), s\geq t \right \}$ the solution of the following SDE
$d X(s) = b(X(s)) d s + ...

**5**

votes

**0**answers

253 views

### Squaring random Schwartz distributions

Let $\mu$ denote the centered Gaussian measure on $S'(\mathbb{R}^d)$ with covariance
$$
\mathbb{E}
[\phi(f)\phi(g)]=\int_{\mathbb{R}^d} \frac{\overline{\widehat{f}(\xi)}
...

**1**

vote

**0**answers

69 views

### Does the expected spreading of sample paths imply increase in variance?

Consider a sample-continuous stochastic process $\left\{ X_t \right\}_{t \in T}$ s.t. each $X_t$ is real-valued and $$\int_\Omega | X_t(\omega) | ^p \, \mathrm{d} P(\omega)< \infty$$ for all $1 ...

**1**

vote

**0**answers

48 views

### Invariant Girsanov Theorem on a Riemannian manifold

This is somewhat a follow-up on this post.
Let $X_t$ be the stochastic process on a compact Riemannian manifold generated by the (possible time-dependent) second-order elliptic operator $L_t$. Let ...

**3**

votes

**0**answers

61 views

### Matroid rank decay

Consider a uniform vector matroid $M(0)=U_{m,n}$ of rank $m$ with $n$ points, $n>m>2$ (you can think of it as a set of $n$ points in general position in vector space $F^m$ for some large field ...

**0**

votes

**1**answer

84 views

### Defining a brownian bridge indexed by angle

I have a random closed curve of the form $(\theta,r_\theta)$, where $\theta\in [0,2\pi]$, is the counter clockwise angle from the x-axis and $r_\theta$ is the radial distance from the origin ...

**5**

votes

**1**answer

195 views

### Average probability that a random cosine polynomial with bernoulli coefficients is small

Let $P_{n}(t)=\sum_{k=0}^{n}\varepsilon_{k}\cos(kt)$ where $\varepsilon_{i}$ are independent random variables taking values in $\left\{-1,1\right\}$ with equal probability. Is is true that for any ...

**1**

vote

**1**answer

50 views

### Can real-valued Markov processes with continuous surjective sample paths admit a non-trivial “forward-invariant” set?

I have both a more general question (concerning stopping times), and then a more specific application (as described in the title).
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t \geq 0},\mathbb{P})$ be ...

**4**

votes

**0**answers

76 views

### Is there a generalization of Polya urns to continuous outcome event?

Take for example the simplest model where there are n blue balls and m white balls in an urn. Then, in a first step realization, a white one has been drawn and then c + 1 of this colour had been put ...

**1**

vote

**0**answers

59 views

### Subclass of semimartingales for which all characteristics can be estimated?

I'm going to ask the question for Ito semimartingales rather than semimartingales in general, but more general answers would be great.
An Ito semimartingale is a martingale for which the ...

**1**

vote

**0**answers

71 views

### Da Prato's notion of Symmetric Operator

For anyone who's familiar with G. Da Prato's books on infinite dimensional analysis, I was wondering if someone could clarify something. In, for instance, "An Introduction to Infinite Dimensional ...

**2**

votes

**2**answers

267 views

### Existence of strong solution to SDEs with non-Lipschitzian drift

Consider the SDE:
$$dX_t=b(X_t)dt+dW_t\quad X_0=x$$
If $b$ is bounded Borel function, using Zvonkin's Transform, one can prove there exists a unique strong solution.
I want to know if we assume $b$ ...

**2**

votes

**0**answers

57 views

### Bounds on moving average process

Let $X_1,X_2,\dotsc$ be a sequence of i.i.d. random variables and define the average process $\{Y_t\}$ as
$$
Y_t = \sum_{i=1}^p a_k X_{t-i}
$$
with some constants $a_1,\cdots,a_p \in \mathbb{R}$. This ...

**10**

votes

**2**answers

334 views

### Minimal expected absolute value of linear combinations of Gaussian random variables

I am interested in the following question. Consider $n$ independent standard normal random variables $g_i$. Cosider a linear combination $w_1g_1+\cdots+w_ng_n$. Can one give a "decent" upper bound for ...

**3**

votes

**1**answer

147 views

### Does $E^{x,t}(f(X_T))$ solve a PDE if $f$ is not continuous?

Many books [see below for references] explore the connections between partial differential equations and expectation values.
Assume $X$ is a diffusion with generator $A$, then they conclude, that ...

**3**

votes

**2**answers

117 views

### Existence of an invariant measure on an infinite dimensional space via Lyapunov functional

Set-up.
Assume that we have a complete separable metric space $\mathcal{X}$ that is not locally compact. Let $V: \mathcal{x} \to [0; +\infty]$ be a functional such that $K_r :=\{x \in \mathcal {X} : V ...

**1**

vote

**2**answers

103 views

### Lyapunov exponents of dual / adjoint / transpose random dynamical system (RDS)

Consider the the state of a system at time $n$, $X_n$, as the action of a product of i.i.d. $d\times d$ random matrices acting on a $d$ dimensional vector $X_0$, so we have
$$X_n = A_n \cdots ...

**0**

votes

**0**answers

122 views

### How to decide a value of learning rate for Stochastic Gradient Descent?

I'd like to know how to decide a value of learning rate for Stochastic Gradient Descent (SGD), such as $\eta$ on the following parameter update iteration equation,
$w_{i+1} = w_i + -\eta \nabla ...

**2**

votes

**0**answers

95 views

### Do the Birkhoff averages of a measurable stationary homogeneous Markov process in continuous time “converge to the right limit”?

[I've decided to rewrite the question, to make the essential point clearer.]
Let $\,\mathbb{R}^{[0,\infty)}:=\{(x_t)_{t \geq 0} : x_t \in \mathbb{R} \ \, \forall t\}$. We say that a set $Y \subset ...

**1**

vote

**2**answers

233 views

### $\lim_{t\rightarrow 0}P\left(X_t >0\right)=\frac 1 2$ for continuous semimartingales?

I am trying to prove the following Lemma, which seems intuitive, but I still have doubts:
Lemma
Given a Brownian motion $\{W_t,\mathcal F_t:0\le t \le1\}$, two bounded processes, $\mu$ and $\sigma$, ...

**1**

vote

**0**answers

76 views

### How to fit a stochastic matrix to given data.?

Given a data sequence of noisy observations of a 3-state Markov chain $X$ -- $y_1$,$y_2$,...$y_n$, with two transition matrices $A_1$ and $A_2$ corresponding to different regions (**) in the (unit) ...

**5**

votes

**0**answers

101 views

### Support of a Measure with Characteristic Functional Continuous in $L_p$, $1\leq p <2$?

Let $\mathcal{S}(\mathbb{R})$ be the space of smooth and rapidly decaying functions and $\mathcal{S}'(\mathbb{R})$ its dual, the space of tempered distributions. Let $\mu$ be a probability measure ...

**0**

votes

**0**answers

80 views

### Rolling map as a diffeomorphism?

Let $M$ be a (compact) Riemannian manifold and $x \in M$. For a piecewise smooth path $\gamma: [0, T] \longrightarrow M$, we can define Cartan's development map (or rolling map)
$$(\Phi\gamma)(t) = ...

**0**

votes

**0**answers

122 views

### When an integral with respect to a Poisson point process is finite?

Let $N(ds,dv)$ be a Poisson measure on $\mathbb{R} _+ \times \mathbb{R} _+$ with intensity $dsdv$. Let $N = \sum\limits \delta_{(s_i,v_i)}$. Assume that $N$ is compatible with a filtration $\{ ...

**2**

votes

**0**answers

69 views

### probabilistic interpretation of elliptic equation with mixed boundary condition

I would like to understand the probabilistic interpretation of the following elliptic problem with mixed Dirichlet-Neumann boundary conditions:
Let $B := \{ x \in \mathbb{R}^n, \quad \| x \|_2 \leq 1 ...

**1**

vote

**1**answer

257 views

### Change of time variable in Wiener process

I'm following a solution of an SDE from here
http://www.math.ethz.ch/~delbaen/ftp/preprints/CEV.pdf
Start with the SDE
$$
dX_t = \delta dt + 2\sqrt{X_t} dW_t
$$
consider a deterministic time change
...

**0**

votes

**0**answers

35 views

### References for symmetric α-stable process (SSP) for $a>2$

Many properties of Brownian motion have been extended to SSP's for $0\leq \alpha\leq 2$ and so it is quite easy to find literature on them. However, I am currently studying the SSP for $\alpha>2$ ...

**2**

votes

**0**answers

102 views

### Equivalence of two non-degenerate Gaussian measures on Banach space

The motivation of this question is to show that two probabilities on
$C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process
on $[0,1]$ starting from zero) induced by two ...

**2**

votes

**0**answers

54 views

### Killing a Feller Process

Given a canonical Feller process $(X_t,P_x)$ with Feller semigroup $P$. Let $T$ a (good) stopping time, for example $T=\inf\{u\ge 0 : X_u=0\}$. I'm looking for a proof of the following claim
...

**0**

votes

**1**answer

324 views

### Version of Ito's lemma applied to a stochastic function

The Ito's formula stated in most books in stochastic calculus is in the form $F(t,X_t)$, where $F: \mathbb{R}^{d+1} \rightarrow \mathbb{R}$ is a $d+1-$dimensional deterministic $C^{1,2}$ function and ...

**5**

votes

**2**answers

181 views

### Origins and Industrial Applications of stochastic processes (eg. Brownian motion) on Riemannian manifolds

I am studying BM on Riemannian manifolds and I am curious how this theory started. In the references below (esp. in Hsu's exposition), you will find many applications of that theory such as a ...

**1**

vote

**1**answer

185 views

### Correlation between two continuous-time stochastic processes

Consider two continous-time stochastic processes $\{A(t)\}_{t \ge 0}$ and $\{B(t)\}_{t \ge 0}$ with $A(t)=t$ and $B(t)=t$. Each process starts at $t=0$ and emits "ticks" at increasing time slots. For ...