Questions tagged [stochastic-processes]
A stochastic process is a collection of random variables usually indexed by a totally ordered set.
2,354
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Show that if $A_{0}(t)+A_{1}(t)W(t)=0$ for all $t$ with $A_{0}$ and $A_{1}$ differentiable in $t$ and $W(t)$ a Wiener process, then $A_{0}=A_{1}=0$
I am learning the quadratic variation of stochastic process, and I am working on an exercise stating that
If for all $t$, we have $$0=A_{0}(t)+A_{1}(t)W(t),$$ where $(A_{0}(t),\mathcal{F}_{t})$ ...
user157103
2
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2
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Process with covariance $E[Y_{t}Y_{s}]=a_{1}-a_{2}|t-s|$
We have a centered Gaussian process $X_{t}$ where we have exact equality $$E[X_{t}X_{s}]=a_{1}-a_{2}|t-s|$$ for $|t-s|<\epsilon_{0}\ll \frac{a_{1}}{a_{2}}$ and $a_{i}>0$.
Q: I am curious if ...
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Proof of extended supermartingale convergence theorem
There is a supermartingale convergence theorem which is often cited in texts which use Stochastic Approximation Theory and Reinforcement Learning, in particular the famous book "Neuro-dynamic ...
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On the level of measure theory, what does it mean for a drift to be deterministic?
Given a drift $F\in W^{1,2}([0,T])$ adapted to the filtration of a Brownian motion $B(t)$ on Wiener space $(C[0,T],\mathcal B(\|\cdot \|_\infty)$ with Wiener measure $\mu_0$, there is another measure $...
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SDE conditional expectation
Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
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Showing an "obviously-optimal" control is optimal (without smoothness assumptions)
Let $\mathcal{A}\subseteq\mathbb R$ be a compact interval, $T\in\mathbb R_+$ be a finite horizon, and $g:\mathbb R\to\mathbb R_+$ be a continuous function with $g\leq 1+|\cdot|$. Consider an optimal ...
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Absolute value of a diffusion
Suppose $B_t$ is a standard Brownian motion on a filtered probability space $\langle \Omega, \mathcal F, \{\mathcal F_t\}_t, \mathbb P\rangle$. Consider two SDEs below.
Suppose, $X_0 = Y_0 = 0$
\...
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Question about the exit time of a time-homogeneous Itô diffusion
Consider a one-dimensional Itô diffusion:
$$\mathrm{d} X_{t}=b\left(X_{t}\right) \mathrm{d} t+\sigma\left(X_{t}\right) \mathrm{d} B_{t}$$
where $X_0 = 0$ and $B_t$ is the standard Brownian Motion. ...
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About the role of total variation measure on boundary reflected stochastic processes
I am reading this paper about stochastic differential equations with reflecting boundary conditions. In page 165, an example equation with an explicit solution is presented. However, I can't see that ...
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2
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Counterexample for absolute summability of autocovariances of strictly stationary strongly mixing sequence
Suppose $(X_i)_{i\in\mathbb{Z}}$ is a strictly stationary, strongly (i.e. $\alpha-$)mixing sequence of real random variables. If we have $\mathbb{E}[|X_1|^{2+\epsilon}]<\infty$ for some $\epsilon&...
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Local time as a measurable map from Wiener space
Let $B$ be a Brownian motion on $[0,1]$. The local time of $B$, which I will denote by $L$, is defined as the process on $\mathbb R$ such that
$$\int_0^1 F(B_t)~dt=\int_\mathbb R F(x)L(x)~dx,\qquad\...
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conditional expected value and in Stochastic differential equations
Let's suppose I have a bidimensional SDE of the form:
\begin{equation} \label{eq:system}
\begin{cases}
dX_t=b(t,X_t,Y_t)dt+\sigma(t,X_t,Y_t)dW_t^1 \\
X_0=x_0 \\
dY_t= B(t,X_t,Y_t)dt+C(t,X_t,Y_t)dW_t^...
- 149
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Markov semigroups and resolvents, difference of continuity
Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define
\begin{align*}
...
- 701
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Spitzer's condition, a slowly varying function and its behavior
Let $S$ denote a random walk that satisfies Spitzer's condition $$ \frac{1}{n} \sum _{k=1}^n P (S_k > 0 ) \to \rho$$ for some $\rho \in (0,1)$. From the book Regular Variation (Bingham, Goldie, ...
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Tightness of Hilbert-space-valued arrays
Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}...
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Martingale optional stopping before a stopping time
Here’s an easy one, I hope:
Suppose $\tau$ is a stopping time and $(M_t)$ is a martingale which together satisfy the hypotheses of the optional stopping theorem so that $\mathbb{E}[M_\tau]= \mathbb{E}...
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Law of large numbers for Harris recurrent Markov chains
I'm trying to familiarize myself with the details of the proof that the Markov chains produce by the Metropolis-Hastings algorithm have a law of large numbers. I've found a half dozen or more ...
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Stochastic invariant subset
Let us consider a stochastic differential equation (SDE),
$$
dx_{t}=f\left( x_{t}\right) dt+\sigma\left( x_{t}\right) dW_{t}%
$$
and a compact set $C\subset\mathbb{R}^{n}$.
Given a stochastic ...
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Constructing uncountably many independent random variables with same distribution from Brownian motion?
It is well known one cannot construct uncountable many independent random variables on $([0, 1], \mathcal{B}[0, 1], \lambda)$. ($\lambda$ Lebesgue measure.)
Also, one can clearly construct infinitely ...
user152718
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How to find the PDE for the following transition density
Suppose I have the following two stochastic differential equations ($t\geq 0$)
$$dX_t = \mu(X_t)dt + \sigma(X_t)dW_t \ \ \text{ and } \ \ dZ_t =dt,$$
where $X = (X_t)$, $Z = (Z_t).$
Note that
$W=(...
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Trace Differentiation with Pauli operators, finding $\frac{d x}{d t}$ and $\frac{d z}{d t}$ from the master equation [closed]
I am trying to derive the Bloch vector $dr$ for a measurement of a observable in any arbitrary direction $\theta$. For context this is the setup and derivation I have for continuous measurement along ...
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votes
2
answers
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Convergence of fraction of expectation values
Let $X_1,...,X_n$ be iid normal random variables.
I am looking for a strategy to establish the following limit for fraction of expectation values
$$\lim_{N \rightarrow \infty} \frac{E(\prod_{1\le i ...
- 506
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1
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Renewal functions inequalities
I came along the statement that for $x \geq z$, if $U(x)$ is a renewal function, there exists a constant $K$ such that
\begin{align}
U(x) - U(x-z) \leq U(z) \leq K (z+1).
\end{align}
This is not ...
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Prove that fractional Brownian motion is not a semimartingale using the p-variation
What follows, up to the horizontal line, is taken from Rogers "Arbitrage with fractional Brownian motion".
Consider an interval $[0,T]$ on which is defined the fractional Brownian motion $B$, and ...
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3
answers
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Continuity of Brownian motion constructed from Kolmogorov extension theorem?
I'm trying to construct Brownian motion using the Kolmogorov extension theorem.
I am happy with the construction of a process with the required FDDs as (the canonical process associated with) a ...
user152718
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Minima of a random walk and an equality for a fraction
Let $S_n := X_1 + \dots + X_n$ denote a random walk with zero mean and finite variance and write $L_n := \min \{ 0, S_1, \dots, S_n\}$. The tail distribution of $L_n$ are well-known and in particular,
...
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Existence of strong couplings for Brownian motion
I have two different standard one-dimensional Brownian motions on different filtered spaces, $\langle\Omega,\mathcal F, (\mathcal F_t)_{t\geq 0}, \mathbb P, (W_t)_{t\geq0}\rangle$ and $\langle\hat\...
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Chernoff-type bound for sum of Bernoulli random variables, with outcome-dependent success probabilities
Let $X = (X_1, X_2, \ldots, X_n)$ be a sequence of (not necessarily independent) Bernoulli random variables where for each $i$, the success probability $\Pr[X_i = 1]$ itself is a random variable ...
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Problem arising from martingale solutions to SPDE: $Law(u)=Law(v)$ on $C([0,T]; X)$, can $Law(u)=Law(v)$ on $C([0,t]; X)$ for $t<T$?
I ask this question because I found in some papers of martingale solutions to SPDE, to prove the approximate solutions $u_n$ is a convergent sequence, one can use "stochastic compact" method to find ...
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Sum of sequences of random variables, with variable success probabilities
Consider two sequences of (not necessarily independent) Bernoulli random variables $X_1, X_2, \ldots, X_n$ and $Y_1, Y_2, \ldots, Y_n$. Suppose that for any $i$, we have $\Pr[X_i = 1] = \Pr[Y_i = 1] = ...
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"Expected Value" of a solution to a differential equation
I'm going to write this question in a very informal way as I'm looking for guidance, rather than a specific answer to a specific problem. So I took a course on stochastic processes and Martingales ...
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Are the increments of a stochastic process driven by fractional Brownian motion independent?
I'm studying the following equation in the context of population dynamics
$$\tag1
dX_t = \mu X_t dt + \sigma X_t dB^H_t
$$
where $B^H$ is the fractional Brownian motion (fBm) of Hurst parameter $H\...
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3
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Chernoff-type bounds for a stopped sum of independent random variables
Let $Y_1, \ldots, Y_n$ and $X_1, \ldots, X_n$ be i.i.d. $p$-Bernoulli random variables and let $T \in \{0, \ldots, n\}$ be a stopping time for the process. From Wald's equation, we know
$$
E\left[\...
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A question about Stroock's notes on the Weyl lemma
On p.4 of these notes, D. Stroock gives a quick and efficient construction of the Markov transition functions of a certain diffusion. The idea of his construction (on page 4) is to 'freeze' the ...
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Is the integral of an adapted, measurable process adapted?
Let $X_s(\omega)$ be measurable and adapted.
Under what conditions will the process
$$
F_{t}(\omega) = \int_0^t X_s(\omega) \, ds
$$ also be adapted?
To me it seems that adaptedness and ...
user152718
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answers
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Definition of random measures
Introducing the notion of a random measure, textbooks usually start with a locally compact second countable Hausdorff space. Where does this requirement come from?
I would like to have a motivation ...
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Construct a random time such that the strong Markov property of Brownian motion fails
Let $\{B_t, \mathcal{F}_t; t\ge 0\}$ be a standard, one-dimensional Brownian motion. Can we construct a random time $S$ such that $P[0\le S < \infty] = 1$ and $W_t = B_{S+t} - B_S$ is not a ...
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The mean of a running maximum
Suppose $𝑊$ is a one-dimensional standard Brownian motion defined on some probability space $(\Omega, \mathcal F, P)$ and let $𝑋(𝑡):=\exp\{𝑊(𝑡)−\frac{1}{2}𝑡−\frac{1}{𝑡+1}\}$ for $𝑡\ge 0$. Note ...
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Tightness of a uniformly bounded sequence of functions integrated with respect to a semimartingale
I am reading this paper by Jacod, Jakubowski and Mémin. In the proof of Theorem 1.3 the authors define, for each $n\geq1$ the function $\phi_n$ by
$\quad\phi_n(s)=i+1-ns,\quad\text{if } \frac{i}{n}&...
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Dirichlet problem for a subharmonic function
Suppose $K$ is a compact subset of $\mathbb R^n$ , $V_0$ and $V_1$ the complements of $K$ in $\mathbb R^n$ a and $\mathbb R^n_\infty$ (one point compactification), respectively. Let $u$ be ...
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“Chapman-Kolmogorov”-convolution vs. smoothness
Let $K:\mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ be a so-called "integral-kernel": we certainly require $K(x,.)$ and $K(.,y)$ to be Lebesgue measurable for almost all $x,y \in \mathbb{R}^n$. An ...
- 1,451
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Mehta integral and orthogonality
The Mehta integral is the following expression:
$$\frac{1}{(2\pi)^{n/2}}\int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} \prod_{i=1}^n e^{-t_i^2/2}
\prod_{1 \le i < j \le n} |t_i - t_j |^{2 \...
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Show that this process is not a martingale
I am cross-posting this question from MSE since I did not received any answer, furthermore I tried asking some professors in my university but still we could not find an answer.
The most surprising ...
- 485
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1
answer
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Writing path-dependent conditional expectation in terms of distribution
Let $x$ be an $\mathbb{R}$-valued random variable, then for any bounded and continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ one may write
$$
\mathbb{E}[f(X)] = \int_{x \in \mathbb{R}} f(x)\pi(...
- 4,969
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0
answers
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Approximation of adapted continuous random function by adapted polynomials
Given filtration $\mathcal{F}_t$. Let $f(t,x)$ is adapted with respect to $\mathcal{F}_t$ continuous in $t$ and $x$ function. Question: Is there sequence of adapted functions $f_n(t,x)$ such that $...
1
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1
answer
172
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Rough paths theory for Non-Markovian processes
I would like to know whether there is a suitable extension of the theory of rough paths that could be useful to solve Non-Markovian problems.
I would appreciate any example or also any other theory (...
- 13
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votes
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365
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Matrix positive semi-definite
We construct a non-random matrix using random variables as follows:
We fix the vector $v=(1,1).$
Let $X$ be a $\mathbb R^2$-valued random variable such that $X$ is distributed according to
$$d\mu(...
- 192
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votes
1
answer
100
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Sign of expectation value
Consider a multivariate Gaussian-type measure $$d\lambda(x):=\nu_{\mu,\Sigma} e^{-\langle (x-\mu), \Sigma^{-1}(x-\mu) \rangle - \vert x \vert^2} $$
with vector $\mu \in \mathbb R^n$ and $\Sigma$ ...
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votes
2
answers
427
views
Constant Martin kernel and amenability
Consider a finitely supported random walk on a discrete group G such that the support generates $G$ as a semigroup. The Martin kernels are then non-negative functions on the product $G \times M$ where ...
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1
answer
380
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Law of large numbers for random Dirac measures
Suppose $\{X_1,...X_n\}:\Omega \to \mathbb{R}^p$ be i.i.d. random vectors with common probability law/measure $p$, i.e. $Prob(X_i^{-1}(E))=p(E) \forall E \subset \mathbb{R}^p $ Borel measurable.
...
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