Tagged Questions
1
vote
0answers
32 views
Total set of functions in $L^2(\Omega)$
Are the sets of functions $\{e^{\int_0^T h(s)dB_s -\frac{1}{2}\int_0^T h(s)^2 ds}\}$ and $\{e^{\int_0^T h(s)dB_s}\}$ total in $L^2(\Omega)$? What is the difference? What should one use to prove weak ...
1
vote
0answers
38 views
maximize the expected value of the logarithm of the weighted average of random variables
I'm trying to do the following.
$$\max_{m\in\mathbb{R}} \mathbb{E}\left[\log (wA + (1-w)B_m)\right],$$
where $0<w<1$ and $A, B_m > 0$ are correlated random variables. $A$ does not depend ...
-1
votes
0answers
26 views
Covariance function of primitive of Brownian process
Let W the Brownian motion. Calculate the covariance function of the process $$\left\{\int_0^t X_s\,ds,t\geq0 \right\}$$
and $X_t:=tW_t-W_t$.
0
votes
1answer
34 views
Second derivative of Brownian motion?
My question is, we give a meaning to the following expression:
$$dX(t) = \mu(t,X(t))dt + \sigma(t,X(t))dW(t), \ \ X(0)=x.$$
where $W$ is a Wiener process.
This equation can be thought as
...
1
vote
1answer
37 views
Example of a martingale and a stopping time with $E(T)<\infty$ but $E(X_T) \neq E(X_0)$
Is there an example of a martingale in discrete time $X_0, X_1, X_2,\ldots$ and a stopping time $T$ so that $E(T) <\infty$ but $E(X_T) \neq E(X_0)$?
With added assumptions on how $X_n$ behaves, ...
1
vote
2answers
44 views
why is this Markov Chain aperiodic
I have this Matrix:
$$P=\begin{pmatrix} 0 & 1 \\ 0.3 & 0.7 \end{pmatrix}$$
this markov chain is said to be aperiodic, I dont understand how it comes to it. Period $\delta$ is the gcd of ...
0
votes
0answers
34 views
Integrable local martingale is a supermartingale
Let $M_t \in L^1(\mathbb P)$ be a local martingale. Hence exists an increasing sequence of stopping times $\tau_n$, for each of which the process $M_t$ is a martingale.
\begin{align}
\mathbb E [M_t ...
1
vote
0answers
29 views
Why can't I use the variance of the sample average in the Central Limit Theorem for the weak-stationary process?
Under mild conditions $\dfrac{\bar{X}-\mu}{\sqrt{\sigma^2/n}}$ approaches the standard normal (where $\sigma^2$ is the process variance, not the marginal variance $\sigma^2_x$).
Why is the ...
3
votes
0answers
42 views
Convergence of a Subordinator.
Let $\left( X_{t}\right) _{t\geq0}$ be a subordinator with the Laplace
expoent given by
$$
\Phi\left( \lambda\right) =d\lambda+\int_{0}^{\infty}\left( 1-e^{-\lambda
x}\right) \nu\left( ...
1
vote
1answer
29 views
Poisson process (simple question)
Imagine you have two events starting at the same time. The duration time for each event is exponential, with different parameters. Knowing that one of the events is finished (we don't know which) at ...
2
votes
0answers
45 views
A equivalent definition of the Feller Process.
I saw this on Liggett's Book (P.95).
Let $S=%
%TCIMACRO{\U{2115} }%
%BeginExpansion
\mathbb{N}
%EndExpansion
,$ and suppose $\left( X_{t}\right) _{t\geq 0}$ is a continuous-time Markov
process with ...
1
vote
1answer
33 views
A question about Infinitesimal generator of Feller Process
Let $S=%
%TCIMACRO{\U{211d} }%
%BeginExpansion
\mathbb{R}
%EndExpansion
$, and consider the Feller process $\left( X_{t}\right) _{t\geq 0}$ with
state space $S$ such that $X_{t}=t+X_{0}$ for all ...
3
votes
1answer
48 views
question about Ito's formula
I'm currently learning about the Ito's lemma / formula
In my textbook, a direct application of the formula is to compute quantities like that :
(W is a Brownian motion)
While trying to prove these ...
0
votes
1answer
50 views
Implications between $\mathbb P [\tau < \infty] =1 $ and $\tau \in L_1 (\mathbb P)$
We've got the usual filtered stochastic basis $(\Omega, \mathcal F, (\mathcal F_n). \mathbb P), \space \tau : \Omega \to \mathbb{N}\cup \{\infty\}, [\tau \le n] \in \mathcal F_n$
($\tau$ is an ...
1
vote
1answer
27 views
how do I model probable time until simultaneous availability?
Short question: If several people, all of whom have limited availability, need to meet, how far in the future will I need to schedule the meeting?
I was hoping there was a readily available answer ...
2
votes
1answer
68 views
When is the following local martingale strict local martingale?
By Section 5.5 of the book [Karatzas and Shreve 1991],
the following 1-d SDE has unique weak solution
in the form of
\begin{equation}
d X_{t} = X_{t}^{\gamma} \cdot I_{\{X_{t}\ge 0\}} dW_{t}, \ ...
-1
votes
0answers
32 views
Wiener process integration [duplicate]
Let W =(Wt;Ft)t$\ge$0 be a standard Wiener process and let f : [0;$\infty$) $\to R$ be a
deterministic, continuously di fferentiable function. Prove the following integration
by parts formula:
...
0
votes
1answer
55 views
How to show $t{B^{2}_t}+t^2$ is $\mathcal{F}_t$-addapted process?
How to show $t{B^{2}_t}+t^2$ is $\mathcal{F}_t$-adapted process? Here $B_t$ is Brownian Process.
Please Help
-1
votes
0answers
39 views
Markov chain and irreducible, aperiodic graph [closed]
In order for markov chain's distribution to converge to a unique stationary distribution $\pi $ no matter what the original distribution $P_0$ is, the Markov Chain graph must be irreducile and ...
1
vote
0answers
57 views
Poisson point process convergence
Let Π be a Poisson point process on [0,∞) with intensity measure $\mu$. Assume $μ([0,t])<∞$ for all $t<∞$ and $μ([0,∞))=∞$. Also assume $μ({x})=0$ for all x. Prove ...
1
vote
1answer
62 views
Random walk probability non-symmetric steps
I currently have a probability class tutorial question that I have no idea where to begin. At first instinct, I thought it may have been a CTMC question or branching question, but now I have no idea, ...
2
votes
1answer
47 views
A theorem about the Poisson Point process.
In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson
point process.
The theorem states that if $\Pi$ is a poission point process on $S$ with
intensity measure $\mu.$ Let ...
1
vote
1answer
64 views
Prove $\mathbb{E}[X_t | \mathcal{F}_s] = \mathbb{E}[X_t | \sigma(\mathcal{F}_s \cup \mathcal{G}_s)] $
We want to prove that if $X_t$ is an $\mathcal{F}_t$ - martingale: $\mathbb{E}[X_t | \mathcal{F}_s] = X_s$ for $s<t$, then it's also a $\sigma(\mathcal{F}_s \cup \mathcal{G}_s)$- martingale. ...
0
votes
1answer
31 views
Examples of convergence of random variables
First, let's recall the definitions of 4 different types of convergence:almost surely, in $r$th mean, in probability and in distribution:
$X_n\xrightarrow{a.s.}X$ if $\{\omega \in ...
0
votes
1answer
43 views
Continuous Non negative martingale converging to 0
Is there any (non trivial) continuous non negative martingale which converges to 0?
1
vote
2answers
36 views
Generalization of Doob Dynkin for Stochastic processes
Let $\{X_t\}_{t\geq 0}$ be continuous time stochastic process and $\{\mathcal{F}_t^X\}_{t \geq 0}$ be the filtration generated by it. If the process $Y$ is $\{\mathcal{F}_t^X\}_{t \geq 0}$ adapted, is ...
5
votes
1answer
47 views
lower bound of expectation of stochastic differential equation
I'm looking for a lower bound on the expected value of a smooth, non-negative, increasing function $\mathbb{E}f(X_t)$, $f(0)=0$ of the solution to a stochastic differential equation $X_t = x + ...
1
vote
1answer
48 views
finding the probability density function of $ dY_t = - Y_t X_t dW_t$
Could someone point me to where I can learn how to derive the stationary distribution for the martingale $Y_t$ which itself has stochastic volatility drive by $X_t$:
\begin{align}
dY_t &= - Y_t\ ...
1
vote
1answer
41 views
The weighted distribution function for combination of two variables
For example, we have two random variables $a$ and $b$. And they have cumulative distribution function $F(x)$ and $H(x)$. We have number $0 < p < 1$.
Suppose, some machine get this random ...
1
vote
0answers
27 views
Girsanov kernel moments
Let $Z_t=e^{\int_0^tq_tdB_t-\frac{1}{2}\int_0^tq^2_tdt}$, where $(q_t)_{t\geq0}$ is a predictable process, and $(B_t)_{t\geq0}$ a $\mathbb{P}$-Brownian motion. In particular, Novikov's condition ...
0
votes
1answer
39 views
Continuous time Stochastic Process stopping time measurability
Let $\{X_t,\mathcal{F}_t;0\leq t < \infty\}$ be continuous time stochastic processes and $T$ be $\{\mathcal{F}_t\}_{0\leq t < \infty}$ stopping time. How to prove $X_T$ is $\mathcal{F}_T$ ...
0
votes
0answers
13 views
fGn asymptotic claim correlation
Let $(X_{i})$ be the fractional Gaussian noise for $H\in(0,1)$.
Since it is stationary $\mathbb{E}(X_{i}X_{j})$ only depends on $|j-i|$.
How can I prove for $\rho(|j-i|)=\mathbb{E}(X_{i}X_{j})$ that ...
1
vote
1answer
43 views
Moment generating function of two non-independent Brownian increments
I am writing to ask if it is possible to get closed-form solution to the expression to the following expression:
$\mathbb{E}[e^{\sigma^2(W_t-W_u)(W_s-W_u)}]$ where $W$ is a standard Brownian motion, ...
0
votes
0answers
17 views
Submartingale bounds
Let $X_1,X_2,\ldots$ be a submartingale with respect to the filtration generated by it. Is it possible to get any bounds for the probability $\mathbb{P}(X_2 < 0\mid X_1 >0)$ ?
1
vote
1answer
87 views
A Boundary crossing result for discrete brownian bridge
Let $S_n$ be a random walk with gaussian increments with $S_0=0$, i.e. $S_n-S_{n-1}\sim N(0,1), n\geq 1$. Fix $a>0,b\in \mathbb{R}$ and $c<a+bn$. Define the new process
$$
...
2
votes
1answer
83 views
Sum of stationary process
Suppose you have two stationary process $A_{t}$ and $B_{t}$. Suppose $Z_{t} = A_{t} + B_{t}$. Show that $Z_{t}$ is stationary. I am unsure how to solve this without knowing if the processes are ...
0
votes
1answer
41 views
Product of stationary stochastic process
Suppose $z_{t} = x_{t}y_{t}$ where $x_{t}$ and $y_{t}$ are 0 mean, independent stationary stochastic process. What is the autocovariance function of $z_{t}$? Show that the spectral density can be ...
1
vote
1answer
37 views
Finding expectation of stochastic process
Suppose $\sigma_{t}^{2} = w + \alpha_{1}y_{t-1}^{2} + \beta_{1}\sigma^{2}_{t-1}$ where $\alpha_{1} + \beta_{1} = 1$ and $y_{t} = \sigma_{t}e_{t}$ and $e_{t}$ is $ N(0,1)$. How do you show that
...
1
vote
1answer
32 views
Inequality related to Doob's martingales
I have the following question on Doob's martingales.
Let $A$ be an integrable $\mathcal F$-measurable random variable on the
stochastic basis $(\Omega, \mathcal F, \mathcal F_t, \mathbb P)$. ...
0
votes
1answer
38 views
Integral: Is there a closed form?
I wonder whether there is a closed form or way to compute explicitly:
$$\int_0^t e^{\alpha s} dB_s$$
where $\alpha$ is just a real number and the integral is in the Itô sense.
Thank you very much!
1
vote
1answer
68 views
Using a Brownian martingale to compute the second moment of a hitting time
Prove $ W_t=B_t^4 -6B_t^2t+3t^2$ is a martingale, and compute $E(T^2)$ where $T=\inf(t\ge0,B_t=-a, B_t=b)$ if $a=b$.
Ok, if $0\lt t\lt s$, $W_t$ is a martingale if $E(W_s|[B_r]_{r\le t})=W_t$
So ...
0
votes
1answer
23 views
Basic brownian motion computation
Let $B_t$ denote a standard 1-d Brownian Motion. Find $P(B_2 \gt 2)$.
My sol.
$B_2 ~ N(0,2)$ so $P(B_2 \gt 2)=1-P(B_2\le 2)=1-\frac{\int_0^2e^{-\frac{x^2}{4}}}{\sqrt{4\pi}}$, but where do i go from ...
1
vote
0answers
32 views
Cylindrical sigma algebra answers countable questions only.
I got a missing link in some in the following (standard) textbook question:
Show that the cylindrical sigma algebra $\mathcal{F}_T$ on $\mathbb{R}^T$ (equals $\bigotimes_{t\in ...
4
votes
2answers
82 views
Prove that a random walk on $\mathbb{Z}_+\cup \{0\}$ is transient
Prove that a random walk on $\mathbb{Z}_+ \cup \{0\}$ is transient with $p_{i,i+1}=\frac{i^2+2i+1}{2i^2+2i+1}$ and $p_{i,i-1}=\frac{i^2}{2i^2+2i+1}$.
So since this Markov chain has only a single ...
2
votes
1answer
65 views
A bound for the probability that a Brownian motion stays in an interval
Suppose I have a Brownian motion $X_t$ with $X_0=0$. Let $T$ be the first exit time of the interval $[-1,1]$.
I'm trying to get a "quick" lower bound for the probability that $T$ is very large which ...
0
votes
0answers
29 views
Computing spectral density of process
Suppose you have a stochastic process $Y_{t} = \frac{1}{2}(X_{t-1} + X_{t} + X_{t+1})$. $X_{t} = 0.4X_{t-1} + \omega_{t}$. How would you compute the spectral density of the process? I know that ...
1
vote
1answer
49 views
Is geometric Brownian motion stationary?
I was just wondering if the solution to
$$dX(t) = \mu X(t) dt + \sigma X(t) dB(t)$$
gives a stationary process for any $\mu,\sigma$ and what the distribution would be.
Thanks a lot!
1
vote
1answer
31 views
Normal probability and Brownian motion
Let $X_t$ be a Brownian motion with parameter $\sigma$. Find the probability in terms of $$\Phi(x)= \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^x e^{- \frac{ \alpha ^2}{2}}d\alpha$$
How would I do this for ...
3
votes
0answers
55 views
Prove the 2 definitions of the periodicity of Markov Chain are equivalent.
In many textbooks, there are basically 2 ways of defining the periodicity of Markov Chain. One is by partitioning the graph in to subgraph such that transition in one group of state leads to the other ...
1
vote
0answers
10 views
Measuring time with a clock that monitors decay events occurring with a known mean time (though sampling from an unknown probability distribution)
Imagine I have some hypothetical particle that decays over time, where $\mu$ is the mean decay time, and where the probability of each decay event is governed by some unknown probability distribution. ...
