Hello, I have a question about using Ito's lemma to solve SDEs: solve $dX_t=X_tW_tdt+dW_t$, $X(0)=X_0$, where $W_t$ is the standard brownian motion. I try to use $exp(\int_0^t W_sds$ as the integrating factor, but I cannot get the $dW_t$ term
Consider the 1-dimensional Ornstein-Uhlenbeck process
$$X(t)=\mu+e^{-\lambda t}(x-\mu)+\frac{\sigma}{\sqrt{2\lambda}}e^{-\lambda t}W^{(e^{2\lambda t}-1)}$$
with mean $\mu+e^{-\lambda t}(x-\mu)$ and variance $\frac{\sigma^2}{2\lambda}(1-e^{-2\lambda t})$. I'd like to show the infinitesimal generat...
This is a well-known result in functional analysis, in terms of dual of quotient spaces and annihilators of subspaces. Let me formulate the problem first and a new attempt to prove it:
Let $W$ closed vector subspace of $V$, $V/W$ the quotient space, and $W^\perp$ the annihilator of $W$.
(1) Show ...
I am doing some exercises in E-Li-Vanden-Eijnden's Applied Stochastic Analysis and I meet this problem:
(Exercise 3.19) Consider an irreducible Markov chain $\{X_n\}$ on a
finite state space $S$. Let $H\subset S$ and define the first passage
time $T_H=\inf\{n:X_n\in H\}$, $h_i=P_i(T_H<\infty)$. ...
Hello, I have a problem regarding first hitting time for irreducible Markov chains. Someone has given a solution, but I'd like to know other approaches. Thank you.
I am reading E-Li-Vanden-Eijnden's Applied Stochastic Analysis, and the author claims:
The recurrence relation of $\boldsymbol{\mu}_n=\boldsymbol{\mu}_{n-1}\boldsymbol{P}$, where $\boldsymbol{\mu}_n$ is the distribution of $X_n$, and $\boldsymbol{P}$ is the transition matrix, can be written as $$...
Hello, I have a question about proving the following is true in Markov chain. I'm trying to prove by Chapman-Kolmogorov equations, but I failed. Any hints and suggestions are welcome!
I am reading E-Li-Vanden-Eijnden's Applied Stochastic Analysis, and the author claims:
The recurrence relation of $\boldsymbol{\mu}_n=\boldsymbol{\mu}_{n-1}\boldsymbol{P}$, where $\boldsymbol{\mu}_n$ is the distribution of $X_n$, and $\boldsymbol{P}$ is the transition matrix, can be written as $$...
Can we move the discussion to my post? I think your epsilon-approach is interesting, can you go to my post and write an answer if you are willing to?@Thorgott
@user2103480 Personally, I find the Taylor expansion argument not a rigoruous one. Do you have other approaches to prove that? Note the open set is arbitrary.
I am doing an exercise in the book "Applied Stochastic Analysis" by E-Li-Vanden-Eijnden, and I meet this problem (on Page 26):
(Exercise 1.19) Prove that if the moment generating function $M_X(t)$ can be defined
on an open set $U$, then $M_X(t)\in C^{\infty}(U)$.
Here is my approach:
Use Taylor...
Hi, I have a problem about smoothness of moment generating functions on arbitrary open sets. Someone suggested an answer, but it does not have generality and I am still stuck at it. Can anyone take a look? Thanks. math.stackexchange.com/q/4018712/792125
Problem:
$\alpha$ and $\beta$ are two mutually orthogonal probability measures in the sense that for some Borel subset $E\subset[0,1]$, $\alpha(E)=0$ and $\beta(E)=1$. If $F(x)=\alpha\{[0,x]\}$ and $G(x)=\beta\{[0,x]\}$ for $0<x\leq 1$, show that for any $\varepsilon>0$, there is finite collectio...
Problem:
$\alpha$ and $\beta$ are two mutually orthogonal probability measures in the sense that for some Borel subset $E\subset[0,1]$, $\alpha(E)=0$ and $\beta(E)=1$. If $F(x)=\alpha\{[0,x]\}$ and $G(x)=\beta\{[0,x]\}$ for $0<x\leq 1$, show that for any $\varepsilon>0$, there is finite collectio...
Problem:
$\alpha$ and $\beta$ are two mutually orthogonal probability measures in the sense that for some Borel subset $E\subset[0,1]$, $\alpha(E)=0$ and $\beta(E)=1$. If $F(x)=\alpha\{[0,x]\}$ and $G(x)=\beta\{[0,x]\}$ for $0<x\leq 1$, show that for any $\varepsilon>0$, there is finite collectio...
Hello, I have this question regarding orthogonal probability measures. Can anyone take a look and help if possible? Thank you. math.stackexchange.com/q/3928794/792125
I have a question about showing that a Toeplitz operator is Fredholm:
Show that the operator $T_{e^{inx}}:L_2^+\to L_2^+$ acting on $L_2^+=\sum_{k\ge
0}a_ke^{ikx}$ with $\sum_{k}|a_k|^2<\infty$ is Fredholm for every
$n\in\mathbb{Z}$ and find its index.
Denote $S^1$ to be the unit circle. The sp...
I'm not sure. Maybe you are right. I have show that Lambda^+ can be represented uniquely by mu^+, and Lambda^- can be represented uniquely by mu^-, then I can apply decomposition to get the desired the result. The problem is that I cannot represent the negative part with the negative variation of the measure
The first problem is just proving that a bounded linear functional on C[0,1] can be decomposed into the difference of two nonnegative linear functionals, which I can handle. The problem is from a problem solving session in real analysis.
Here is the statement I'm trying to prove:
If $\Lambda(f)\in C([0,1])$ (space of bounded continuous functions on
$[0,1]$) satisfies $\Lambda(f)\leq C\sup_{x\in[0,1]}|f(x)|$, and
$\mu=\mu^+-\mu^-$ is a signed measure with
$\mu^+([0,1])+\mu^-([0,1])\leq C$, prove that $\Lambda(f)$ has the
represen...
I think the proof should be two sided: One side is just |Lambda(f)| \leq ||\Lambda|| * ||f||_sup (as you mentioned), but the other side should be harder.
Okay. I might post this problem to the community. I applied Riesz theorem and Hahn decomposition to it, but I find it hard to use the control inequality. @user2103480
@user2103480 Did you use Hahn Banach to complete the extension? And how to do you complete you last step-"finite measure \lambda for which |\Lambda(f)| \leq D*int_[0,1] |f|^2 d\lambda"?
What I've tried is to present the bounded linear functional Lambda by difference of two nonngative bounded linear functionals Lambda^+ and Lambda^-, but I failed to control the measure decomposition. Any ideas or thoughts? I may post this on the community if I have further progress. Thank you.
