Recent content by Paalfaal

I would appreciate it if someone could clear up the martingale problem: The SDE ## X_t = X_0 + \int_0^t b(s,X_s)ds + \int_0^t \sigma(s,X_s)dB_s ## as I understand it, is NOT a martingale wrt ## \{ \mathcal{F}_t:t\in [0,T] \} ## unless the drift term ## b(t,x) ## is zero. In Theorem 10.2.2...

Sorry, I was wrong in two spots; Klöden-Platen does not use Doobs martingale inequality, but rather the 'Markov inequality': ## P(|X| > a) \leq \frac{1}{a^r}E[|X|^r] ## for ##a,r > 0 ##. And thus I don't need to show that ##X_t^n## is a martingale. Moreover, Kuo uses 'Doobs submartingale...

I'm currently working my way through the existence theorem of strong solutions for the stochastic differential equation ## X_t = X_0 + \int_0^t b(s,X_s)ds + \int_0^t \sigma(s,X_s)Bs ##, Where ## \int_0^t \sigma(s,X_s)Bs ## is the Ito integral. The assumptions are: 1: ## b,\sigma ## are jointly...

A fellow student of mine asked a question to our teacher in functional analysis, and the answer we got was not very satisfactory. In our discussion on Banach spaces the student asked "Why is it interesting/important for a normed space to be complete?". To my surprise the teacher said something...

I certiantly will! Thank you for your help! Much appreciated :smile:

Hmmm.. When is it true that A \subseteq B \Rightarrow \bar{A} \subseteq \bar{B} ? I ask because I've been working on a similar problem: Prove that \bar{\ell}0 = \ell2 in \ell2. With a similar approach of the attempt above I got \bar{\ell}0 \supseteq \ell2. Since \ell0 \subseteq \ell2...

Homework Statement Define: - c0 = {(xn)n \in \ell\infty : limn → \infty xn = 0} - l0 = {(xn)n \in \ell\infty : \exists N \in the natural numbers, (xn)n = 0, n \geq N}Problem: Prove that \overline{\ell}0= c0 in \ell\infty Homework EquationsThe Attempt at a Solution I want to find the...

Of course! Thank you very much :smile:

Hi there! I'm working on a random walk-problem my professor gave me. Given a MC with the following properties: P0,0 = 1 - p P0,1 = p Pi,i-1 = 1 - pi Pi,i+1 = pi i \in[1,∞) Xn: state at step number n The chain is irreducible and ergodic and 0 < p ≤ 1 introducing N: N = {min n...

Thank you! Got it now :)

Ok, this should be quite simple. I've been looking at this problem for quite some time now, and I'm tired.. Please help me! The equation to solve is r' = r(1-r^2) The obvious thing to do is to do partial fraction expansion and integrate(from r_{0} to r): ∫ (\frac{1}{r} +...

I think it should be an A2 instead of A this line.. However, it made sense now. Thank you so much! Very helpful :smile:

I'm trying to prove eA eB = eA + B using the power series expansion eXt = \sum_{n=0}^{\infty}Xntn/n! and so eA eB = \sum_{n=0}^{\infty}An/n! \sum_{n=0}^{\infty}Bn/n! I think the binomial theorem is the way to go: (x + y)n = \displaystyle \binom{n}{k} xn - k yk = \displaystyle...