Tagged Questions

1answer
39 views

My understanding of “$\sigma-$algebra represents information”.

In stochastic process $\{X_t\}_{t\ge0}$ adapted to $\{\mathcal F_t\}_{t\ge0}$ where $\mathcal F_s\subset\mathcal F_t,\forall s<t$. Many textbook say that $\{\mathcal F_t\}_{t\ge0}$ represents a ...
1answer
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Adapted and backward adapted?

I understand the following: Consider a probability space $(\Omega, \mathcal{A},P)$ and a Brownian motion $B=\{B_t, t\in [0,1]\}$ on this space and denote $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
0answers
40 views

Card Shuffling and Convergence in Probability

There are $4n$ cards, and we denote the set of cards with number $4k,k \in \{1,2,\ldots,n\}$ as $S$. The we shuffle the whole cards randomly, which means that each permutation will happen with the ...
1answer
38 views

law of iterated logarithm

Wikipedia claims see this link that the law of the iterated logarithm marks exactly the point, where convergence in probability and convergence almost sure become different. It is apparent from the ...
1answer
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Return time Markov chain

I have been wondering about this for quite a while now that I found in a textbook in the proof that an irreducible positive recurrent markov chain $(X_n)$ has a stationary distribution Let $t_i$ ...
1answer
201 views

Properties of Markov chains

We covered Markov chains in class and after going through the details, I still have a few questions. (I encourage you to give short answers to the question, as this may become very cumbersome ...
1answer
56 views

Eigenvalue markov chain

I have a questions: We said that if we have a positive recurrent Markov chain, then there is a unique stationary distribution. 1.) Does this mean that if I have several positive recurrent classes, ...
0answers
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Transient/Recurrent Markov chain

I am currently studying the concept of recurrent and transient states and was wondering about the following: Is this concept dependent on the initial distribution? Let me take this example: You can ...
0answers
31 views

Strong Markov property and its meaning

Given a sequence of random variables $(X_n)_n$ (fulfilling the Markov property) and a stopping time $\tau$ such that $P(\tau < \infty)=1$, we have that ...
2answers
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Brownian Bridge conditional probability

The problem is to show that the density $P[W_{t_1} \in dx_1,...,W_{t_n}\in dx_n | W_T = b]$ is the density of a Brownian bridge from $a$ to $b$. $W$ is Brownian motion. The density of a Brownian ...
1answer
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Superposition of simple birth process

Theorem Suppose $X$ and $Y$ are independent simple birth processes with birth rates $\lambda n$ with the same $\lambda$. Then $X+Y$ is a simple birth process rate $\lambda$ Proof let $Z=X+Y$ ...
1answer
150 views

certain stochastic process as martingale

Let $(\Omega, \mathcal{F}, (\mathcal{F}_n)_{n\in\{0,1,2,\ldots,N\}}, P)$ be a stochastic basis, carrying an adapted and integrable stochastic process $X=X_n$. Show that X is a martingale iff ...
2answers
1k views

Prove the time inversion formula is brownian motion

Let $B=(B_t)_{t\geq 0}$ be a brownian motion. Show the time inversion formula $\hat{B}=(B_t)_t\geq0$ is a brownian motion, where for $t \geq 0$ we set $\hat{B}=0$ for $t=0$ and $\hat{B}=tB_{1/t}$ for ...
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1answer
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Relationship between Hölder continuity and differentiability of Brownian motion

I came across the following exercise: Let $(B_t)_{t\geq 0}$ be a Brownian motion. Show that, almost surely, there is no interval $(r,s)$ on which $t\to B_t$ is HÃ¶lder continuous of exponent ...
1answer
219 views

Kolmogorov Backward Equation Boundary Value Problem

I need to solve the backward equation $u_t - \gamma x u_x + \frac{1}{2} b^2 u_{xx}$ subject to the final condition $u(x,T) = (x-a)^2$. Here a and b and $\gamma$ are constants. I am given a strong ...
0answers
681 views

First order variation and total variation of a function/stochastic process

The notions of first-order variation and total variation of a function or a stochastic process are equated in this book. However, I found their definitions different from two other sources: In ...
1answer
129 views

A question about poisson processes

Reading through the book "Brownian Motion & Stochastic Calculus" by Karatzas and Shreve, I found the following exercise (problem 3.9, page 15): Let $\ N \$ be a poisson process with intensity ...
3answers
288 views

Stochastic integral and Stieltjes integral

My question is on the convergence of the Riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. Here ...