2
votes
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 ...
1
vote
1answer
24 views

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 $\mathcal{F}:=(\mathcal{F}_t)_{t\in [0,1]}$ ...
0
votes
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 ...
2
votes
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 ...
0
votes
1answer
23 views

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 $t_i$ ...
6
votes
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 ...
1
vote
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, ...
1
vote
0answers
42 views

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 ...
0
votes
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 ...
0
votes
2answers
40 views

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 ...
0
votes
1answer
40 views

Fubini Question in context of Independence

I am trying to show that if $X_t$ is some process and there is a function $p$ such that $$P[(X_{t_1},...,X_{t_n}) \in A_1 \times...\times A_n] = \int_{A_1 \times...\times A_n} ...
1
vote
1answer
32 views

American Put question

If the interest rate is zero. Then show that the optimal exercise for an american put option is always the terminal time. That is, it is equivalent to a european put option.
0
votes
1answer
42 views

Question on complex valued local martingales

So I was reading and found that the following was given as an example of a complex valued local martingale: $M_t = e^{\int_0^t f(\omega,s)dB_s - \frac 12\int_0^tf(\omega,s)^2ds}$ with $f(\omega,s) = ...
1
vote
1answer
58 views

Path Continuity and Stochastic Integration

In a book I'm working through there is a proof that $$\int_0^{\tau(\omega)\land t}f(\omega,s)dB_s(\omega) = \int_0^t f(\omega,s)1\{s\le \tau(\omega)\}dB_s(\omega)$$ The proof begins by claiming that ...
1
vote
1answer
51 views

Compute expectation (Ito integral/calculus)

I am having trouble computing this expectation. Does anyone know how to proceed? $$E\left[e^{2B(t)} \int_0^t s dB(s) \right].$$ Is it 0? I tried expressing $e^{2B(t)}=1+ 2\int_0^t ...
0
votes
1answer
44 views

Where is the error? Expectation, independent random variables

Let $X,Z$ be two correlated variables and $Y,Z\sim N(0,1)$ where Y is independent of $X,Z$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly ...
1
vote
1answer
58 views

Expectation/ independence of random variables

Let $X,Y$ be two correlated variables and $Z\sim N(0,1)$ independent of $X,Y$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly $E[f(X,Y)Z]=E[f(X,Y)]E[Z]=0$ ...
0
votes
2answers
164 views

Conditional Expectation given joint distribution

Given 2 random variables $X,Y$, is it possible to write conditional expectation $\mathbb{E}[X|Y]$ in terms of their joint distributional function $F_{X,Y}(x,y)$?
1
vote
1answer
67 views

Does $\sup_{t \leq T} |M_{n_k}(t)-M_{m_k}(t)|\to 0$ imply $\lim_k M_{n_k}(t)$ exists and is continuous?

This came up in proving that $\mathcal{M}^2_c$ is a complete metric space using the invariant metric induced by $$ ||M|| = \sum_k \frac{||M(k)||_2\wedge 1}{2^n}. $$ Suppose $M_n(t)$ is a sequence of ...
1
vote
0answers
110 views

A query on Palm Khintchine Theorem's proof

I was searching for a good reference on Palm Khintchine theorem proof. When I googled it, I got the following reference (as a Google book) here. It states that a superposition of independent "low ...
2
votes
0answers
89 views

Absolute Continuity and simple discontinuity

I am reading a book called Stochastic Process, Estimation, and Control, in P.32 it states that a function with finite simple discontinuities can still be absolutely continuous, which confused me, I ...
1
vote
0answers
22 views

Analytic random function

Let $t\in[0,1]\mapsto X_t\in[0,1]$ an analytic random function. I'd like to say that the deterministic function $t\mapsto \mathbb{E}[X_t]$ is still analytic. What are the minimal conditions needed? ...
1
vote
0answers
104 views

Lebesgue Stieltjes Integral for unbounded variation functions

Why can not we define Lebesgue Stieltjes integral $\int_0^T f dg$ when $g$ is not a bounded variation function on $[0,T]$ ?
1
vote
1answer
164 views

Brownian Motion and the Functional CLT

Suppose we have a time series $(x_t\mid t\in \mathbb{Z})$ for which the partial sum process $X_T$ defined on the unit interval by $$ X_T(\xi)=\omega_T^{-1}\sum_{t=1}^{[T\xi]} ...
0
votes
1answer
95 views

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 $Z=X+Y$ ...
1
vote
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 ...
5
votes
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 ...
1
vote
0answers
58 views

Equation Involving Bilateral Laplace Transform

Assume that $f(x,y)$ is a compactly supported, joint probability density function on $\mathbb{R}^2$ and nice enough for the following function to make sense: $$P_t(y):=e^{ty}-\int_{-\infty}^\infty ...
4
votes
2answers
110 views

boundedness of a martingale sequence?

Let $F_n(t)=\frac{1}{n} \sum_{k=1}^n 1_{X_k \leq t}$ be the emprical distribution function of i.i.d. random variables $X_k\sim U[0,1]$. Define for $t\in [0,1)$ $$M_n(t)=\frac{F_n(t)-t}{1-t}.$$ I have ...
2
votes
2answers
227 views

Convergence of $\frac{a_n}{n}$ where $a_0=1$ and $a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$

Given $a_0=1$ and:$$a_n=a_{\frac{n}{2}}+a_{\frac{n}{3}}+a_{\frac{n}{6}}$$Find convergence or divergence of $\frac{a_n}{n}$. Such a weird problem; I don't know how to attack it. I'm also fairly ...
2
votes
1answer
210 views

Pointwise convergence almost surely of an Approximation sequence

Unfortunately I've trouble to see the following: If you work with stochastic process $X$ you often want to approximate this in the following sense, define: $$ X^{(n)}(s,\omega) = ...
1
vote
0answers
76 views

Continuity of the function under the integral

Let $S,T\in\mathbb R$ such that $0\leq S<T<\infty$. For a function $X:\mathbb R\times \Omega\to\mathbb R$ I know that $$ \lim\limits_{t\to s}\int_\Omega (X(t,\omega)-X(s,\omega))^2\mathsf ...
3
votes
1answer
133 views

Levy processes with no positive jumps

Let X be a Levy process with no positive jumps and $\tau_y:=\inf\{t> 0: X_t > y\}$ then we have $$X_{\tau_y}=y\text{ on }\{\tau_y <\infty\}.$$ Could you explain that why? and does it hold ...
5
votes
1answer
493 views

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 Hölder continuous of exponent ...
1
vote
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 ...
4
votes
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 ...
5
votes
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 ...
5
votes
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 ...