0
votes
1answer
14 views

Application of Ito's Lemma to integral expression

I have a problem applying Ito's lemma. I know that if: $dX_t= \mu_t \, dt + \sigma_t \, dB_t$ then for $f(t,x)$: $df(t,X_t) =\left(\frac{\partial f}{\partial t} + \mu_t \frac{\partial ...
0
votes
0answers
17 views

Branching processes, extinction probability

Why do we assume that a branching process starts with one ancestor. What happens to extinction probability if we have more than one ancestor in generation Y(0)?
0
votes
1answer
15 views

Showing a process is a martingale from Ito's lemma

Suppose that we have the process $t-W^2(t)$ where $W(t)$ is a Brownian motion with filtration $\mathcal{F}_t$. It is easy to show that this is a martingale by computing ...
0
votes
0answers
26 views

Is there a standard proof for $\mathbb P(S^X_n\text{ hits }A\text{ before }B) >\mathbb P(S^Y_n\text{ hits }A\text{ before }B)$?

Let $X_i$ and $Y_i$ be two continuous random variables on $\mathbb{R}$ having distribution functions $F$ and $G$, respectively satisfying $G(y)>F(y)$ for all $y$. Let futhermore $S^X_n=\sum_{i=1}^n ...
2
votes
1answer
22 views

arbitrage free price in martingale measures

Consider a one-period market with $S^1_t,\cdots,S^n_t$, with $t=0,1$ the price process of $n$ assets, where $S_1$ is a risk-free asset: $S^1_0=1$,$S^1_1=1+R$. Assumes that this market satisfies the ...
0
votes
1answer
31 views

Gaussian vectors and covariance matrix.

The following is a part of a question I was given in stochastic processes course. It goes like this - I am given a series of gaussian iid random variables $\{V_i\}_{i=1}^N$ , the variable $X_0 \sim ...
0
votes
2answers
34 views

$1/(1+X_n)$ bounded in probability

I am trying to prove that if $X_n\rightarrow 0$ in probability, then $1/(1+X_n)$ is bounded in probability. My attempt is: $$P(\frac{1}{1+X_n}<\frac{1}{1-\epsilon})=P(|1+X_n|>1-\epsilon)\\ \geq ...
2
votes
0answers
12 views

Convergence of Quantiles moments.

QUESTION: Let $F$ be an absolutely continuous distribution function with density f, and $F_{n}$ be its nth empirical distribution. Suppose that $t\in (0,1)$ is constant. Is true the convergence ...
2
votes
1answer
53 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
0
votes
0answers
10 views

A Question on the independence of the sample mean and sample variance

The aim of the following question is to show the given random variable follows a student T distribution. Although it seems quite straightforward at the first sight, I am quite confused about the ...
0
votes
2answers
25 views

What is Poisson Point Process?

"Points $\{A_j\}_{j\in\Phi(\lambda)}$ are assumed to be distributed according to a homogeneous PPP with intensity $\lambda$, denoted $\Phi(\lambda)=\{X_j\}$, where $X_j$ is the location of the $j$th ...
3
votes
0answers
34 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
2
votes
0answers
33 views

Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...
1
vote
1answer
56 views

Use Ito's Lemma to show:

I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ ...
1
vote
1answer
35 views

Fixed-time Jumps of a Lévy process

If one defines a Levy process as a stochastic process $(X_t)_{t\geq0}$ that has independent and stationary increments with (a.s.) cadlag paths (hence a def. withouth stochastic continuity). How can I ...
0
votes
0answers
22 views

Girsanov's theorem for OU process

Say that I've got the process $dr_t=a(b-r_t)dt+\sigma dB_t$ and that I want to calculate $E[\exp(-\int_0^T r_s ds)]$ by using Girsanov's theorem. How do I do this? I cannot seem to find an explicit ...
1
vote
1answer
43 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
3
votes
0answers
101 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
2
votes
1answer
51 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
0
votes
1answer
32 views

a martingale equality

Let $X_{t}$ a positive continuous martingale satisfying: $\lim_{t\longrightarrow \infty}X_{t}=0 $ ps and $X_{0}=a \in {R_{+}}$ Show that $\textbf{P}(\sup_{t\geq0}X_{t} \geq b)=\frac{a}{b}$ , a < ...
0
votes
1answer
38 views

Representation of Markov process adapted to given Filtration

Let $X$ be continuous Markov process adapted to a filtration generated by Brownian motion $B$. Does there exist a function $f$ such that $X_t = f(t,B_t)$? My guess is that it should have such ...
2
votes
2answers
143 views

Traversing an array and counting the number of distanct number from the given elements in an array.

You are given an array $A[0 \ldots n-1]$ of $n$ numbers. Let $d$ be the number of \emph{distinct} numbers that occur in this array. For each $i$ with $0 \leq i \leq n-1$, let $N_i$ be the number of ...
2
votes
1answer
35 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
1
vote
1answer
41 views

If $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$ then $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s

Let $X_t$ be an Ito's process where $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$. Prove $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s Here my solution for ...
0
votes
2answers
28 views

If $u(z),$ where $Z_t=W^1_t-iW^2_t$, is a complex anlyt. fx, show $du(Z_t)=u'(Z_t)dZ_t$

If $u(z)$ is a complex analytical function, where $Z_t=W^1_t-iW^2_t$ is a complex Wiener process, show $du(Z_t)=u'(Z_t)dZ_t$.
0
votes
1answer
30 views

Quadratic Variation for $X_t= \int \sigma_s dW_s$ where $\sigma_s \in S$

Let $\sigma_s \in S$. Setting $X_t=\int^t_0 \sigma_s dW_s$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ such that $d_n=\max_i |t^n_{i+1}-t^n_i| \rightarrow 0$ as $n \rightarrow ...
1
vote
1answer
62 views

An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
7
votes
2answers
172 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
0
votes
1answer
40 views

Quadratic Variation for $X_t= \int b_s ds$ where $b_s$ is an $F_t$ adapted process.

Let $b_S$ be an $F_t$ adapted process, Borel measurable in $t$ st $\int |b_s|^2ds < \infty$ (a.s). Setting $X_t=\int^t_0 b_sds$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ ...
1
vote
2answers
59 views

Does a Brownian motion remain in any given open set for a given interval of time with positive probability?

Let $B$ be a standard $d$-dimensional Brownian motion. Given $b>a>0$ and an open ball $U$ in $\mathbb{R}^d$, I want to be able to comment on the probability that $B$ remains in $U$ during the ...
3
votes
1answer
116 views

Proving the reflection principle of Brownian motion

The reflection principle of Brownian motion states that Brownian motion reflected at some stopping time $\tau$ is still a Brownian motion. The proof found in Mörters & Peres (as well as in ...
0
votes
1answer
50 views

Conditional expectation of a stochastic process in filtered space

It was suggested* to me that if we have a stochastic process with independent increments, and $T > t$, then $$ E(X_{T-t}| \mathcal{F}_t) = X_{T-t} $$ where $\mathcal{F}_t$ is the filtration at time ...
1
vote
0answers
34 views

Find an example such that $\tau$ is a stopping time and $\mathcal{F}_\tau$ and $\mathcal{F}_\infty$ differ on $\{\tau = \infty\}$.

I need to find an example such that the following is true: $\tau$ is a stopping time and $\mathcal{F}$ is a filtration defined on $\mathbb{R}_+$. Let $\mathcal{F}_\tau$ denote the stopped ...
0
votes
0answers
14 views

Calculating the joint distribution of an affine stochastic process

I have a recursively defined system given by $$X_i = X_{i-1}H_i+N,$$ where $H_i$s are i.i.d. exponential random variables and N is a constant. At the $n$th iteration I have $$X_n = ...
3
votes
0answers
63 views

Measurability of one set of measures

Let $X,Y$ be a standard Borel spaces (a Borel subset of a complete separable metric space), and let $\mathcal B(X),\mathcal P(X)$ denote collection of Borel sets and Borel probability measures on $X$ ...
0
votes
1answer
24 views

Stopping time problem - Show that T is bounded

Let $a< 0 < b$ and $W_t$ is Brownian motion $T_a$=inf{$t\ge$0|$W_t\le a$} $T_b$=inf{$t\ge$0|$W_t\ge b$} T=min{$T_a$,$T_b$} $1)$ Show that $T$ $<$ $\infty$ My attempt : ...
2
votes
1answer
35 views

Skorohod convergence (space of right continuous functions with left limit)

If $f_n$ is a sequence of functions of the Skorohod Space $D([0,\infty),E)$, where $E$ is a separable Banach space, such that $f_n \to f$ in the Skorohod topology. Is it possible that there exists a ...
0
votes
0answers
28 views

Why define shift invariant set as $\Lambda=\tau^{-1}\Lambda$?

Can we define shift invariant set as $\Lambda = \tau\Lambda$ instead of $\Lambda = \tau^{-1}\Lambda$, where $\tau$ is the shift operator? Can permutable set be defined as either $\Lambda = \pi ...
0
votes
0answers
41 views

The relationship of $\sigma(f(X))$ and $X$

If X is a random variable and f is a measurable function, 1) Is f(X) measurable with $\sigma(X)$ ? 2) Is X measurable with $\sigma(f(X))$ ? Please give proof & example or counter example. Ok ...
1
vote
1answer
59 views

problems with probability kernels

Let $(S,\mathcal{S})$ and $(T,\mathcal{T})$ be measurable spaces and consider a measurable function $\phi: S\to T$. Define a probability kernel $\Phi$ from $S$ to $T$ by $\Phi(x,\cdot) = ...
1
vote
1answer
33 views

Showing that a Bessel-squared process is time homogeneous

I'm trying to show that a squared Bessel process is time homogeneous. So far, I've shown that $Y_t=f(X_t)$ is a time-homogeneous Markov process with transition semigroup $(\nu_t)_{t\geq 0}$ w.r.t. ...
0
votes
1answer
46 views

Proving a chain is aperiodic, and finding a stationary distribution.

We have an irreducible Markov chain with a not necessarily finite state space. It has a transition matrix $P$ such that $P^2=P$. Prove (1) the chain is aperiodic, and (2) prove $p_{ij}=p_{jj}$ ...
0
votes
1answer
17 views

Question involving an invariant measure on a Markov chain

Suppose $\mu$ is an invariant measure for a Markov chain with state space $S$ with $\mu(i)p_{ij}=\mu(j)p_{ji}$ $\forall i,j \in S$. Describe a Markov chain with this property. Also, show that $\mu$ is ...
0
votes
0answers
21 views

Showing transition function is a transition semigroup

Suppose we have two measurable spaces $(S,\mathcal{S}),\,(T,\mathcal{T})$ and a measurable function $f:S\to T$. Define a probability kernel $\phi$ from $S$ to $T$ by $\phi(x,\cdot) = \delta_{f(x)}$. ...
3
votes
1answer
62 views

Filtrations and Sigma-Algebras

I have been practising a question set by my lecturer and try to verify the answer, unfortunately I am unable to understand the following question and answer. $\textbf{Question:}$ Let ...
1
vote
0answers
40 views

About one stoping time definition in Chung's book (A Course in Probability Theory)

In Chung's book, he defines the stopping time $ \alpha^k $ in the following way. $\alpha^1 = \alpha$; $\alpha^{k+1}(\omega) = \alpha^k(\tau^\alpha \omega)$; where $\tau^\alpha$ is the ...
0
votes
1answer
35 views

Invariance of Brownian motion under orthogonal transformations

Let $\left(B_t\right)_{t \in [0,\infty)}$ be an $n$-dimensional Brownian motion with start at $x \in \mathbb{R}^n$, and let $A$ be an orthogonal $n \times n$ real matrix. I'm trying to show that $AB$ ...
1
vote
0answers
14 views

Hitting time for a planar diffusion

Let $A$ be an open subset of $\Bbb R^2$, and let us consider a diffusion $\mathrm dX_t = f(X_t)\mathrm dt + g(X_t)\mathrm dW_t$ where $f$ and $g$ are globally Lipschitz continuous maps. Suppose I am ...
0
votes
0answers
27 views

Continuous time Markov chains, how would this definition be expanded from time-homogeneous to time-inhomogeneous.

Below I have a picture of how we can view a continuous time Markov chain that is time-homogeneous. Now, I am wondering what happens when we have a inhomogeneous continuous Markov chain. I have ...
0
votes
1answer
34 views

Markov processes and semimartingales

Semimartingales and Markov processes are two fundamental families in probability theory. There are many specific processes that belongs to the intersection of those two families, e.g. Levy processes. ...