A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

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First-passage probability with absorbing boundary at origin (No Laplace)

I have the following problem which I would like to solve without using Laplace transform. Can you possibly help or provide pointers? What is the first-passage probability, and mean first-passage time ...
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302 views

Which is a good textbook on stochastic processes which takes measure theoretic approach?

I was looking for an intermediate-advanced textbook on stochastic process. I have graduate level probability knowledge.
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139 views

Stochastic differential

Im really new in the stochastic procceses please help me. How can I solve this stochastic differential equation? $$dX = A(t)Xdt$$ $$X(0) = X_0$$ If $A$:[0,$\infty$]$\to$ $R$ is continous and $X$ is ...
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62 views

Lower bound for stochastic process

Suppose the non-negative stochastic process $(X_t,Y_t)$ is such that $E\{X_t - X_a | Y_u \in A \,\,\forall u \in [a,t] \} \geq Z(A)(t-a)$. Let $T_{A}$ be the time of a visit to $A$. Assuming that the ...
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46 views

How can I determine this equation?

I have an equation that have been formulated as: $$\Delta w_{ki}=\eta (y_{k}-y_{o})(x_i-x_o)+a_1,$$ where$$y_{k}=\sum_{j}w_{kj}x_{j}+a_{2}$$ and where $a_1,\eta,x_o ,y_o$ and $a_2$ are constants and ...
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101 views

Compute $d(\log(S_t))$ using Ito's Formula

We are given the following: d$S_t$ = $\sin(S_t)t^2dt + e^{\sqrt{S_t}-t}dB_t$ And are asked to compute several different things, one of which is $d \log(S_t).$ If I'm understanding Ito's formula ...
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88 views

Atypical exponential martingale

Process $\{M\}$ is a pure-jump martingale, with finite number of jumps on any finite time interval, and a compensator $a_t$ at every time $t$. It can be thus written: $$ M_t = \sum_{0<s\leq t} ...
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31 views

Exercise about stationary processes [duplicate]

I need some help with this exercise: $\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the processes $X(t)$ and $X' (t)$ are orthogonal. $\bullet$If ...
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1answer
65 views

Stochastic processes with non-zero higher order variations

I'm under the impression that how non-zero quadratic variation of the Brownian motion results in Itō's lemma or in general, the creation of the Itō's calculus. I'm also aware that stochastic integral ...
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982 views

What is the transition matrix and stationary distribution of ${T}$

Let $T = (X_n:n \in \mathbb{N})$ denote a homogeneous Markov chain with state space $E=\lbrace 1, 2, 3\rbrace$ and $$\mathbb{P}(X_1=2\vert X_0=1) = \mathbb{P}(X_1=3\vert X_0=1)=\frac{1}{3}$$ as well ...
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172 views

A question about predictable stochastic process.

Let $X_t$ be predictable with respect to filtration $(\mathscr{F}_t:t\in[0,T] )$. If I observe the process over an interval $[0,s],0<s<T$, does that mean I can tell the value of $X_t$ over ...
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74 views

Time after last jump and waiting time before the next jump of Poisson process

Consider $N =(N_t)_{t\geq0}$ a Poisson process of intensity $\lambda > 0$ and $(T_n)_{n\geq 1}$ its jump instants. Then consider for all $t \geq 0$, $Z_t = t- T_{N_t} \mathbb 1 _{\{ t \geq ...
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85 views

Conditional distribution of compounded Poisson process

Consider a Poison a process $N = (N_t )_{t\geq 0}$ of intensity $\lambda >0$ whose instants of jumps are $(T_n)_{n\geq0} $ $(T_0 =0)$ and a process $\tilde{N} =(\tilde N_t )_{t\geq 0}$ defined as ...
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103 views

Adaptation of sum of arrival times of Poisson process

Let $\{N_t\}_{t\geq0}$ be a Poisson process and $\{F_t\}_{t\geq0}$ be its nautral filtration so that $\{N_t\}_{t\geq0}$ is adapted. $T_i$ be the $i$th arrival time of Poisson process of arrival rate ...
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3k views

What is the definition of a sample path of Brownian motion?

My question has been asked before at beginner's question about Brownian motion . There was only one answer, which was not accepted. It was probably incorrect, because nothing was said about ...
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67 views

Approximation of Stochastic integral with Stieltjes integrals

Let $V^n(t,\omega)$ be a sequence of continuous, adapted and bounded variation processes such that with probability 1, $V^n$ converges to $B$ uniformly on compact intervals of $[0,\infty)$ ($B$ is ...
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174 views

Survival Probability of a Population

A population starts with one amoeba. In each generation, each amoeba divides in two with probability $\frac{1}{2}$, or dies, with probability $\frac{1}{2}$. Let $p_n$ be the probability that the ...
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113 views

Skorohod space measurable function

Consider the space of RCLL (Cadlag) functions on the domain $[0,1]$ and endowed with the Skorohod topology. Let us consider the set $S := \{x: \omega_x (\delta) \leq \epsilon\}$, where $\omega_x ...
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292 views

A book/text in Stochastic Differential Equations

Somebody know a book/text about Stochastic Differential Equations? I'm in the last period of the undergraduate course and I have interest in this field, but my university don't have a specialist in ...
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1answer
100 views

Show $\sup_{s \in [0,t]} \mathbb{E}|X_s|<\infty$ for any submartingale $(X_t)_{t \geq 0}$

My problem: For any submartingale $(X_s)_{s\geq0}$ and for all $t\geq0$ show that $\sup_{s\in[0,t]}\mathbb{E}[|X_s]|]$ is finite. What I have until now: I know that $\mathbb{E}[X_s]$ is ...
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1answer
250 views

Need help with Sigma-algebra

I am confused on how to determine a Sigma-algebra. The following partitions of a set are given: $$ A1 = \{1,3\} $$ $$ A2 = \{2,4,6,8\} $$ $$ A3 = \{5,7,9\}$$ And Omega is $$ \Omega = ...
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376 views

About stationary and wide-sense stationary processes

I have just started with stochastical calculus, and I need some help with a pair of problems: $\bullet$If $X(t)$ is a mean square differentiable wide-sense stationary stochastical process then the ...
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73 views

A minor clarification on completion of $\sigma$-algebras

This is from Karatzas + Shreve Definition: The stochastic process $X$ is adapted to filtration $\{\mathcal{F}_t\}$ if, for each $t\geq 0$, $X_t$ is an $\mathcal{F}_t$-measurable random variable. ...
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224 views

measurability question with regard to a stochastic process

Here are two related exercise from Karatzas and Shreve Let $X$ be a process, every sample path of which is right continuous with left limits. Let $A$ be the event that $X$ is continuous on $[0,t_0)$. ...
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1answer
31 views

Distribution F in a renewal process

Referring to the Ross textbook stochastic processes, define the inter arrival times $x_i$ follow a distribution $F$. What is F representing here the PDF or CDF? What would $\overline{F}$ represent?
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212 views

Basic (continuous) martingale properties

I just learned about martingales in continuous time and solved some basic exercises. But unfortunately there are some seemingly easy and surely basic things I still have problems with. 1) Let ...
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1answer
132 views

Causality MA(1) process

Given is the MA(1) process: $X_t = Z_t + \theta Z_{t-1}$ Where, $Z_t \sim WN(0,1)$ For what values of $\theta$ is $X_t$ a causal function? I know how to show causality for a AR(p) process with ...
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2answers
133 views

Can we define probability of an event involving an infinite number of random variables?

Consider a collection $(X_a)_{a\in[0,1]}$ of i.i.d. random variables following the uniform distribution on [0,1]. That is, for every real number $a \in [0,1]$ we have a random variable $X_a$. Can we ...
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53 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 ...
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85 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 $E[f(X,Y)Z]=E[f(X,Y)]E[Z]=0$ ...
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60 views

Locally Nondeterministic Property of Brownian Bridge

Could anyone please give ideas or point me out references where I can find any result concerning the locally nondeterministic (LND) property (in the sense of Berman: ...
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1answer
453 views

Expectation of sum of arrival times of Poisson process in $[0, t]$

Let $T_i$ be the $i$th arrival time of Poisson process of arrival rate $\lambda$, given $t>0$, how to calculate $$ E(\sum_{i=1}^\infty T_i 1_{\{T_i<t\}}) $$ I think since this is equal to $$ ...
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133 views

Find the distribution of $T_a=\inf\{n\ge 0: R_{n}\gt a\}$ for fixed number $a\gt 0$

Suppose $R_{n}=\sum_{i=1}^{n} X_{i}$ for $n\ge 1$ and $R_{0}=0$ , where $X_{i}\gt 0$ are independent and identically distributed. Find the probability law of the stopping time $T_a=\inf\{n\ge 0: ...
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1answer
117 views

Showing that a hitting time is $\mathbb P-\text{a.e.}$-finite

Let be $\alpha, \beta \in \mathbb R$ such that $\alpha < \beta $ and $x \in [\alpha, \beta ]$. Consider the random time $$T_x = \inf \{ t\geq 0 : x+ B_t \notin [\alpha, \beta]\},$$ where ...
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151 views

Intensity Function of Stochastic process`

I'm fitting some financial data to a model based on a stochastic process and evaluating the fit of it by looking at the compensator. However, I cannot understand well what does it mean to take the ...
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1answer
263 views

First hitting time on a element of $\mathcal B ( \mathbb R^d) $ a (right, left) continuous path stochastic process

It's known that, given $\Gamma \in \mathcal B (\mathbb R ^d)$ and $X = > (X_t)_{t\geq 0}$ with right-continuous path, the random time $$T_{\Gamma} = \inf \{ t\geq 0 : X_t (\omega) \in ...
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54 views

Probability of winding number in 2D Brownian motion

Let $B_t$ be a 2D Brownian Motion with $B_0 = (1,0)$. Now, express $B_t$ in polars, that is, $B_t = (r(t), \theta(t))$. Let $\tau = \inf\{t > 0 : \theta(t) \geq 2 \pi \}$. What is $\mathbb{P}[\tau ...
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Percolation and number of phases in the 2D Ising model.

Update. As my previous figure had conceptual mistakes I decided to change the picture to another, more instructive After a long time I came back to try to understand an article on the Ising ...
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1answer
55 views

A small question in stochastic process(sth. related to number theory)

Let $J$ be a set of nonnegative integers whose greatest common divisor is $d$. And suppose that $J$ is closed under addition, then J contains all but a finite number of integers in the set $ \{ ...
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44 views

is this solution correct about joint distribution?

The question is if $x,y,z$ are independent $x\sim\exp(\lambda), y\sim\exp(\mu), z\sim\exp(\gamma)$ and define $u=\min(x,y), v=\min(y,z)$ what is the probability $p(U>u,V>v)$. Consider the cases ...
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1answer
45 views

Probability of $i$ elements in a Branching process

I have a branching process that's family size each generation is Binomially distributed. How do I calculated the probability of the family size $Z$ in stage $n$ is $i$: $P(Z_n=i)$. At the begining ...
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1answer
392 views

Probability of 2D Brownian motion passing through a particular point.

Let $B_t$ be a two-dimensional Brownian motion at time $t \in [0,\infty)$. Fix a point $p \in \mathbb{R}^2$. Is the probability that $B_t = p$ at some $t > 0$ equal to zero? If so, why?
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1answer
48 views

Definition of partition of set regarding countability

In Stochastic Processes, we define a partition of a set as: A countable collection of sets $U=\{A_1,A_2,A_3,...\}$(which may be a finite collection) which are pairwise disjoint is a partition of a ...
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141 views

Forming a local martingale with continuous increasing process

If $M_t$ is continuous martingale, we know that there exists quadratic variation process which is continuous and increasing. I am interested to know if the converse is also true. To make it precise ...
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2answers
192 views

Cover information theory 7.21 tall, fat people

I am stuck on Thomas Cover information theory 2nd edition, problem 7.21 Fat, tall people. The problem is like following: 7.21 Tall, fat people. Suppose that the average height of people in a room is ...
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79 views

limit of sup of a stochastic integral [duplicate]

Let $W$ be a standard, one-dimensional Brownian motion and $0 < T < \infty$. Show that $$\lim_{\beta \to \infty} \sup_{0\leq t \leq T} |e^{-\beta t }\int_0^t e^{\beta s } dW_s| = 0$$ a.s.
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2k views

Expected stopping time of brownian motion

I've been looking at this for some time now and still have no sensible solutions, can somebody help me out please. Say I define the stopping time of a Brownian motion as followed: $$\tau(a) = \min (t ...
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1answer
272 views

Random Dynamical Systems: Intuitive Understanding

I am having trouble understanding this definition, [Arnold: Random Dynamical Systems]: Definition: A measurable random dynamical system on the measurable space $(X,\mathcal{B})$ over a metric ...
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1answer
124 views

Branching process question.

Suppose we have a branching process, where at each time $n$, each individual produces offspring independently with the distribution $\{p_k\}$ and then dies with probability $0 < q < 1$. For ...
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2answers
585 views

$L^1$ bounded martingale

If $(M_t)_{0\leq t<\infty}$ is continuous martingale and it is $L^1$ bounded, does it imply that quadratic variation $\langle M\rangle_\infty$ is finite a.s. ?