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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449 views
Maximum Likelihood Estimation of an Ornstein-Uhlenbeck process
I am wondering whether an analytical expression of the maximum likelihood estimates of an Ornstein-Uhlenbeck process is available. The setup is the following: Consider a one-dimensional ...
6
votes
1answer
245 views
Does Itō isometry have different versions?
Itō isometry from Wikipedia:
Let $W : [0, T] \times \Omega \to \mathbb{R}$ denote the canonical
real-valued Wiener process defined up to time $T > 0$, and let $X :
[0, T] \times \Omega \to ...
5
votes
2answers
334 views
Decomposition of semimartingales
From Kal97, pg. 446:
Theorem 23.14 Any semimartingale $X$ has an a.s. unique decomposition $X=X_0 + X^c + X^d$ where $X^c$ is a continuous local martingale with $X_0^c=0$ and $X^d$ is a purely ...
5
votes
1answer
1k views
Distribution of hitting time of line by Brownian motion
I came across the following question:
Let $T_{a,b}$ denote the first hitting time of the line $a + bs$ by a standard Brownian
motion, where $a > 0$ and $−\infty < b < \infty$ and let $T_a ...
5
votes
1answer
314 views
Bounds for submartingale
Let $x$ be a positive number and $X_n$ be real-valued submartingale such that $X_0 = x$. I am interested in upper bounds on probability
$$
\psi(x) = \mathsf{P}_x\left\{\inf\limits_{n\geq 0}X_n \leq ...
5
votes
3answers
1k views
Distribution of compound Poisson process
Suppose a compound Poisson process is defined as $X_{t} = \sum_{n=1}^{N_t} Y_n$, where $\{Y_n\}$ are i.i.d. with some distribution $F_Y$, and $(N_t)$ is a Poisson process with parameter $\alpha$ and ...
5
votes
1answer
474 views
Recommendation on stochastic process books
I was wondering if someone could recommend good books on stochastic processes
with measure theory treatment
with not much or no measure theory
treatment
for each, it would be nice to have some ...
4
votes
3answers
203 views
Diverging random walk
I have a process $X_{n+1} = X_n\xi_n$ where $\xi_n\sim\mathcal N(1,1)$ and $\xi_n$ is independent of $X_n$. I need to prove that if $X_0\neq0$ then
$$
\mathsf P\{|X_n|>1\text{ for some }n\geq0\} = ...
4
votes
1answer
586 views
Questions about geometric distribution
I have some trouble understanding the record value for a sequence of i.i.d. random variables of geometric distribution. Following quotation is from Univariate discrete distributions By Norman Lloyd ...
4
votes
1answer
1k views
Interpretation of sigma algebra
My question is how to interpret sigma algebra, especially in the context of probability theory (stochastic processes included). I would like to know if there is some clear and general way to interpret ...
3
votes
0answers
96 views
Uniqueness of a local martingale problem
First, some notation: let $X = (X_{t})_{t\geq 0}$ be some strong Markov process in $E = \mathbb R^n$ with cadlag paths. Let us denote by $P_t$ the transition semigroup of $X$ and by $\mathbb B$ the ...
3
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1answer
195 views
what's the difference between RDE and SDE?
what's the difference between random differential equation and stochastic differential equation?
does stochastic differential equations include random differential equation?
3
votes
2answers
202 views
What is the average rotation angle needed to change the color of a sphere?
A sphere is painted in black and white. We are looking in the direction of the center of the sphere and see, in the direction of our vision, a point with a given color. When the sphere is rotated, at ...
3
votes
1answer
185 views
Poisson arrivals followed by locking
following is my problem:
Pulses arrive at a processor according to a Poisson process of rate λ.
Suppose each arriving pulse that is processed by the processor locks the processor for a fixed time T, ...
3
votes
1answer
293 views
About stochastic continuity
One of the definitions of stochastic continuity is: $\forall a > 0$ and $\forall s \geq 0$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$ What does it mean intuitively? I know that it implies the ...
2
votes
1answer
95 views
Sum of random subsequence generated by coin tossing
Let $(\pi_1, \pi_2, \cdots)$ be an infinite sequence of real numbers such that $\forall i\; \pi_i > 0$ and $\sum_i \pi_i = 1$. This can be thought of as a probability over natural numbers.
Let ...
2
votes
1answer
176 views
joint distribution of random vector
I want to find the joint distribution of the random vector $(W_t, \int_0^t W_s \; \mathrm ds)$
where $W_t$ is Brownian motion. I know $W_t \sim N(0,t)$, but I don't know how to calculate the ...
2
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3answers
358 views
How to show Martingale property for sum of $S_k-E(S_k)$-summands where $S_k$ is a function of two RV's
EDIT: new formulation of the question (old version below).
In a paper I found the statement that a certain sum $M_n =Y_1+\dotsb Y_n$ is a martingale, $Y_i=f (X_k, Z_k) - E ( f(X_k, Z_k) | X_k)$. (The ...
1
vote
0answers
54 views
$dX_t=1_{X_t\not=0} dW_t$
Given The SDE : $dX_t=1_{X_t\not=0} dW_t$ with $ X_{0}=\xi $
how can I construct two obvious strong solutions to prove that SDE has non pathwise uniquenss
Indeed
Consider the stopping time $$ ...
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vote
3answers
329 views
What is more elementary than: Introduction to Stochastic Processes by Lawler
I have trouble to reading this book!
What book is more elementary/preliminary than this book: Introduction to Stochastic Processes by Lawler
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1answer
694 views
Poisson Process
Customers arrive at a certain facility according to a Poisson process of rate lambda. Suppose that it is known that five customers arrived in the first hour. Each customer spends a time in the store ...
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1answer
254 views
Does this modified random walk (2D) return with probability 1?
Pólya showed that a random walk (with the directions at each step uniformly distributed) on the integer lattice returns with probability 1.
What if instead we consider the random walk where we are ...
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votes
2answers
628 views
Is this local martingale a true martingale?
Using the Ito's formula I have shown that $X_t$ is a local martingale, because $dX_t=\dots dB_t$, where
$$X_t = (B_t+t)\exp\left(-B_t-\frac{t}{2}\right),$$
$B_t$ - is a standard Brownian motion
I ...
5
votes
1answer
108 views
Poisson Process - Courts
IITK sports facility has $4$ tennis courts. Players arrive at the courts at
a Poisson rate of one pair per $10$ min and use a court for an exponentially
distributed time with mean $40$ min. Suppose ...
5
votes
2answers
406 views
Joint moments of Brownian motion
My approach to this SE question uses the following joint moments of
Brownian motion. For $n=1,2$ they are obvious and well-known, the others
are not terribly hard to work out. Is there a reference ...
4
votes
2answers
127 views
Can we prove directly that $M_t$ is a martingale
Suppose we define the stochastic process
$$M_t:=e^{\int_0^t\phi_s dW_s -\frac{1}{2}\int_0^t\phi_s^2ds}$$
where $\phi\in L^2[0,T]$, $t\in [0,T]$. Note that $M_t$ is just the stochastic exponential of ...
4
votes
2answers
45 views
Show $ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.$
How to show that for all $t\geq 0$ $$ \int _0^t \frac{\left|B_u \right|}{u}du < \infty \ a.e.,$$
where $ \left( B_t \right)_{t\geq 0}$ is the real standard brownian motion starting from zero ?
4
votes
1answer
151 views
Circular random walk
Suppose we have a circumference divided in N arcs of the same length. A particle can move on the circumference jumping from an arc to the adjacent, with probability $P_{k \to k-1}=P_{k\to ...
4
votes
1answer
361 views
Expectation value of a product of an Ito integral and a function of a Brownian motion
this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated.
Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the ...
4
votes
2answers
634 views
How do you check if a sequence of numbers is truly random?
Suppose a source produces an indefinite sequence of positive integers. How can you check whether the numbers are generated truly randomly?
3
votes
2answers
130 views
Generalization of a product measure
Let $(X,\mathfrak B(X))$ and $(Y,\mathfrak B(Y))$ be measurable spaces and further let $\mu$ be a measure on $\mathfrak B(X)$ and let $K$ be a kernel, i.e. for any $x\in X$ we have $K_x$ is a measure ...
3
votes
1answer
505 views
Finding the stationary distribution of a markov chain
I am asked to compute the stationary distribution of the markov chain with state space $E=\mathbb{N}_0$, $q_n >0$ for all $n \in \mathbb{N}_0$ and transition matrix below:
\begin{bmatrix}
q_0 ...
3
votes
1answer
301 views
Relations between Order Statistics of Uniform RVs and Exponential RVs
Say we have $U_1 \dots U_n$ i.i.d. random variables uniform on $[0,1]$ and $Y_1 \dots Y_{n+1}$ i.i.d. random variables distributed as $Y_i \sim Exp(1)$. I know that the joint distribution of the order ...
3
votes
1answer
123 views
Measurability question for martingale (Is $S(X_i, Z_i)- E(S(X_i,Z_i) \mid \mathcal{F}_i)$ $\mathcal{F}_i=\{X_1, \dots, X_i \}$-measurable?)
One condition for a martingale $M_k$ with a general filtration $\mathcal{F}_k$ is that the involved random variables $M_k$ are $\mathcal{F}_k$-measurable.
Now I have $M_n=Y_1+\dots +Y_n$ and ...
3
votes
2answers
181 views
Examples: invariant events
In a couple of books I'm reading chapters devoted to the Ergodic theory. As a setting:
$(\Omega,\mathcal F,\mathsf P)$ is a probability space,
$X:(\Omega,\mathcal F)\to (S,\mathcal S)$ is random ...
3
votes
3answers
404 views
2
votes
1answer
60 views
$\sigma$-algebras and independent stochastic processes
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a complete probability space. We consider a Wiener process $W$ with respect to his standard filtration $(\mathcal{F}_t^W)_{t \geq 0}$ and a process $X$ with ...
2
votes
1answer
62 views
Are these two some kinds of generalized Ornstein–Uhlenbeck processes?
An Ornstein–Uhlenbeck process is
$$
d X_t = (\mu - X_t) dt + d W_t
$$
We try to build a model using some generalized Ornstein–Uhlenbeck processes.
The first one is
$$
d X_t = \exp(-|X_t- \mu|) ...
2
votes
1answer
53 views
Continuous local martingales with same crochet have the same law?
Consider $M= \left(M\right )_{t \geq0}, \ N=\left(N\right) _{t \geq0} \in \mathcal M_{c,loc} $ starting both from zero, such that, a.e.$ \langle M \rangle_t =\langle N \rangle_t, \ \forall t\geq 0$.
...
2
votes
2answers
97 views
Relation between $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$
Let $B_t$ be a standard Wiener motion. What can we say about $\text{d}M_t$ and $\text{d}B_t$ when $M_t=\max_{0\leq s\leq t}B_s$? Is there a relation?
2
votes
1answer
232 views
Expectation of exponential martingale and indicator function.
Let $W$ be a Wiener process, $r,\sigma \in \mathbb{R}_+$ and $S(T) = S(t)e^{(r-\frac12 \sigma^2)(T-t) + \sigma(W(T)-W(t))}$.
I want to evaluate
$$A:=E[e^{- \frac12 \sigma^2 (T-t) - ...
2
votes
2answers
456 views
How to prove this property in a Poisson process?
For a Poisson process show, for $s < t$, that
$P(N(s)=k|N(t)=n) = \binom{n}{k} (\frac{s}{t})^k (1-\frac{s}{t})^{n-k}$
2
votes
1answer
146 views
Hitting open sets
Let $(\Omega,\mathscr F,(\mathscr F_t)_{t\geq 0},\mathsf P)$ be a complete filtered probability space and $X = (X_t)_{t\geq 0}$ be a cadlag stochastic process with value in a Polish space $E$. Is it ...
2
votes
1answer
236 views
Joint distribution of non homogeneous Poisson event times?
I am trying to calculate the density of $(T_1,T_2)$ where $T_1$ is the time of the first event and $T_2$ is the time of the second event. I have been looking at the Wiki article on Poisson process and ...
2
votes
0answers
65 views
Stochastic stability and convergence
Consider a Markov process $X$ on $\mathbb R$. Suppose that $X^2$ is $\mathsf P_x$-supermartingale for any $x\in \mathbb R$. If we want that for some neighborhood $U_0$ of $x=0$ holds: for each $x\in ...
2
votes
1answer
166 views
exponential stochastic process
Let $T$ be a random exponentially distributed time. $P \left(T > t \right)=e^{-t}$. Define $M$ via $M_t = 1$ if $t-T \in Q^+$, $M_t = 0$ otherwise. Where $Q^+$ being positive rationals. let ...
2
votes
1answer
1k views
Average waiting time in a Poisson process
Sheldon Ross's Introduction to Probability Models, exercise 5.44.b: cars pass a certain street location according to a Poisson process with rate $\lambda$. A woman who wants to cross the street at ...
1
vote
1answer
73 views
Stopping time proof
Let $\{X_t, t \ge 0\}$ be a continuous stochastic process and adapted to the filtration $\{\mathcal{F}_t,t\ge 0 \}$ and consider
$$
\alpha = \inf\{t, |X_t|>1\},
$$
the first time the the process ...
1
vote
0answers
46 views
Determining the spectral density?
Suppose you have a process $X_{t} = 0.5X_{t-1} + w_{t}$ where $w_{t}$ is $WN(0,\sigma^{2})$. How does one determine the spectral density of the process? Do you first find the ACF of the process and ...
1
vote
1answer
37 views
Is independence preserved in this special setting under a change of measure?
This is a question due to the answer of Did in this post Independent increments of $X_t:=\int_0^t\phi(s) dW_s$. Precisely, we assume that the dynamics of a stock prices follows
...
