Questions tagged [stochastic-processes]
For questions about stochastic processes, for example random walks and Brownian motion.
15,781
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Probability of getting given pass generated by Gaussian process
Suppose I have a Gaussian Process $f \sim GP(\mu, k)$ with given mean function $\mu(x)$ and covariance function $k(x, x')$. I also have a trajectory $\textbf{p} = (p_1, \dots, p_n) \in \mathbb{R}^n$.
...
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Ito-Formula for a Poisson-Process
I am new to Stochastic Theory and trying to understand (Prop 20.13) of this Article
https://personal.ntu.edu.sg/nprivault/MA5182/stochastic-calculus-jump-processes.pdf
(The Ito-Formula for a Poisson-...
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Solution manual for "Continuous Time Markov Processes" by Liggett
Is there a solution manual for "Continuous Time Markov Processes: An Introduction" by Thomas M. Liggett? Thanks.
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Behaviour of a spectral measure - question regarding certain terms used to describe its behaviour
Good morning,
I'd like to ask a question based on this paper, which is about Gaussian Stationary Processes.
One of the main concepts is the concept of a spectral measure. After Theorem 4.1 of this ...
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3
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Yule process intuitive question
I have a Yule process with $n$ individuals.
There is no death, so the death rate is $\mu_n$ $=$ $0$ for all $n$.
Each individual gives birth to a new individual independently after waiting for $\text{...
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1
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How does $\Omega$ figure in stochastic processes?
So I read this page for clarification on trajectories and $X(\omega, \cdot): T\to \mathbb R$ maps while going through lectures on stochastic processes. I still have doubts which are described as ...
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function of Gaussian process?
I am not very versed in stochastic calculus and I need help with this. I have a stochastic variable describing noise in a physical system, $\phi(t)$, whose derivative is white and Gaussian, i.e. it ...
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Watanabe characterization of a Poisson process
There is an implication that I am not able to find by myself in my lecture notes.
I consider $X_t$ a cadlag process with values in $\mathbb{R}_{+}$ such that $X_t$ is locally integrable (with respect ...
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i need a stochatic differential equation that its solution will represent this data of wind direction in radians.
enter image description here
hi everyone
I have this data of wind direction in radians from -pi to pi (picture is added).
any idea which stochastic differential equation (SDE) can represent it? i need ...
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1
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Find the stochastic differential equation that is solved by $e^{Y_t}$
Let $Y_t=\left(\int_0^t e^{t-s}dB_s\right)^2$. Find the stochastic differential equation that is solved by $e^{Y_t}$. The answer must be given in differential notation.
What I did was rewrite $Y_t$ as
...
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How to compute $E[B_{1}^2B_{2}^2]$ where $B_{t}$ is a standard Brownian motion starting at 0 [closed]
We are given that $E[X^4] = 3\sigma^2$ for gaussian X, and I've tried solving for $E[(B_1^2+B_2^2)^2]$ to use this fact, but I really am not clear on what to do
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Prove equality without Ito's formula
Prove the following equality without Ito's formula:
$$tB_t=\int_0^t sdB_s+\int_0^tB_sds.$$
I was thinking of moving $\int_0^t sdB_s$ to the left side and then taking the expectation on both sides to ...
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Variation of St. Petersburg Paradox
I was discussing the the St. Petersburg paradox and the following question came up:
Suppose the game doesn't end within nine rounds, then the player directly receives $2^{10}$ dollars , while ...
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1
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Girsanov-type Theorem that alters the variance of a Wiener process
Consider a general probability space $(\Omega, \mathcal{F}, \mathbb{S})$, on which two or more other probability measures, $\mathbb{P}_1$, $\mathbb{P}_2$,...,$\mathbb{P}_j$,...,$\mathbb{P}_n$ are ...
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What is the formula for the Mode of the Log-Normal distribution followed by the Geometric Brownian Motion?
What is the formula for the Mode of the Log-Normal distribution followed by the Geometric Brownian Motion?
This question was reformulated because of claims in the comments and closing votes.
...
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Prove that $\{ f \in \mathbb{R}^{[0,1]} : \sup_{t \in [0,1]} f(t) < 1 \}$ is not measurable?
Let $\mathbb{R}^{[0,1]}$ be the set of all functions $f : [0,1] \to \mathbb{R}$. The infinite dimensisonal product $\sigma$-algebra $\mathcal{T}([0,1], \mathbb{R})$ is generated by the cylindrical ...
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Application of some estimate to stopped process
I have question regarding this step:
Assume you have a stochastic process $X_t$ and a stopping time $\tau$.
Furthermore assume that some estimate like
$\mathbb{E}f(X_t)\leq \mathbb{E}g(X_t)$ holds for ...
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Prove that $ \int^t_0X_s dA_s$ is progressively measurable.
Let $(\Omega, \mathcal F_\infty, \mathcal F= (\mathcal F_t)_{t\geq 0})$ be a filtered probability space, let $X = (X_t)_{t\geq 0}$ be a progressively measurable process and $A= (A_t)_{t\geq 0}$ be a ...
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Understanding the sum of Bernoulli variables, and functions defined on such sums.
Let $y_1,y_2$ be independent Bernoulli variables, and let $f(y_i)=ay_i+1$ be a function defined on both Bernoulli variables. Also, let $B_i(f(y_i))=ay_i$ be an operator acting on the function $f$. ...
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Maximum value of the sum of sine functions with random phases
There are infinite sine functions that have the form:
$$
f_i(x) = A_i\sin(k_i x +\phi_i),
$$
where $A_i$ is the amplitude, $k_i$ is the frequency ($i$ times the fundamental frequency $k_1$) and $\...
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Ornstein-Uhlenbeck Operator as an Infinitesimal Generator of a Stochastic Process
This question continues the discussion from an earlier post on this website found here :
Why the operator is termed as Ornstein–Uhlenbeck operator? I am interested in the relationship between the ...
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Solve a quite easy Stochastic Differential Equation
Given $du=\mu dt +\sigma dB$ where B is an one-dimension standard Wiener process, then what's $u(t)$? Is there a general solution when $t\to 0$?
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How to prove that $(N-\lambda t)^2 -\lambda t$ is a martingale where $N$ is Poisson process [closed]
Let $N$ be a Poisson process with a parameter $\lambda$ >0. Can anyone help to show that $(N-\lambda t)^2 -\lambda t$ is a martingale? You already answered to this question, but I don't get why $\...
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Donsker's theorem for multivariate Brownian motion [closed]
Let $B = (B^1, \dots, B^n)$ be an $n$-dimensional Brownian motion (i.e. $B = (B_t)_{t \geq 0} \in \Omega \rightarrow (\mathbb{R}^n)^{[0,\infty)})$.
Do we have something as Donsker's theorem to show ...
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How to find an invariant measure for continuous-state discrete time Markov chain?
I have a continuous-state discrete-time Markov chain and I want to find the density of an invariant measure when I already know by some theorems that there exists a stationary measure.
In a discrete-...
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Correcting term to variance OU process when using Euler Maruyama and time step of 1
I am trying to simulate the following OU process.
$$
dX_t = -\theta X_t dt + \sigma dW_t
$$
I then simulated the process using the Eulers-Maruyama discretization approach i.e
$$
X_{t+\Delta t} = X_{t} ...
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Derivative of the quadratic variation of Levy process
Let $L(t)$ be a n-dimensional Levy process having the decomposition
$$
L(t) = \int_{B} x \widetilde{N}(t,dx)
$$
where $B=\{ |x|<1 \}$ and $\widetilde{N}(dt,dx) = N(dt,dx) - \nu(dx)dt$ is the ...
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Gaussian processes as elements of a Hilbert space
Let $(\Omega, \mathcal{F}, P)$ be a probability space. We define a random variable on a Hilbert space $H$ as a measurable function
$$
X : \Omega \to H
$$
where $H$ is equipped with its associated ...
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Moments of waiting time in a G/G/1 queue
Are there any results for moments of waiting or sojourn time(total time spent by a job in the system including its own service) for a G/G/1 queue. I know that in the special case of M/G/1 queues the ...
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Brownian motion X(t) is with probability 1 a continuous function of t
Here is an excerpt from "An Elementary Introduction to Mathematical Finance" by Sheldon Ross, 3rd edition:
I understand this is not meant to be rigorous, but I'm having trouble ...
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Basic Poisson Process
This is Exercise 2.1 of Varadhan's Stochastic Processes. Let $\tau_i$ be a sequence of independent identically distributed random variables with a common exponential distribution $e^{-\tau}\textrm{d}\...
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Computing expected value of hitting time for a Feller process
We consider a Feller-Dynkin Markov process $X$ with generator $G$, which, when restricted to $C^2$ functions with compact support, is given by $Gf(x) = \frac{c(x)}{2}f''(x)$, where $c$ is a positive ...
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Sample Space of such a random variable
Suppose there is a random experiment in which a person is asked to flip a coin $3$ times. The coin has $2$ sides (numbers and pictures). In the sample space of the random experiment, $3$ random ...
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Meaning of the terms in the infinitesimal generator formula
In the following formula
$$Af(x) := \lim_{t \downarrow 0} \frac{\mathbb{E}^x(f(B_t))-f(x)}{t} $$
If $B_t$ is the Brownian motion, what are $f(B_t)$ and $f(x)$? here some explanation was given. Can ...
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1
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Is continuous, uniformly integrable martingale indistinguishable from BMO martingale?
Definition:
BMO: Let $M$ be a martingale in $\mathcal{H}^2$. $M$ is said to be in BMO if there exists a constant c such that for any stopping time T we have
$$
E\{(M_\infty-M_{T_-})^2 \mid\mathcal{F}...
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Moment Generating Function of $\bar{M}−\bar{N}$
What is the Mean value of $\bar{M}−\bar{N}$; Moment Generating Function of $\bar{M}−\bar{N}$; and Variance of $\bar{M}−\bar{N}.$ Given $M_1,M_2,\dots,M_n$ is a random sample of size $p$ from the Gamma ...
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Calculate Levy triplet of a scaled Levy process
Let $X$ be a Levy process.
From the Levy-Khinchine formula and basic properties of Levy process, it follows that
$$\mathbb{E}e^{iuX_t}=exp\{t[ibu-\frac{1}{2}au^2+\int (e^{iux}-1-iux\mathbb{I}_{B_1}(x))...
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Consider a discrete time Markov chain
Determine the value of $$𝑃(𝑋_{3000000}=2 | 𝑋_0=1); 𝑃(𝑋_{3000001}=2 | 𝑋_0=1); 𝑃(𝑋_{3000002}=2 | 𝑋_0=1).$$ If consider a discrete time Markov chain $X_1,X_2,\dots$ with a state space set $S=\{1,...
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Union over preimages of projections form an algebra
Let $(S,\mathcal{S})$ be a measurable space, $J\subset T$ and denote
$$S^T=\{x=(x_t)_{t\in T} \ \lvert \ x_t\in S \}.$$
Let $\mathcal{S}^{\otimes T}$ be the smallest $\sigma$-Algebra that contains all ...
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answer
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How to calculate probabilities in a Poisson process with exponential lifetime of arrivals?
I have a Poisson process where people arrive at the rate of $λ$ -- so when an event occurs, a new person arrives. This means that the times between successive arrivals are $T_i$ ~ Exponential $(\...
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sde for brownian bridge
Let $B(t)$ be a standard Brownian motion and
\begin{align*}
Y(t)=B(t)-tB(1) &&
Z(t) = \left\{
\begin{array}{ll}
Z(t)=(1-t)B\left(\frac{t}{1-t}\right)& t\in [0,1)\\
0 & t=1 \\
\end{...
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Finding rates by setting up a birth and death process
I have the following scenario, where I am trying to set up a birth and death process.
There are $10$ bulbs, and the bulbs have independent Exponential $(\lambda)$ lifetimes. If a bulb stops working, ...
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How to prove the equivalence of the following Markov property?
$X$ is a stochastic process, $\mathcal{F}_t=\sigma(X_s: s\leq t), \mathcal{G}_t=\sigma(X_s: s\geq t)$, prove that the following statements are equivalent:
$X$ has Markov property.
$\mathbb{P}(A\cap B|...
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1
answer
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Can we show a process has bounded mean?
Assume $X_0=0$. The process evolves with the following rules.
$X_{t+1} = X_{t} +1 $ if $X_t<10$;
If $X_t\ge 10$,
$$
X_{t+1} = \begin{cases}
X_t + 1 &\text{with probability } 1/2\\
X_t -3 &...
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1
answer
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Hitting time of Brownian motion past a given point in time
The random variable whose distribution I am interested in is defined as follows:
$$\tau := \inf\{u > 1: W_u = 0\}$$
where $W$ is Brownian motion.
I derive the distribution below but it doesn't ...
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0
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$\alpha$-mixing properties and convergence in distribution
I have a stochastic process $\{W_t\}_{t\geq 1}$, of uncorrelated but not indipendent random variables, with $\mathbb{E}(W_t) = 0$ and $Var(W_t)=\frac{t-1}{2}$ $\forall, t\geq 1$ (The $\{W_t\}_{t\geq 1}...
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Calculating the expected hitting time of a specific birth and death chain
I am working with a specific birth and death chain, defined as follows.
Consider a set of states $X = \{0,1,2,...,n\}$, where $x^* \in (0,n)$ is a recurrent state. Transition probabilities are defined ...
3
votes
1
answer
75
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Infinitesimal generator of Brownian motion on the unit sphere
The infinitesimal generator of a standard Brownian motion (as Markovian process) in $\mathbb R$ can be computed with
$$Af(x) := \lim_{t \downarrow 0} \frac{\mathbb{E}^x(f(X_t))-f(x)}{t} = \lim_{t \...
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0
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Analyzing A Random Process
Consider the following random process. For some $m \in \mathbb{N}$, we have a set of $m+1$ items $I = \{i_1,\ldots,i_{m+1}\}$. In every time $t \in \mathbb{N}$, there is a set of items $S_t \subseteq ...
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Show that $Z(t)=\exp{\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t}$ is a martingale
I'm trying to show that $Z(t)=\exp{(\sigma X(t)-(\sigma \mu + \frac{1}{2} \sigma^2)t)}$ is a martingale.
Attempt:
I want to show that $E[Z(t)|\mathcal{F}(s)] = Z(s)$
$E[Z(t)|\mathcal{F}(s)] = E[Z(t)/ ...
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