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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Optimization of stochastic differential equations

Is there a way to optimize or maximize a set of differential equations. such that each equation is represented by a time series S_((t+1),μ) = μ*(S_(t+1)-S_t) + S_t and μ = 2/(i+1), i=1,...,n. Then I ...
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37 views

How do I solve this SDE (stochastic differential equation)?

I am stuck in trying to solve this equation \begin{align} d X_t = - b^2 X_t (1 - X_t)^2 dt + b \sqrt{1 - X_t^2} dW_t \end{align} Here, $b$ is a constant. I am trying to apply my usual methods for ...
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17 views

Smoothness of marginal distribution of a diffusion in the initial condition

Let $\{p_t\}_{t \ge 0}$ be a one-dimensional diffusion process (on [0,1] ) with drift $\mu(p) = C_1(1-p)-C_2p+p(1-p)s(p),$ where $s$ is a Lipschitz function and $C_1,C_2 \in (0,1)$, and diffusion ...
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1answer
14 views

Power spectral density of the system output

$w(t)$ and $z(t)$: two stationary random processes $z(t) = Pw(t)$. $P$: a stable, LTI system. How to show: $$ S_z(jw) = P(jw)S_w(jw)P(jw)^*$$ $S_z(jw)$ is the power spectral density of ...
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1answer
25 views

Convergence of random variables in $L^1$

So $g$ is a continuous real-valued function and are given that the sequence of random variables $Y_n$ converges to $Y$ in $L^1$, $E[|g(Y_n)|]<\infty$ and $E[|g(Y)|]<\infty$. Show that $g(Y_n)$ ...
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6 views

Formula for $X_t - X_{t+h}$ where $X_t$ is $MA(q)$ process

Lets say that we have $MA(q)$ process $X_t = Z_t + \psi_1 Z_{t-1} + \dots+\psi_q Z_{t-q}$. $Z_t$ are IID (normal with mean $0$ and standard deviation $\sigma$). Now I need to find form of $X_t - ...
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1answer
42 views

Stochastic integral where the integrator is zero in probability

We are given a continuous semimartingale $Y$ and a continuous process $B$ of finite variation. Hence, we know that $\langle B \rangle$, the quadratic covariation of $B$, is zero in probability. I now ...
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1answer
50 views

Expectation and Convergence of Sum of Random Variables [closed]

Let $X_1, X_2, ...$ be a sequence of independent random variables with $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Let's now consider the sum $S_n=\sum_{k=1}^{n} X_k$. I need to show three ...
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1answer
53 views

limit distribution of weak convergent sequence

Let $\{X_n(t) \mid t \in T\}$ be a bounded stochastic proces for some non-empty set $T$, we assume that the finite dimensional distributions converge in distribution and we let for $k\in \mathbb{N}$, ...
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26 views

Conditions for a Markov process to have independent increments [duplicate]

I consistently see "Let $\{X(t)\}$ be a stochastic process with independent increments..." in various texts, though I have yet to find any conditions under which we can guarantee a process to have ...
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0answers
57 views

Conceptual Understanding of a Simple Random Process

I have a simple discrete time random process that with probability $0.5$ chooses a deterministic sequence so that $X(t) = -1$, for $t<1$ and $X(t) = +1$ for $t \geq 1$, similarly with probability ...
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1answer
38 views

The Pólya urn model describes a martingale

Suppose an urn contains one blue and one red ball and that we perform the following random experiment: In each round $n\in\mathbb{N}$ we randomly draw a ball If the drawn ball is blue, we replace it ...
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1answer
91 views

$P^n$ transition matrix of a Markov chain

The setup: We have an unlimited supply of balls and $k$ boxes. In every step, we randomly (all of them have the same probability) choose a box and put a ball in it. Let $X_n$ be the number of ...
3
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1answer
46 views

Squared Bessel Process and Ito Lemma

$dX_t = \delta dt+ 2\sqrt{X_t} dW_t$, where $W_t$ is a standard Wiener process, Define $\tau =\frac{\sigma ^2}{2\nu(2 − \delta)}\left(1 − \exp \left(−\frac{2\nu t}{2−\delta}\right)\right)$ If ...
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2answers
16 views

Do you think this transition Matrix is correct?

Here is the situation we are trying to model: given a car that has 3 states, labeled 1, 2 and 3. state 1: is when the vehicle is in good operating condition. state 2: repairs may be required to ...
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0answers
28 views

Random sum a.s convergence and convergence in probability

Let $X_n$ be a sequence of independent random variables such that $$\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$$ Consider the sum $S_n=\Sigma_{k=1}^nX_k.$ How can we show that for any ...
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0answers
10 views

How to decompose a Markov Modulated Poisson Process (MMPP)

I have two questions to ask here. The superposition of two independent MMPPs is also a MMPP. How to calculate the rate of a new burst and the rate of requests within one burst if these two MMPPs are ...
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0answers
29 views

Riesz decomposition of a nonnegative supermartingale

$X=(X_n,\mathcal{F_n})$ is a nonnegative supermartingale, and moreover $EX_n\to 0$, i.e., it is a potential. If $X_n=M_n-A_n$ is the Doob decomposition, then $$EX_n=EM_n-EA_n=EX_0-EA_n,$$ so by the ...
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1answer
44 views

Law of Iterated Logarithms

I know the Law of Iterated Logarithms states the following almost surely: $$\limsup_{t\to\infty} \frac{B(t)}{\sqrt{2t\log\log t}} = 1 $$ I was wondering if there are similar ones. For example, if I ...
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0answers
16 views

Predictive Density Independent in Gaussian Process Regression?

I am a little confused in Gaussian process regression. In a GP regression, let $Y=[Y_a, Y_b]\sim \mathcal{N}(0, K+\sigma^2I)$, where $Y_b$ is the target of training samples. The task is to predict ...
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28 views

Quadratic Variation of Increasing Process?

I am looking through my notes and I came across the following statement: Let $X_s$ be a positive local martingale and let $M_t = max_{0 \le s \le t} X_s$. Then since $M_t$ is an increasing process, ...
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39 views

Solving an expectation related to CIR process

I encounter the following question Let $X$ satisfy the SDE $$dX_s=k(\alpha-X_s)ds+\sigma\sqrt{X_s}dW_s$$ for $s\geq t$ with $X_t=x$, where $k,\alpha,\sigma$ are positive constants. Find the ...
2
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0answers
36 views

A characterization of quadratic variation for $L^2$ martingales

I am trying to prove the following statement but I am totally at a loss. Let $(A_t)$, $t \in \mathbb{R}^+$ be an adapted (with respect to the filtration $(\mathcal{F}_t)$) continuous integrable ...
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2answers
34 views

Stochastic Differential Equations - A Few General Questions

I just have a few questions about stochastic differential equations. I generally did a lot of pure math but signed up for a course on probability models and stochastic differential equations because I ...
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1answer
30 views

A distribution of a stopped Wiener process

Let $(W_s)_{s \geq 0}$ be a Wiener process and $\tau$ be a random variable with an exponential distribution with parameter $1$. Suppose that $W$ and $\tau$ are independent. Determine the distribution ...
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0answers
31 views

A Doob-Meyer decomposition related question

First I will state the question and then I will show my answer, which I obtained by imposing an additional condition on the processes involved. I would like to get some help on how to solve the ...
4
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1answer
144 views

What is the difference between Calculus and Analysis? In Stochastic processes?

I guess one could say that Calculus is just a non-rigorous version of Analysis. What about in subjects involving stochastic processes? I took up masteral classes called stochastic calculus. I plan to ...
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29 views

capacity of biased random walk in $\mathbb{Z}^2$

Let $P_{x,y}$ the probability that a random walk starting from $x$ will ever visit $y$. Consider a biased random walk in $\mathbb{Z}^2$. Let $A_k$ be the set of vertices having a distance less than ...
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1answer
21 views

How do we sample from a Gaussian process

I have one particular question on Gaussian processes. A Gaussian process is fully characterized by $\mu$ and $\Sigma$. However, I do not understand how can we sample a (random) function from the so ...
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18 views

Brownian motion reflected on a sphere

Consider a standard Wiener process (in 3 dimensions) $(W_t)_{t>0}$, such that $W_0 = x_0 \neq 0$. I am trying to determine the transition density of $(W_t)$ reflected on a sphere of radius $a < ...
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0answers
27 views

Convergence in finite-dimensional distributions of some integral

Let $(X^n_t)_{t \geq 0}$ be a sequence of random real-valued processes that converges in finite-dimensional distributions, i.e. for all $k \in \mathbb{N}$ and for all $0 \leq t_1 < \dots < t_k$ ...
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21 views

Mean and variance of Gamma distribution

How do I calculate the mean and the variance of a Gamma distribution? I was told to prove the variance was sigma/lambda(^2), I don't know how to find the variance much less the variance.
3
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44 views

Brownian motion with drift (stopping time and supremum)

Suppose $(B(t))_{t \geq 0}$ is a Brownian motion and $(B_{\mu}(t))_{t \geq 0}$ is a Brownian motion with drift, which is defined by $$B_{\mu}(t) := B(t) + \mu t, \ \ \ \mu <0. $$ With $T_{a} := ...
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1answer
13 views

Connection between transition probability and SDE?

Can someone highlight what is the connection between the transition probability of a continuous time stochastic process $X_t$, i.e. $p(x,t\vert x_0,0)$ and the stochastic differential equation of the ...
3
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1answer
56 views

Why is $\mathbb{P}(F\geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g$?

For random variables $F,G$ I have problems with understanding the equation $$\mathbb{P}(F \geqslant G) = \int_{\mathbb{R}} \mathbb{P}(F \geqslant g | G=g) \, D_G(g) \text{d}g, $$ where $D_G$ is the ...
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1answer
49 views

Help with Semimartingale decomposition.

I'm having trouble with the following question: Let $\{W_t\}_{t\geqslant0}$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal ...
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0answers
22 views

A “Fourier Phase” for (stationary) random processes?

Let $X_t$ be a real w.s.s. random process. Its spectrum is given by $S(f)=\mathcal{F}R_X(\tau)(f)$ where $R_X$ is the process autocorrelation. As $X_t$ is real, the spectrum will be real and ...
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0answers
18 views

Local time for reflected random walk

Say I have a process starting from 0, and last for 100 steps, each step either moves up or down by one unit, within the boundary -10 and 10. My understanding is that since this is a martingale, the ...
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1answer
26 views

M/G/1 queueing problem

I need to prove that in the M/G/1 queueing system with Poisson arrivals with parameter lambda and exponential service time with parameter mu, that q_k = (lambda/(lambda+mu))^k (mu/(lambda+mu)).
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40 views

Prove that this space of stochastic processes is complete

See page 17 on http://www.stat.cmu.edu/~cshalizi/754/notes/lecture-19.pdf We define $\mathcal{QM}(T)$ to be the space of all non-anticipating processes $X$ such that the norm $||X||_{{QM}(T)}$ ...
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34 views

Markov chains: Condtitional independence implies independence?

In one proof, I encountered the following reasoning: $$P(T_1=n,T_2=m\mid X_0=j)=P(T_1=n\mid X_0=j)P(T_2=m\mid X_0=j)$$ Where $T$s are waiting times between returns to a state, $X_0$ is the state at ...
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21 views

Does the expected spreading of sample paths imply increase in variance?

Consider a sample-continuous stochastic process $\left\{ X_t \right\}_{t \in T}$ s.t. each $X_t$ is real-valued and $$\int_\Omega | X_t(\omega) | ^p \, \mathrm{d} P(\omega)< \infty$$ for all $1 ...
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1answer
41 views

Variance and expectation of the stochastic intergal [closed]

Compute the unconditional expected value and variance, and describe, as far as possible, the distribution of the random variable $Y_{t} = \int^{t}_{0} W_{s} ds $ with the hint below $\int^{t}_{0} ...
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0answers
29 views

Conditional distribution on arrival time (Poisson process)

Suppose that $\{N_t: t\geq 0\}$ is a Poisson process of rate $\lambda$ and $T_1< T_2< \dotsb\ $ are its arrival times (i.e. $T_i := \min \{t\geq 0 : N_t \geq i\} $). What is the conditional ...
3
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38 views

the 2D fractional Gaussian noise as derived from the 2D fractional Brownian motion

Let $X_n$ be a 1D discrete fBm. Then, its 1st order difference, $W_n=X_n-X_{n-1}$ is fractional Gaussian noise (fGn). This case is simple. But what happens in 2D? Let $Y(m,n)$ be a 2D fBm, then we ...
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1answer
53 views

Relation between Hermite polynomials and Brownian motion (on martingale property) [duplicate]

Let us define Hermite polynomials as $H_n(x)=(-1)^n e^{x^2/2}\frac{d^n}{dx^n} e^{-x^2/2}.$ One can prove that $e^{\theta x-\frac{1}{2} \theta^2}=\sum_{n=0}^{\infty} \frac{1}{n!}H_n(x)\theta^n \quad ...
2
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2answers
29 views

Let $T_{a}$ denote the first time the Brownian motion process hits $a$. When $a>0$, then $P\{X(t)\ge a|T_{a}\le t\}=\frac{1}{2}$

Let $T_{a}$ denote the first time the Brownian motion process hits $a$. When $a>0$, then $P\{X(t)\ge a|T_{a}\le t\}=\frac{1}{2}$ I cannot see how it can be true, anyone could help me? Thanks very ...
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1answer
17 views

A question involving inter-arrival times of a Poisson process

I can't demonstrate that the inter-arrival times of a Poissom process are i.i.d. How can I demonstrate it? Or, where can I find the demonstration? Thank you!
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10 views

Generality the random function

soit X1, ..., Xn of the random function ondescosinus. Calculer les lois fini-dimensionnelles de (X1+...+Xn)/√n?;X is a ondescosinus if X (t) = √(2)cos(2w(t) + Z (t)) with w(t) ~ U [0; 1]; U [0; 1] is ...
2
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0answers
28 views

Question about zero set of Brownian motion

I was reading the posted to solutions to one of the questions on a probability midterm and couldn't figure out how to justify one of the steps. Let $\{B_t\}_{t\geq 0}$ be a Brownian motion and ...