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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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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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 ...
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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
54 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 ...
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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 ...
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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 ...
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12 views

No. of points in a Poisson process

Can anyone suggest a proof of the following: If the mean measure of a Poisson process on $\mathbb{R}$ is infinite, then the no. of points in the process are almost surely infinite.
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44 views

writing down markov chain transition matrix

Question: An experimental animal can stay in room-A until 1 minute,and it can stay in room-B until 2 minutes. There exist deadly gases in room-C. One room among these three rooms is being randomly ...
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1answer
36 views

Time sampling an ordinary poisson process

My questions will be given at the end, let me just give some definitions first. The counting process $\{ N(t), t \geq 0 \} $ is said to be a non homogenous Poisson process with intensity function ...
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2answers
264 views

Could someone explain rough path theory? More specifically, what is the higher ordered “area process” and what information is it giving us?

http://www.hairer.org/notes/RoughPaths.pdf here is a textbook, but I am completely lost at the definition. It is defined on page 13, chapter 2. A rough path is defined as an ordered pair, ...
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1answer
54 views

proving that $\lim_{n\to \infty}P(A_n)$ exists and $\lim_{n\to \infty}P(A_n) =P(\lim\sup A_n)$

Note: Borel-Cantelli Lemma says that $$\sum_{n=1}^\infty P(A_n) \lt \infty \Rightarrow P(\lim\sup A_n)=0$$ $$\sum_{n=1}^\infty P(A_n) =\infty \Rightarrow P(\lim\sup ...
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17 views

Time Index of Lévy Process

Consider (for all $t\geq 0$) a linear time transformation function $\nu(t)=at+b$ with the following properties: $\nu(0)=-1$ $\nu(t)$ is an increasing function of the time index $t$ i.e. $a>0$. ...
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1answer
51 views

Prove that discrete first hitting time is a stopping time

I have problems with the proof that a first hitting time is a stopping time: Let $\tau$ be the first hitting time into the set A, for a process $\{ X_n \}$ adapted to a filtration $\mathcal F_n$. I ...
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1answer
43 views

What is an intuitive explanation for Birkhoff's ergodic theorem?

If I'm not familiar with measure theory, what is a good way to understand the idea behind the definitions involved, the interpretation of the theorem, and the proofs thereof? Particularly, it's not ...
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3answers
138 views

Standard deviation of mean of a set of numbers, which are imprecise

I have a problem which seems very simple, but for some reason I can not find out what I have to do exactly. Let's say I have a set of derived values, where each of them has an individual error: ...
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1answer
50 views

Prove increment of Brownian motion is Brownian motion

I am trying to solve the following exercise in Oksendal's book: Let $B_t$ be Brownian motion and fix $t_0\ge 0$. Prove that $$\bar{B_t}:=B_{t_0+t}-B_{t_0};\quad t\ge 0$$ is a Brownian motion. I ...
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1answer
39 views

Impossible stochastic process

I am trying to prove that a stochastic process with the following properties cannot exist. Let $\{X_t: 0 \leq t \leq 1 \}$ be a stochastic process such that i) $X_s$ and $X_t$ are independent ...
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2answers
50 views

Making it possible to do a Fourier transform on it: $\frac{1}{(k+w)^2(a^2 +w^2)}$

Sorry for all the edits, I'm very stressed and not so used to Latex. Full question: consider a filter with impulse response $$h(t)=e^{-bt} u(t)$$ where $u$ is the unit step function. The input ...
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25 views

MCMC/E-M limitations?MCMC over E-M?

I am currently learning hierarchical bayesian models using JAGS from R, and also pymc using ...
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1answer
65 views

Cross Power Spectral Density from Individual Power Spectral Densities

Let $X$ and $Y$ be two zero-mean, wide-sense stationary random processes. The power spectral density of a process is the Fourier transform of the process's auto-correlation function. The cross power ...
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7 views

Approximation of Stochastic Integral with Integration by Parts

I am trying to approximate the solution to: $\int_{0}^{t} f(s) db(\omega,s) = f(s)b(\omega,s)|^{t}_{0} - \int_{0}^{t} f'(s) b(\omega,s) ds$ where $f(t) = sin(t)$ and $t \in [0,2\pi]$ for both sides ...
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1answer
43 views

Question about random walk markov chain

For a random walk, let $a$ denote the probability that the markov chain will ever return to state $0$ given that it is currently in state $1$. Because the markov chain will always increase by $1$ with ...
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1answer
117 views

A textbook for a rigorous introduction to Stochastic Analysis with emphasis on stochastic differential equations

I'm looking for a good textbook for an introduction to Stochastic Analysis, preferably one that focuses on rigour. I am familiar with measure theory and basic probability theory. The direction I am ...
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11 views

From power density spectrum to difference equation of an AR process

I am trying to solve the following question, but I am stucked at the beginning. I think I have to take inverse fourier then apply yule walker equation, but I can't take the inverse fourier transform ...
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2answers
132 views

Further Reading on Stochastic Calculus/Analysis

I'm looking to read up more on Stochastic Analysis/Calculus (whatever it's called?) for PhD proposal. So far, I've had 2 courses on Stochastic Calculus, mainly focusing on Finance, 1 course on ...
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18 views

The differential of the trace of two random matrices

I have two random matrices evolving in time, $X_{t}$ and $Y_{t}$. I know that $dX_{t} = X_{t}Adt + X_{t}dB_{t}$ and $dY_{t} = AY_{t}dt + Y_{t}dB_{t}$, where $A$ is a constant matrix and $dB_{t}$ is ...
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16 views

What is a process with an independent killing rate?

''The process $X$ with an independent killing rate $\alpha$'': generally, is it the process $\{X_{t\wedge\theta}\}$, where $Pr(\theta>t)= e^{-\alpha t}$ ?
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1answer
39 views

How to interpret the covariance matrix of Brownian motion

I'm reading Bernt Oksendal's "Stochastic Differential Equations". It says, Brownian motion $B_t$ is Gaussian Process, i.e. for all $0 \leq t1 \leq \cdots \leq t_k$ the random variable $Z = ...
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9 views

Brownian Bridging Time Series Variance

Suppose I have a time series of daily levels $(X_t)_{t\geq 0}$. I want to create Brownian Bridges between these levels, such that variance is preserved. I assume that $X_t$ diffuses as, $dX_t=\mu ...
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35 views

Derivation of Kolmogorov Backward Equation for Inhomogeneous CTMC

I'm trying to clear something up regarding inhomogeneous CTMCs, and I just can't seem to get a proof working. So, I'm hoping that someone here could maybe give me some pointers :) I'm considering a ...
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1answer
48 views

Distribution of Brownian motion before stoping time.

Let $B_{t}$ be a standard Brownian motion. Stopping time $\tau_{a} = \inf \{t \ge 0: |B_{t}| = a\}$. How to find $E[B_{\frac{\tau_{a}}{2}}]$? Or where is it possible to read about it? Thanks in ...
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34 views

Find the probability that P(X(t)=1,3,5…) that a Poisson process having rate,m, is odd

I have a confusion X(t) has rate mt But, for any odd number ,how can I generalize probability of P(X(t)=1.3.5....) ?
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1answer
35 views

Azuma's/Hoeffding's inequality for geometric series

Let $X_1,X_2,\dotsc$ be a sequence of a.s. bounded, zero-mean random variables. For $\alpha \in (0,1)$ define $Z_t$ as the geometric series with $Z_t = \sum_{i=1}^t\alpha^{t-i}X_i$ and $\mathcal{F}_k ...
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15 views

a problem of linear optimization

I struggle with the following problem: Given equation: $y = Hx$ where -> $x$ is a complex random process of $N$X1 dimentions. $E(x_i(t_1)x_j(t_2)^*)=0 \space \space \forall t_1,t_2, \space i\neq j$ ...
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How to calculate $E[\sup_{s\in[t,t+a]}(B_s-s)]$?

At first, I was trying to calculate the $E[\sup_{s\in[t,t+a]}(B_s-s)^+]$, here "a" is a constant. since $Y:=\sup_{s\in[t,t+a]}(B_s-s)^+$ is a positive r.v, so ...
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1answer
44 views

Two-sided hitting time of Brownian motion

I am trying to compute the hitting time of a linear Brownian motion on a two-sided boundary. More specifically, let $W_t$ be a (one-dimensional) Wiener process. Let $T = \inf \{t: |W_t| = a \}$ for ...
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1answer
26 views

Conditional Probabilities Poisson Process

If I let ${X(t); t>=0}$ be a Poisson process having rate parameter $\lambda = 2$. I'm supposed to determine the probability: Pr{${X(1)>=2 | X(1) >=1}$} My solution: I looked at this as ...
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1answer
22 views

Conditional Distribution Poisson Process

In class, our professor told us to verify this solution on our own time. The problem is: Let $\left\{X(t),t \geq 0 \right\}$ be a Poisson process of rate $\lambda$. For $s,t >0$, determine the ...
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1answer
18 views

How to find mean of random process with no limits given?

I've got a quick question. I've been set this question: To find the mean, cov, var I guess you consider them as two separate RP's so: $$E(Y(t)) = \int \cos(\omega t+\psi)p_1(\psi)\,d\psi + \int ...
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1answer
45 views

Is the geometric series of a set of $n$ RVs a martingale?

Let $X_1,\dotsc,X_n$ be independent, zero mean random variables and define $Y_k = \alpha^{n-k}X_k$. Is $\{Z_k\}$ with $Z_k = \sum_{i=1}^k Y_i = \sum_{i=1}^k \alpha^{n-i}X_i$ a martingale? I suppose ...
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1answer
26 views

Is $\sup_{t \ge 0}E[M_t^*]<\infty$ for a supermartingale $(M_t)_{t \geq 0}$? [closed]

Assume $M_t$ is a supermartingale, $M_t^*=\sup_{s \le t}M_s$. Is $\sup_{t \ge 0}E[M_t^*]<\infty$ holds? If not, give me an example plz. Thx!
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34 views

Are the following Stopping Times?

I've been working through the following list of stopping time questions. I am have problems with the final two (e and f). I appreciate any assistance offered. $\textbf{Question:}$ Let $S,T : ...
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24 views

Distribution of Brownian motion at a stopping time

Let $X,Y$ be independent Brownian motions, and let $$ T_a = \inf\{t\geq 0 : Y_t = a\} $$ for some $a > 0$. My question is how can I find the distribution of $X_{T_a}$? The hints I have are to use ...
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1answer
27 views

Analytical method to compute probability of an event given other event(s)

Suppose event $P(H1)$ denotes the probability of getting exactly one head and $P(T1)$ denotes the probability of getting exactly one tail after tossing two fair coins simultaneously. I am trying ...
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25 views

How to show that stochastic exponent is integrable?

I need to prove that if $u: [0,T]\rightarrow \mathbb{R}$ is a deterministic square integrable function then stochastic exponential process defined : $M_{t} = exp(-\int_0^t \! u(s) \, \mathrm{d}W_{s} ...