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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Infinitesimal Generator of Poisson process

I would like to compute the infinitesimal generator of a Poisson process $N$ with intensity $\lambda$. So I can write: $$\mathbb{E}[\ f(N_{t+s})-f(N_s)\ |\ \mathcal{F_t^0} \ ] = \mathbb{E}[\ ...
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38 views

If $(X_t,t\in I)$ is a process with values in $(E,\mathcal{E})$, are $\sigma(X_t,t\in I)$ and $\sigma(X)=X^{-1}(\mathcal{E}^{\otimes I})$ equal?

Let $(\Omega,\mathcal{A},\operatorname{P})$ be a probability space $E$ be a Polish space and $\mathcal{E}$ be the Borel $\sigma$-algebra on $E$ $I$ be an index set $X_t$ be a random variable on ...
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Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that

Let $\{X_n; n\geq 0\}$ be a martingale with respect to $\{Y_n\}$. Prove for any set of integers $k\leq l<m$ that the difference $X_m-X_l$ is uncorrelated with $X_k$, that is, $$E[(X_m-X_l)X_k]=0.$$ ...
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2answers
43 views

Exchanging expectation and limits

Exchanging expectation and limits I have a stochastic process, ${b_t} \, (t=0, 1, 2, \ldots)$, which follows a random walk. Specifically, ${b_0} = 0$ and for $t$ greater than zero, $\displaystyle ...
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13 views

Doob's submartingale stopping theorem in the context of the submartingale problem

Let $$X^\omega_f (t, w) = f(w(t)) - f(w(t \wedge \tau)) - \frac{1}{2} \int_{t \wedge \tau}^t \Delta f(w(s))\, ds$$ be a $P^\tau_\omega$-submartingale. 1) Why Doob's submartingale stopping theorem ...
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39 views

Applying the Multivariate Ito Formula

I want to show that the stochastic process $$ S_t^i = S_0^i \exp\left( \int_0^t \left(\mu_s^i - \frac{1}{2} \sum_{j=1}^m (\sigma_s)^{ij} \right)^2 d s + \sum_{j=1}^m \sigma_t^{ij} S_t^i dW_t^j ...
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17 views

Aggregate arrivals from a renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process". The inter-arrival time of a renewal process, t, conforms to a general distribution, denoted by PDF $f(t)$. ...
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33 views

Aggregate arrivals from a Poisson Process

The inter-arrival time of a Poisson Process, $t$, conforms to the exponential distribution, so the probability density function for $t$ is $f(t)=λe^{−λt},~t>0$. ($λ$ is the arrival rate of the ...
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36 views

Simulation of interacting Ornstein-Uhlenbeck processes

I would like to simulate the following system of interacting OU processes on $[0,T]$: $$dX_t^1=(X_t^2-X_t^1)\,dt+\sigma_1 \,dW_t^1,\quad X_0^1=x_1$$ $$dX_t^2=(X_t^1-X_t^2)\,dt+\sigma_2 ...
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40 views

Mean-Square Ergodicity of Certain Quantities?

I apologize in advance for my lack of mathematical knowledge, especially in the field of stochastic processes, but I will try my best to formulate my question in a mathematical way. Is it possible ...
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1answer
24 views

Working with the random variable $\log X$ instead of $X$

Suppose I have a positive stochastic process $X_t$. I'd like to compute certain properties about $X_t$, but suppose I can't and instead I can compute properties about $\log(X_t)$. Can I say anything ...
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25 views

Local Martingales in a Finite Time Horizon setting

I apologise if this question has been answered somewhere else. Consider the following definition. Let $T \in [0, \infty), d \in \mathbb{N}$, let $(\Omega, \mathcal{F}, P, (\mathbb{F}_t)_{t \in [0, ...
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15 views

process stochastics and branching process [duplicate]

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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1answer
33 views

Is a martingale with bounded variance therefore bounded in $L^2$?

If a martingale $W_n$ has bounded variance, does this mean that $W_n$ is automatically bounded in $L^2$? I feel like this ought to be obvious but I don't see how to prove it and I haven't been able to ...
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10 views

discontinuous Gaussian field

I am trying to build an example of a discontinuous Gaussian field. The simplest I could come up with is the following: Let $Y,Z$ be two independent brownian motions on $[0,1]$, and $T$ a uniform ...
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24 views

Local martingale and integral condition

Suppose $M^i_t = X^i_t - X^i_0 - \int_0^t b_i(s,X)\, ds$ where $b_i:[0,\infty)\times \Omega \to \mathbb{R}$ is a progressively measurable functional and $X^i_t: C[0,\infty)^d \to \mathbb{R}$ ( ...
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1answer
61 views

Conditional probability branching process

Consider a discrete time branching process $X_{n}$ with $X_{0}=1.$ Establish the simple inequality $$P\{X_{n}>L\ \textrm{for some}\ 0\leq n\leq m\ |\ X_{m}=0 \}\leq [P\{X_{m}=0\}]^L$$ Note: This ...
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39 views

$W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$?

This is a sample question for the actuarial exam MFE. Let $Z(t)$ be a standard Brownian motion. Let $W(t)=t^2 Z(t)-2\int_0^t sZ(s)ds$. What is $dW(t)$? The only thing I know is Ito's Lemma. So I ...
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26 views

Simple Markov property

I want to prove the simple Markov property but I come to a point where I do not see how to conclude. I want to prove $\mathbb{E}_\nu[Z\circ\Theta_t\mid ...
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0answers
27 views

Eigen function of one Stochastic Process from the eigen function of another Stochastic Process

Let us consider a centred square integrable stochastic process $\{X_t:t\in [0,2]\}$. Also let the eigen values and the eigen function of the kernel of the covariance operator of $X_t$ are ...
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30 views

Proofing Analytic continuation and stationary increments of an exponential Family

In U.Küchler "Exponential Families of Stochastic Processes" 1997 Theorem 4.2.1 we have the following setup. Let $(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq0})$ be a filtered measurable space. Let ...
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20 views

Represent stochastic process as conditional expectation

I try to reduce my problem to the following question, which is stated rather sloppy (without possibly necessary additional conditions). Let $Y_t$ be a real stochastic process for $t \in [0, T]$ and ...
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1answer
217 views

Conditional expectation $\mathbb E\left(\exp\left(\int_0^tX_sdB_s\right) \mid \mathcal F_t^X\right)$

I have found a theorem (see below) in two papers an I try to figure how it could be proved. The result seems to be intuitive, but I'm not able to prove it in a rigorous way. Assumptions: Consider a ...
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27 views

Prove that an integral is zero (from Gardiner's Handbook of stochastic methods)

I have troubles in one proof of the book Handbook of stochastic methods by Gardiner. In the paragraph 3.7.3 he writes this integral $\sum_i\int d\vec x \frac{\partial}{\partial ...
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1answer
68 views

what does `ensemble average` mean?

I'm studying this paper and somewhere in the conclusion part is written: "Since this rotation of the coherency matrix is carried out based on the ensemble average of polarimetric scattering ...
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1answer
30 views

How to get the basis of $L^2[0,1]$ from the basis of $L^2[0,2]$

Is there any way to derive orthonormal basis of $L^2[0,1]$ from the orthonormal basis of $L^2[0,2]$? Here $L^2[0,2]$: is space of square integrable centered stochastic process on $\Omega\times[0,2]$, ...
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1answer
34 views

Why use stopping times rather than a deterministic sequence to localise a martingale?

I am a beginner on stochastic processes I am wondering why , to localise a martingale, require the existence of one non-decreasing sequence of stopping times [$ \tau_1 , \tau_2$,...] such that the ...
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7 views

How to show the symmetry of a special green's function? (without defining the class of green's functions in general)

Given a two-dimensional simple random walk $ (X_i)_{i\in\mathbb{N}}$ on $ \mathbb{Z}^2 $, a square $ S_N :=\{1,2,\dots, N\} \times \{1,2,\dots, N\} $, and the stopping time $ \tau_{\partial ...
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16 views

Ito Isometry on Multivariable indicator function

The background of this question is a paper written by Morten O.Ravn and Harald Uhlig, titled "On Adjusting The HODRICK-PRESCOTT Filter For The Frequency of Observations" I will very much appreciate ...
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1answer
30 views

Calculating $ \mathbb E \left[e^{-\mu W_T } 1_\left( {\min W_t \leq a} \right) \right]$ for a Wiener process

Let $W_t$ be a standard Wiener process, $a$ some real number, and $\chi (x)$ the indicator function. I am trying to calculate the following expectation: $$ \mathbb E \left[e^{-\mu W_T } \chi \left( ...
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0answers
13 views

Customers and Anti-Customer Queueing Problem: What is the Customer delete probability

Hello may ask for your help? First the setting: I have got a problem with some queueing theory. The whole problem would be a grid of nodes, all nodes have an operation intensity $\mu_{i,j}$. ...
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1answer
26 views

“The well-known formulas that gives the relation between the generating functions of a sequence and the sequence of its 'tails'”

I'm reading a paper on Branching Processes and the Theory of Epidemics, and the fourth page (p. 262 of the book) the author refers to "the well-known formulas that gives the relation between the ...
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1answer
39 views

$ P(W_t - W_\tau > 0 \text{ and } \tau <t) = \frac{1}{2}P(\tau < t) $ for a stopping time $\tau$

Let $W_t$ be a standard Wiener process and $\tau = \min \lbrace t \geq 0 :W_t \geq a \rbrace$, the first time the process reaches level $a$. By symmetry of the Gaussian distribution we have $$ P(W_t ...
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13 views

Mean and variance regime-switching model

Suppose we have the following model for stock price: $$ X_{t}=X_{0}\exp\left(\int_{0}^{t}(r-\frac{1}{2}\sigma_{\epsilon(s)}^2)ds+\int_{0}^{t} \sigma_{\epsilon(s)}dW_{s}\right) $$ This follows a normal ...
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Mathematical philosophical questions about the general theory of stochastic processes.

After 6 months spent on what is termed the "general theory" of stochastic processes and after having worked out many nuances of the field, I realized that: The general theory is beautiful ...
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1answer
55 views

Compound Poisson Process function expected value

For the calculus of a financial derivatives, I need to compute the next expectation: $$\mathbb{E}\left((\sum_{i=1}^{N_T} (J_i-k))_+\mid J_1+\cdots + J_{N_t}=x \right)$$ where $$(X_t-k)_+= ...
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46 views

Conditional expectation of stochastic integral with independent components

Let $T$ denote a maturity and $\mathbb{F}$ a filtration. Besides, consider two processes $A$ and $B$ which are mutually independent and are both dependent on (a subfiltration of) $\mathbb{F}$. Does it ...
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31 views

Strong Feller property and uniform continuity ( interpreting Stroock Varadhan 1969 )

In the article Diffusion processes with continuous coefficients II (Stroock Varadhan - 1969), the authors begin a section named Strong Feller property with the following: "Let $P(s, x, t , dy) $ be ...
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43 views

Markov Chains and transition semigoups

I'm trying to figure out what the following statement refers to. A process $X$ is markov with transitions semigroup $(K_t)_{t\geq0}$ and initial distribution $\mu$ if and only if for all ...
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27 views

covariance of a function of Wiener processes

Consider two independent Wiener processes, $W_1$ and $W_2$. The covariance of certain functions of Wiener processes is simple, for example ...
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1answer
18 views

Stochastic ordering functionally invariant

I am studying for an exam in actuarial science, where I have the following exercise: Prove that the stochastic order relation $\leq_{\mathrm{st}}$ is functionally invariant; i.e. show that $$X ...
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1answer
57 views

The quadratic variation of $B \cdot B$, where $B$ is a Brownian motion

Let $B$ be a standard, one-dimensional Brownian motion. Can I show that $[B \cdot B] = B^2 \cdot [B]$, using the "fundamental identity of stochastic integration", namely that $[H \cdot X, Y] = H \cdot ...
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41 views

Impossible Events, Probability Zero Events, Change of Sample Space, Invariant, Canonical Sample Space?

I am reading this post about probability theory and its foundations by T. Tao, and also this and this post, and they say in essence that the underlying sample space is not that much important. Often ...
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0answers
17 views

Convergence of a stochastic integral [duplicate]

Let $(B_t)$ the standard Brownian Motion and $(H_t)$ be an adapted continuous process. Show that $$\frac{1}{B_t}\int _0^tH_sdB_s $$ converge in probability. I guess that the limit is $H_0$ but I ...
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19 views

On stochastic process

What is a linear stochastic process ? is it different from a stationary process ? Can you give me an example of linear discrete stochastic process ? What are the title of good books so that i can ...
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27 views

Charaterize the $\mathcal{F}_\tau$ a sigma algebra for the stopping time $\tau$

Consider a stochastic process $X: [0, \infty) \times \Omega \to \mathbb{R}^d$ We define $\mathcal{F}_t = \sigma(X(s), 0 \leq s \leq t)$ The sigma algebra generated by the sets $\{\omega: X(s,\omega) ...
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50 views

Lagrange multiplier and minimum variance

Looking into a control variate technique of Monte Carlo simulation I have run into a cost-optimization problem that I'm not quite sure I understand. It seems it has to do with Lagrangian multipliers, ...
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2answers
56 views

Mean Square Error of Monte Carlo

Trying to develop the expression for the Mean Square Error (MSE) of Monte Carlo, I found myself a bit lost when going through a simple proof in the literature. I am working in the context of ...
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29 views

Kac Lemma for staionary ergodic processes

Simple question. I have a stationary ergodic process $U$ on a finite alphaet and I want to prove the Kac's Lemma (see Cover and Thomas - Elements of Information Theory (second ed.) page 445). In the ...
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1answer
56 views

Prove that the difference of continuous and monotonically increasing functions has continuous variation

Let $G:[0,\infty)\to\mathbb{R}$ be continuous and $$V^1_t(G):=\sup\bigcup_{n\in\mathbb{N}}\left\{\sum_{i=0}^{n-1}\left|G_{t_{i+1}}-G_{t_i}\right|:0=t_0\le\cdots\le t_n=t\right\}$$ be the variation ...