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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Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
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8 views

Distribution of an autoregressive process

Say that we are given a AR process. Also, lets assume that the residuals of the process come form a distribution $P_R$ which, while known to us, is not necessarily normal. Can I derive the ...
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1answer
32 views

What is the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$?

I am trying to find the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$, but am unable to use Ito's Formula because of the nasty integral. Is there another solution I am missing? Thanks!
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33 views

Stochastic domination

Suppose we have two probability measures on a space $X$, $\mu$ and $\nu$, such that $\nu$ stochastically dominates $\mu$, i.e.there exist a coupling of $\mu$ and $\nu$ on the product space $X \times ...
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57 views

Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$

The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
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9 views

Existence of first passage time density for time-inhomogeneous diffusion

Let $X$ be a time-inhomogeneous diffusion process in $\mathbb{R}^d$: $$dX_t=b(t,X_t)dt+\Sigma(t,X_t)dB_t,$$ where $\Sigma_{d\times d}$ is uniformly elliptic, and coefficients are such that the above ...
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22 views

How to find the mean of $\int_0^t W_s ds$, where $W_s$ is a Wiener process?

am trying to find the expectation of $\int_0^t W_s ds$, with $W_s$ being the Standard Wiener process. I am trying to use Ito's formula, by decomposing as: $$ \frac{W_t^3}{6} = \frac{1}{2}\int_0^t ...
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1answer
43 views

How to solve for the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$?

I would like to find the expectation of the Ito Integral: $\int_0^4 B_t^2 dB_t$. My strategy is to use Ito's general formula with: $$ f(t, B_t) = f(0,0) + \int_0^t \frac{df}{dx}(s, B_s) dB_s + ...
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11 views

In stochastic calculus, what is the importance behind quadratic variation?

I am learning stochastic calculus right now and I came across several mentions of the computation of the quadratic variation of a Wiener process random variable. However, most of the resources I have ...
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12 views

For stochastic differential equations, why do we care if the process is $L^2$ bounded?

I have been studying Stochastic Differential Equations, and one theorem relates to the existence of a solution to the SDE: $$ dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dB_t $$ with $X_0 = x_0$ and $0 ...
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159 views

Can we apply an Itō formula to find an expression for $f(t,X_t)$, if $f$ is taking values in a Hilbert space?

Let $U$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ $(\Omega,\mathcal A,\operatorname P)$ be a probability space ...
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7 views

Normal transitional pdf Wiener process continuity mistake and what other standard pdfs are used

I need you to tell me where I am making a mistake in the following: $$f_{1|1}(x_2,t+\Delta t|x_1,t) = \frac{1}{\sqrt{2\pi\Delta t}}e^-{\frac{(x_2-x_1)^2}{2\Delta t}}$$ If I let $\Delta x = x_2-x_1$, ...
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2answers
58 views

Showing that this is a martingale.(4.13 in Øksendals SDE)

This is an exercise from Øksendals stochastic differential equations, where I get stuck. It is exercise number 4.13.(I simplified the notation a bit.) I have that X is an Itô-process where: ...
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1answer
25 views

If voters arrive according to a Poisson process, how can we find the conditional number of votes of a candidate?

Suppose that voters arrive to a voting booth according to a Poisson process with rate $\lambda = 100$ voters per hour. The voters will vote for two candidates, candidate $1$ and candidate $2$ with ...
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44 views

Hitting Times for Brownian Motion - Levy Process?

Let $X$ be a Brownian motion and let $$H_a = \inf\{ s \ge 0 \mid X_s = a \} \;\ \text{and} \;\ S_a = \inf\{ s \ge 0 \mid X_s > a \}.$$ Now, I've shown that $H_a$ and $S_a$ are equal almost surely ...
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17 views

Working out payoff of a derivative with random interest rates

For this question, I've worked out the payoffs at N=3 but I'm not able to understand how to calculate the the expectation of the terms inside. If anyone could tell me how to find the expectation of ...
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15 views

Properties of Kernel Integral inner Product of Gaussian Process

Can anyone give any reference / suggest how to get the rigorous mathematical properties of the following : $$ Y=\int_{a}^{b} K_{X} (t) \ f(t) \ dt $$ where $$f \sim GP (\mu(\cdot), R ...
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27 views

Construction of a random variable

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda. In Appendix A.2, where they discuss the construction of a random variable, there is the ...
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20 views

Estimate for average probability of Ito diffusion falls into an interval

Denote $E^x(X_t)$ be the solution to a Ito diffusion starting with $X_0=x$. Let $K\subset \mathbb{R}$ be a compact subset. I also assume $X^x_t$ has transition probability $p(t,y,x)$. Currently I am ...
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39 views

Some Kind of Generalized Brownian Bridge

Let $\displaystyle X(t) = \int_0^t f(s)dB(s)$ where $B(t)$ is a Brownian motion and $f(t)\in L^2[0,1]$. What is a simple representation for $Y(t):=(X(t)|X(1))$ in terms of $B(t)$? Note, I am not ...
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24 views

brownian motion and process C1 (order 1 of continuity)

Here is my problem, With probability 1 (ie: a.s) the brownian motion $(B_t)_{t\in[0,T]}$ is continuous (which is define on a classic probability space $(\Omega, \mathcal{F}, ...
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50 views

Approximation of $\int_0^tF_x(s,X_s)Φ_0dW_s$ where $dX_s=φ_sds+Φ_sdW_s$ and $F_x$ is the Fréchet derivative of some $F:[0,t]×H→\mathbb R$

Let $U$ and $H$ be Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric with finite trace $U_0:=Q^{1/2}U$ be equipped with the usual inner product $(\Omega,\mathcal A,\operatorname P)$ ...
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17 views

Estimate for Expectation of Reciprocal Bessel Process

Let $W=(W_{t})_{t\geq 0}$ be a standard $3$-dimensional Brownian motion, and let $a\neq 0\in\mathbb{R}^{3}$. Consider the $3$-dimensional inverse Bessel process defined by ...
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1answer
14 views

Does an embedded discrete-time Markov chain preserve its properties in continuous time?

Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a continuous ...
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2answers
73 views

Itō formula as presented in “Stochastic Equations in Infinite Dimensions” by Giuseppe Da Prato

In Stochastic Equations in Infinite Dimensions, Theorem 4.32 (Google Books), the authors present the following version of an Itō formula: Given Hilbert spaces ...
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1answer
34 views

How to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson Process?

I am trying to show that $(X(t)-\lambda t)^2 - \lambda t$ is a martingale, where $X(t)$ is a Poisson process with rate $\lambda$. So far, what I have done is: \begin{align*} E\left((X(t)-\lambda t)^2 ...
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26 views

Expectation of an Exponentiated Integral of a Brownian Bridge

Given a Brownian bridge $X(t)$ where $X(0)=0$ and $X(1)$ equal to some given constant. What is $\displaystyle \mathbf E\Big[\exp\Big(\int_0^1X(t)dt\Big)\Big]$? I suppose I can always discretize the ...
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13 views

Gaussian filtering

I'm reading a paper and don't get how they tackle the drift of a gaussian process. We are in the setting of isonormal Gaussian processes. Let $Z$ be a Gaussian process with covariance operator ...
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29 views

Integrability of a stochastic process

Let $x(t)$ be some random path $t\in[a,b]\subset\mathbb{R}$. I.e. $x:\Omega\rightarrow\mathbb{R}^{[a,b]}$ etc. When is $\int_a^b x(t)dt$ defined? If $x(t)$ is Brownian motion, I know it's ok. A ...
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I found $E(Σ_{j=0}^{k-1}η_j-Σ_{j=0}^{k-1}E(η_j|G_j))^2=Σ_{j=0}^{k-1}(E(η_j)^2-E(E(η_j|G_j)^2)$ in a book with faulty assumptions on the objects

In Stochastic Equations in Infinite Dimensions (Second Edition) on page 109, the authors state the following: If $\eta_0,\ldots,\eta_{k-1}$ are random variables with finite second moments and ...
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1answer
18 views

Bounding expectation of a supremum process

This is exercise 3.9(c) on page 15 of Karatzas and Shreve's Brownian Motion and Stochastic Calculus. Let $N_t$ be a Poisson process with intensity $\lambda$. In particular, if $t$ is fixed, $N_t$ is ...
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1answer
38 views

Spread of a rumor in a growing population

This is a variation on a classic problem. It occur's in several problems I am researching and I'd like to get some help from folks who may have dealt with this already or can offer insights. Let ...
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45 views

Diffusions: Conditional expectation of Stopping time

Out of Rogers - Williams: Diffusions, Markov Processes and Martingales: Page 279: Let $\{\mathbb{P}^x:x\in I\}$ be a regular canonical diffusion. For $y\in J:=[a,b]\subset I$ let $X_t$ denote the ...
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19 views

Intutitive understanding of the law of a stochastic process

I'm trying to get my head around the notion of "Law of a stochastic process" intuitively. This is what I got for a Brownian motion: Denoting the law of a Brownian motion ...
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27 views

Prove that a sum of random variables converges against an Itō integral

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be ...
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26 views

Derive an Itō formula for $f(t,X_t)$ where $X_t=X_0+tY+W_tZ$ and $f:[0,\infty)\times H\to\mathbb R$ and $H$ is a Hilbert space

Let $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace $f:[0,\infty)\times H\to\mathbb R$ be Fréchet ...
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1answer
45 views

Is this an adapted process?(deterministic integrator in Itô-process)

Assume you have a probability space with a filtration, $(\Omega,\mathcal{F},P,\{\mathcal{F}_t\})$. Assume that the stochastic process $X_t$ is adapted to this filtration, and is jointly measurable ...
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1answer
56 views

Example of a Continuous-Time Markov Process which does NOT have Independent Increments

1. Given a discrete-time Markov chain without independent increments, is the embedding of it into a continuous time Markov chain (i.e. via the use of exponential waiting times) an example of a ...
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7 views

How to normalize sets of scores to have very similar histogram?

I have the output of several stochastic processes I need to combine into a single value. They have similar histogram curves, but not exactly the same. These curves are not perfectly Gaussian (see ...
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16 views

“Local” functional central limit theorem for the empirical distribution function

Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. real-valued random variables such that $\mathbb E[X^2]<\infty$. Denote by $F_X(t) := \mathbb P(X\leq t)$ their common distribution function. ...
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1answer
53 views

A Taylor Expansion of a Stochastic Process

As part of a binomial model of a stochastic process, my professor claims that the Taylor Expansion of: $$x\pm = 1 \pm (e^{\sigma^{2}h} - 1)^{1/2}$$ is: $$x = 1 \pm \sigma \sqrt h + O(h^{3/2}) $$ ...
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25 views

For Ito diffusion, what is the difference between two measures $Q^x$ and $P$?

I am confused about the difference between $Q^{s,x}$ and $P$ for the following ito diffusion: $$dX_t=b(X_t)dt+\sigma(X_t)dB_t,\quad t\ge s;\quad X_s=x.$$ Followings are from most books: given the ...
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19 views

Markov property of ito diffusion [duplicate]

Most books show Ito diffusions satisfy Markov property, that is, $E[f(X_{t+h})\mid F_t]=E^{X_t}[f(X_h)]$. But I was wondering whether it's true that $E[f(X_{t+h})\mid X_t]=E^{X_t}[f(X_h)]$. In this ...
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28 views

expectation of stopping time in Wiener process

Let $(W_t)$ be a Wiener process and for $a>0$ define stopping time: $$\tau = \inf \left\{t>0: W_t + at = 5\right\}$$ a) show $\tau < \infty$ a.s; b) compute $\mathbb{E}\tau$. I have done ...
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22 views

Filtration of path space

Let $W\left(M\right)$ be the path space of $M$. An element of $W\left(M\right)$ is a continuous map $x:\left[0,\infty\right)\to M$ (with some further technical details). I've been trying to determine ...
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2answers
33 views

Finding large deviation bound for binomial distribution

$S \sim Binomial(n, p)$. $\forall a > p$, find large deviation bound for $P( S \geq an)$ In the book, the large deviation bound definition is as follows: $\phi(t)$ is finite for some $t > 0$, ...
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1answer
32 views

How can I demonstrate that my data is sampled from a Gaussian process?

I have an experiment that, I believe, produces data with Gaussian noise. That is, any subset of my data points have a joint multivariate normal distribution with covariance K (i.e., they are sampled ...
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39 views

Probability of finding $n$ individuals in the logistic model

A population has a birth rate proportional to both the actual population, and its difference with a certain saturation population $\sigma$. The equation for the probability of finding $n$ individuals ...
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24 views

Itō isometry in Hilbert spaces

Let $U$ and $H$ be separable Hilbert spaces $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $\mathfrak L:=\mathfrak L(U,H)$ be ...
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1answer
20 views

fisherman proble, adding Poisson processes

A fisherman catches fish of type $A$ and $B,$ determined by Poisson processes of rythms $a$ and $b$ / minute, respectively. (1) If the fisherman caught $10$ fish in $2$ hours, what is the ...