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 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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23 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
42 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
39 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
52 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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22 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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27 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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21 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
31 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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22 views

About equivalent definitions of brownian bridge [closed]

According the definition from Wikipedia, a Brownian Bridge is a conditional random process, $B_t=\{W_t \mid W_1=0\}$, then equivalently, how do I prove that $B_t=W_t-tW_1$. ...
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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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35 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 ...
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6 views

How can we proove that it's a Gaussian system?

$(W_1, W_2)$ are 2 independent Wiener processes and $$B_1= W_1, ~~~ B_2 = a W_1 + \sqrt{1-a^2} W_2,$$ where $a=(a(t, \omega))_t>0$ and is $(F_t=F_t^{(W_1,W_2)})$-measurable. $0<a<1$. It ...
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1answer
75 views

Ito integral of average of the square of a Wiener signal?

How do we evaluate the average of the square of a Wiener signal? Standard case: Typically, the signal average is $S(t)=\frac{1}{T}\int_{0}^{T}s(t)dt$, where we can write the integral in Ito form ...
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10 views

Reflection principle for the modulus of the Brownian Motion

I have the following question. Suppose we define $M(t)=\sup_{0\le s\le t}|B(s)|$, where $B$ is an ordinary Brownian motion in $\mathbb{R}$. How can we compute $P(M(t)\ge a)$? Is it $2P(|B(t)|\ge a)$? ...
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15 views

Does there exist a FCLT for this sequence?

Assume $(X_i)_{i=1}^{\infty}$ is a sequence of i.i.d. random variables such that $\mathbb E[X_1] = 0$ and $\mathrm{Var}(X_1) = \sigma^2$. Define the partial sum $S_n(t)$ as \begin{align} S_n(t) ...
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27 views

Models for Probability Density Functions with unknown parameters and given mean and variance

The PDF $f(x)$ of a non-negative random variable $x$ has the structure $$f(x)=\exp (a-bx-cx^{2})$$ where $a$, $b$ and $c$ are any model parameters. It is assumed that $c\ge 0$ so that $f(x)$ does not ...
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23 views

Mixing time for lazy random walk on hypercube.

I am studying for a probability exam and am having trouble with the following exercise: Let $X_n$ be a lazy random walk on $\{0,1\}^d$ starting at $(0,\ldots,0)$ (stays put with probability ...
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34 views

Why is a discounted price process a local martingale under the Risk Neutral Measure?

I'm familiar with the fact that if the stochastic process $\left( g(t) \right)_{t \in \left[0 , T \right]}$ is almost surely square integrable, i.e. $\mathbb{P}\left( \int_0^t |g(s)|^2ds < \infty ...
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29 views

Predictable quadratic Variation Intervals of Constancy

From Revuz and Yor - Continuous Martingales and Brownian Motion 1999 Chapter IV Proposition 1.13 it is proven, that for a continuous local martingale $M_t$ the intervals of ...
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40 views

Sequence of infinite Backward Products

I am not a mathematician but need to understand an asymptotic property of Backward Products of row-stochastic matrices. Let's say I have the following $\textit{sequence}$ of backward products ...
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1answer
18 views

Calculating the variance of the time until a Markov process jumps to a specific state from a starting state?

A Markov process on $E = {1, 2}$ is constructed according to holding time parameters $λ_1 = 2$ and $λ_2 = 4$; the defining Markov chain has transition probabilities $p_{11} = p_{12} = 0.5$ and ...
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23 views

Brownian Motion hitting time is finite yet has infinite expectation?

I've read that a hitting time of a Brownian motion (defined as $T_a = \inf\{t\ge0:W_t=a\}$ where $W_t$ is a standard Brownian Motion, i.e. a Wiener process), has the following two properties, which I ...
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32 views

stuck !! how to integrate the bottom line…

Find the exact formula for $P(X(5)=3)$, as opposed to using a numerical approximation to $P(t)=e^{\lambda t}$ Hint ; Observe that $P(x(5)=3)= P(R_{1}+R_{2} +R_{3} > 5, R_{1} + R_{2} < 5)$ . i.e ...
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29 views

Find the expectation of the arrival times of a queuing system?

Suppose ${X_1, X_2, \dots}$ are independent identically distributed random variables defined by the density $f(x)=\lambda e^{-\lambda x}$. The renewal process $N={N(t): t>=0}$ is defined by the ...
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1answer
23 views

Reference request for stochastic processes on manifolds

I'm looking for some references on stochastic processes on manifolds. The more introductory the better. Thanks.
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1answer
22 views

SDE Solution: Hull-White extension of Vasicek model

I am trying to figure out the particular ansatz (if that's all there is) for the solution to the SDE: $ dr_t = [v_t - ar_t]dt + \sigma dW_t, $ where $a$ is constant and $v,t$ are, potentially, ...
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14 views

Deciding whether a maximum asset price process is a markov process

I understand how Mn has been drawn. For the second computing part, after computing, I have no idea how to decide if Mn is a markov process I don't understand the solution at all, don't know what ...
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1answer
26 views

Integral representation $B_T^3$

I have to find a $F_t$ such that $B_T^3=E[B_T^3]+\int_0^T F_t dB_t$. I have shown by ito formula that $B_T^3=\int_0^T 3 B_s^2 dB_s+\int_0^T 3 B_s ds$. Could you please help me?
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1answer
40 views

Prove a.s. convergence of random variables.

I need to prove this: Assume that you have a probability space $(\Omega, \mathcal{F},P)$, $X_t$ is a stochastic process which is jointly measurable with respect to $\mathcal{B}(\mathbb{R})\times ...
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20 views

Brownian martingale as time-space changed brownian

Let $M$ be a true real martingale adapted to some brownian motion $B$. What are the most generic conditions on $M$ to find a deterministic map $\Phi:\mathbb{R}_+\times\mathbb{R} \to ...
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19 views

Use of Itô isometry for correlation calculation

When calculating the covariance of the Ornstein-Uhlenbeck process, the Wikipedia article applies implicitly the Itô isometry with the fact of non-overlapping independent increments of the Wiener ...
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2answers
30 views

Checking if $X(t) = \exp(t/2)\cos(W(t))$, with $W(t)$ a Wiener process, is a martingale

This is what I've done: Let $s < t$ and $F_t$ be a filtration adapted to $W(t)$ $$E[e^{t/2}\cos(W(t))|F_s] = e^{t/2} E[\cos(W(t)) - \cos(W(s)) + \cos(W(s))|F_s]$$ $$= e^{t/2} [E[\cos(W(t)) - ...
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21 views

When can an embedded Markov chain X for a Markov process Y be reducible?

It's pretty widely documented that a Markov process Y is reducible/irreducible if and only if the embedded Markov chain X is reducible/irreducible. However I'm not sure this works in reverse. I'm ...
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34 views

Asymptotic distribution of zero-drift Geometric Brownian Motion as $t \to \infty$

If we fix the drift at $\mu = 0$, then my geometric brownian motion will have stationary mean, but it seems that the variance will grow without bound. What does the limiting distribution look like for ...
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20 views

Expected commission over a Poisson process

Customers arrive at a restaurant according to a Poisson process with arrival rate $\lambda > 0$. As the head of the advertising agency for this restaurant, you are paid a comission of $i$ ...
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1answer
19 views

Poisson Process independent Wiener Process using singular measures

I was reading some stochastic calculus of Jump processes and saw the result that if $W_t$ is Brownian and $N_t$ is Poisson both adapted to the $W_t$'s natural filtration then these processes are ...
3
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1answer
60 views

Question regarding Brownian motion

Hello I have two questions regarding the construction of Ito's integral in Øksendals book from here: http://th.if.uj.edu.pl/~gudowska/dydaktyka/Oksendal.pdf On page 25 he lists these 3 properties ...
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1answer
14 views

Correlated Random Walk

If there was a random walk in three dimensions where the angle between any to two connected segments were fixed (rotation allowed around this angle). What would be the best way to show that the ...
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1answer
27 views

Proving an inequality involving conditional probability

Let $(X_t)_{t\ge0}$ be a stochastic process on a probability space $(\Omega,\mathcal F, \mathbb P)$ and let $\mathcal F_t=\sigma(X_s:0\le s\le t)$. Let $\Lambda\in \mathcal F_t$ with $\Lambda\subset ...
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23 views

Does a function with left limits have only countable jumps?

If $f:\mathbb R\to\mathbb R$ has left limits, does it have at most countable jumps? By the set of jumps of $f$ I mean the set $\{x\in\mathbb R:f(x^-)\neq f(x)\}$ This is indeed true if we add the ...
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35 views

Mean Value of a Random Process

Consider a random process $X(t) = Z(t)\sin(wt-Q)$. Here $Q$ is a random variable taking values $q$ in $[-\pi/2,\pi/2]$ with PDF given by $$p_1^Q(q) = \frac{\cos(q)}{2}$$ $Z(t)$ is some random ...
5
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76 views

Derivation of a stochastic Navier-Stokes equation with multiplicative noise

Most of the literature is targeting a special stochastic version of the deterministic Navier-Stokes equation without giving a derivation of the considered equation. I'm searching for such a ...
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26 views

Itō formula for a scalar valued function of the solution of a scalar Itō SODE

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$ $B$ be a real-valued $\mathcal F$-Brownian motion on $(\Omega,\mathcal ...
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1answer
30 views

The Stratonovich Integral and its meaning as the limit in mean square of a sum?

I am studying the Stratonovich Integral and on wikipedia, Stratonovich Integral, it states that the integral, for a process $X:[0,T] \times\Omega \to \mathbb{R}$, as: $$ \int_0^T X_t \circ dW_t $$ ...