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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Subordination of a Levy process when the “subordinator” is not nondecreasing

If $X_t$ is an $\mathbb{R}$-valued Levy process and $Z_t$ is an $\mathbb{R}$-valued subordinator, we know that $X_{Z_t}$ is also Levy process. My question is, are there processes $Z_t$ which are not ...
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26 views

Is exit probability monotonic in drift and diffusion coefficient?

Let $W$ be Brownian motion. Let $b_t$ and $\sigma_t$ be adapted to $\mathcal{F}_t^W$. Consider the SDE $$dx_t=b_tdt+\sigma_tdW_t.$$ Assume that $b$, $\sigma$ are such that $x$ stays non-negative. Fix ...
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27 views

Help understanding a proof of non-differentiablity of Brownian motion

The following statement and proof are taken from the book Brownian Motion by Peter Morters and Yuval Peres. Since I initially didn't fully understand the proof I added some clarifications and I was ...
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17 views

Ratio Distribution of Two Dependent Chi-Squared without Joint Distribution

Assume vectors $\textbf{x}$ and $\textbf{y}$ are two independent Complex Gaussian random vectors with i.i.d elements. What is the PDF of $z$ as the ratio of norms in this form: \begin{equation} ...
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56 views

Martingale Can't be Strictly Increasing

If the sample paths of a martingale are almost surely continuous and not constant on any interval, is it true that they are almost surely not increasing on any interval? Edit for clarity: Let ...
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23 views

Stuck with a problem regarding conditional expectation

I'm stuck with a question regarding conditional expectation. If $X$ denotes the number of successes in n independent Bernoulli trials. Where the success probability is unknown and modelled by the ...
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36 views

Formal examples of Poisson point processes

I am self-studying probability theory and currently the Poisson point process (PPP) gives me hell, firstly because the definition of a point process in general and PPP in particular seems rather ...
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17 views

Bounds for PDF of Sum of Two Dependent Random Variables

Assume $X$ and $Y$ are two dependent random variables and we do not have the joint distribution of these two. Is there an upper/lower bound for the PDF of $X+Y$? I found a paper which provides bounds ...
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45 views

Confusion regarding almost sure events. If given infinite time, will a discrete-time gaussian process cover the entire real line?

This question really pertains to any discrete time continuous-valued, stationary stochastic process on the real line, but the Gaussian process will be adequate for this question. I have this ...
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33 views

Deriving Black Scholes using CAPM

I am referring to http://www.frouah.com/finance%20notes/Black%20Scholes%20PDE.pdf Section 3, which is a bit more detailed version of the original derivation from ...
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42 views

Relation of stopped sigma-Algebra on cadlag sample space to arbitrary sample space

Let $X_t : \Omega \to E$ be a cadlag process with Polish state space $E$, $T$ a stopping time w.r.t. the canonical filtration $\mathcal{F}_t$ of $X$ and $X^T_t$ the stopped process. Then it should ...
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23 views

An equality in SDE.

I read an example in Shreve: How to get the equality in the last line?
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45 views

Mathematical Description for Steam Rising from a Cup

I was staring at a cup of coffee I have on the desk just now. The light shines through the water vapor as they rise from the cup. The shape of the steam is not completely random, as it drift from ...
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39 views

Covariance of m-fold integrated Wiener process

The problem I'm trying to perform a Bayesian approach to the Maximum Likelihood Estimation procedure of Wecker and Ansley (1983). To this end, I need to compute the full likelihood of the data given ...
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14 views

Second Moment of Intensity Function of Stochastic Process

I'm trying to compute the 1st and 2nd moment of the intensity function of a Hawkes Process. The intensity function is of the form $$\lambda(t)=\lambda_0+\int_{- inf}^tv(t-s)dN_s$$ where $\lambda_0$ ...
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64 views

Expected Number of points in Point Poisson Process

Let $\lambda$ be the intensity of points, distributed as point poisson process, in a square grid of area $A$. Then, the Cumulative disributive function is given by: $$ P(r \leq R) = 1 - e^{-\lambda ...
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19 views

Expected number of points within a defined radius [duplicate]

I have the following probability distribution function $$ P(r \leq R) = 1-e^{\lambda \pi R^2} $$ where $lambda$ is the intensity of a point poisson process. I would like to calculate the $\lambda(R)$. ...
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14 views

Empirical Intensity Function

I would like to ask for help determining what other ways are there to compute the "empirical intensity function" of a process. In essence, given that I observe the occurrences of an event in time ...
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39 views

Why a positive recurrent Markov chain implies positive limiting probability?

Let $X=\{X_0,X_1,\ldots\}$ be an irreducible, positive recurrent, and aperiodic Markov chain with the state space $S=\{0,1,2,\ldots\}$ then how do we show that the probability $$ ...
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28 views

Ito formula for integral function

Let $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ where $W_t$ is a Wiener process. Let $$Z_t = e^{-r(T-t)} \int_{t}^{T}{h(u,S(u))du} = g(t,S)$$ where $h$ is a known function of $t$ and $S$. How can we ...
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56 views

Including non-markovian processes in a birth-death process

Current model I want to model a stochastic system with a birth-death (Markovian) model. I therefore have this kind of $n$ times $n$ (where $n$ is the number of possible states) transition matrix: ...
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55 views

Stochastic process gambler's ruin [closed]

This is a gambler's ruin problem I would appreciate if anyone can give me a hint about how to solve it. So A, B play this game by tossing a coin. If H shows then B gets 1 dollar from A and if T shows ...
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48 views

Law of a geometric brownian motion first hitting time (proof checking)

I need to use it in a small step in the middle of a simulation and I think I'm not getting correct results to this probabilities and so for my all subsequent simulation. Could someone ...
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1answer
19 views

If $(X_{t})_{1}^{\infty}$ is a stationary process, would the sigma-rings generated by $X_{t}$ the same for all $t$?

This question is about a stationary process and the sigma-rings generated by its components. Specifically, if $(X_{t})_{1}^{\infty}$ is a stationary process and $\mathscr{F}_{i}$ and ...
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23 views

About measurability of a stopping time.

If $S,T$ are two stopping time w.r.t. $\mathcal F_t$ define $R=S\wedge T$.Then $R$ is a stopping time .How to prove $R$ is measurable w.r.t $\mathcal F_T$? Is there something wrong with this ...
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22 views

Independence of compound poisson process increments

I have compound Poisson process $\{X_t = \sum_{i = 1}^{N_t} \xi_i\, t\geq 0\}$, where $\xi_n$ is all independent and identically distributed random variables, $\{N_t, t \geq 0\}$ is a poisson process ...
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1answer
51 views

Is every PMF on the set of non-negative integers the stationary distribution of some birth-death process?

Let $f(.)$ be a probability mass function on the non-negative integers such that $0<f(n)<1$ and $f(0)+f(1)+...=1$. Then does there exist an irreducible birth-death process with stationary ...
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30 views

What is the probability that a stochastic process that arrived at a final point has passed a specific point in the past?

Consider an ensemble of realizations of a stochastic process that all end at the same final point $x_f$ at time $t_f$! How can I calculate the probability distribution of points at which these sample ...
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40 views

Itô process and covariance of two Brownian motion

I'm a novice in studying the stochastic different equation, and didn't know whether I have describe the question correctly. Here is the question: Suppose $$\begin{array}{rcl} ...
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1answer
68 views

Feynman-Kac representation for a PDE

I have the following PDE: $$ u_t + r x u_x + \frac{\sigma^2 x^2}{2} u_{xx} + h(t,x) u_y - ru =0 $$ $$ u(x,T,y) = y $$ I wanted to check whether the following representation is correct (I used ...
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33 views

Stochastic integral in closed form

Let $(W_t)_{t\geq 0}$ be a Brownian motion and $\alpha>0$ be a constant. Consider the following quantity: $$\mathbb{E}\Big(\int_0^tsdW_s{\bf 1}_{\{t^{-\frac{1}{2}}W_t>\alpha\}}\Big).$$ Can a ...
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33 views

A step in verifying a stopping time.

Suppose $X$ is a cadlag process adapted to $\{\mathcal F_t\}$ and $H$ is a closed set.Verify $\sigma_H\triangleq\inf\{t\ge0:X_t(\omega)\in H\}$ is a stopping time . The first step is: ...
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1answer
25 views

Multi variable Langevin equation

I need to solve the following system $\frac{\partial f(t)}{\partial t}=a_1 f(t)+a_1 g(t)+s_1(t) \\ \frac{\partial g(t)}{\partial t}=a_3 f(t)+a_4 g(t)+s_2(t) $ with $s_i$ being a noise, ...
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27 views

An equality in stopping time.

In a proof,I need the following equality: Suppose $\tau,\sigma$ are two stopping time and $A$ is a event.Then: $$(A\cap\{\sigma\le\tau\})\cap\{\tau\le t\}=(A\cap\{\color{red}{\sigma}\le ...
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98 views

Martingality Theorem: Solving expectation of a stochastic integral

I am trying to prove that: $$ \Bbb E\left[\int_s^t\sigma e^{-k(t-u)}\sqrt{V_u}dW_u\right] =0$$ Where: $$ dV_t=k~(\theta-V_t)~dt+\sigma\sqrt{V_t}dW_t $$ I have attempted to use Ito's formula on the ...
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30 views

Bound for the variance of a stochastic process

Given a random variable $X$ and $N$ realizations of the stochastic process associated to $X$, a theorem gives a bound for the $\sigma^2[X]$: $$\sigma^2[X]\le\frac{1}{4}(A-a)$$ where $A$ and $a$ are ...
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31 views

Central limits without replacement in a finite population.

"Everybody knows" that there are lots of variations on the theme of the central limit theorem. The most frequently seen form seems to be this: Suppose $X_1,X_2,X_3,\ldots$ are i.i.d. random variables ...
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63 views

conditional expected value - Poisson process plus random variable

I've struggled with this actuary excercise for a while and I don't know how to do it: Each claim can be characterized by two random variables $(T,D)$, where $T$ is the moment of reporting the claim ...
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16 views

Recurrence of states in a function of a Markov chain

Suppose $X$ is a Markov chain (or process, for that matter) and suppose further $f(X)$ is also a Markov chain. Let $s$ be a recurrent state in $X$. Is there a general way to determine the recurrence ...
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Distribution of the supremum of a transformed Brownian Motion?

I have a stochastic process given by $z_{t}=w_{t}/\alpha\left(t\right)$ , where $w_{t}$ follows a Wiener process (a standard $\left(0,1\right)$ Brownian Motion) starting from $w_{0}=0$ , ...
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79 views

Exchangeability of random variables and conditional expectation

Let $(\Omega, \mathcal{F}, \mathbb{P} )$ be a product space ${\mathbb{R}}^{\mathbb{N}}$ equipped with the product $\sigma$-algebra $\mathcal{B} ({\mathbb{R}}^{\mathbb{N}})$. Let $(X_n)_{n \geq 1}$ be ...
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30 views

Independent Brownian motions question

Let $B$ and $W$ be independent Brownian Motions and let $\tau$ be a stopping time of $W$. Is it true that $E[\int_0^{\tau} B_s \, dW_s] = 0\text{ ?}$ So far I have tried the following: The integral ...
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1answer
25 views

mean hitting time of a level and growth rate of maximum process

Let $X_t$ be the absolute value of Brownian motion starting at $0$, let $\tau_x$ be it's first hitting time of the level $x>0$, and let $M_t$ be it's running maximum up to time $t$. Suppose we knew ...
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37 views

Ito with the function containing stochastic integral

Statement of problem From Oksendal SDEs question 5.18: The geometric mean reversion process is a solution to: $$ dX_t = k (a - \log X_t) X_t dt + \sigma X_t dB_t $$ In showing that solution is ...
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48 views

Expectation of absolute value of Brownian motion

I'm working on this problem that I can't seem to figure out. The problem involves a 1-dimensional Brownian motion, $B_t$, where the subscript denotes the time, and it asks me to show that the ...
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Optimal stopping problem

Consider the OU process: $dX_t = -X_tdt + dW_t$, $X_0 = 0$. Compute the optimal stopping time for the following problem: $$v = \sup_{\tau} E[|X_{\tau}| - \tau]$$ So far I have set $L\phi = 0$, ...
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30 views

Laplace transform of the autocorrelation of a wss random process

Consider a wide-sense-stationary random process $x(t)$. The autocorrelation function is $r(t-\tau):=E[x(t)x(\tau)]$. Let $S(s)$ be the Laplace-transform of $r(t)$. Can I compute $S(s)$ as ...
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29 views

Correlation and First Order Stochastic Dominance

Suppose we have a random variable $X \sim [0,1]$ with a continuous distribution $F_X(x)$. Suppose $I \in \left\{0,1\right\}$ is a discrete random variable with $\text{Prob}(I=1 \ | \ X=x)$ strictly ...
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12 views

Entropy of non-ergodic process

Two coins have been kept in a box, One is fair while the other is biased. One coin is picked. The probability of either coin being picked is equal. The picked coin is then tossed again and again to ...
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44 views

Well-definedness of the characteristic function of a compound Poisson variable

I am reading about compound Poisson variables and cannot get through the following statement. Let $\nu$ be a non-zero finite measure on $\mathbb{R}\setminus\{0\}$. Assume that $$\int ...