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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Modes of convergence for a *continuous-time* stochastic process

I know that if a sequence of non-negative random variables $(X_n)_{n \in \mathbb{N}}$ satisifies $$\mathbb{E}(X_n) \rightarrow 0 $$ as $n \rightarrow \infty$ implies that a subsequence converges ...
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50 views

Inequality of an expectation (here: perpetual put of an american option)

for a given function $u(x):=\sup_{\tau \in T_{0,\infty}}E[(Ke^{-r\tau}-xe^{\sigma B_{\tau}-(\sigma^{2}\tau)/2})_{+}1_{\tau <\infty}]$ and $x \in [0,\infty)$, K a positive real number, $(B_{t})$ a ...
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79 views

local martingale bounded below by a DL process

Let a continuous adapted process $Z= (Z_t)_{t \geq 0}$ be of class DL if \begin{equation} \{ Z_{\tau \wedge t} : \, \tau \text{ is a stopping time } \} \end{equation} is uniformly integrable, for each ...
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1answer
120 views

Monotone Class Theorem Application

I am trying to proof the following statement. Let $h$ be a bounded, $\mathbb{F}$-predictable process with $\tau$ a $\mathbb{H}$-stopping time, we then like to prove \begin{equation} ...
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45 views

Bivariate GBM - crosscovariance

I have troubles concerning a correlated bivariate GBM with identical drift and diffusion rates. Let $dX^i_t = \mu X^i_t dt + \sigma X^i_tdW^i_t$ and $E[dW_t ^idW^j_t] = \rho_{i,j}dt$ If $X_0^i = ...
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145 views

Lévy's upward theorem and $\mathcal{L}^p$ convergence.

Lévy's upward theorem: Let $Y \in \mathcal{L}^1(\Omega, \mathcal{F}, P)$, $(\mathcal{F}_n)_{n=1}^{\infty}$ a filtration of $\mathcal{F}$ and $\mathcal{F}_{\infty} = \sigma( \bigcup_{n=1}^{\infty} ...
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53 views

Trying to show convergence (in probability) of integrals using Taylor expansion

I've been working for a long time now on how to prove a proposition given in a paper about the asymptotic normality of POT-quantile estimators. Hope somebody can help me out. Proposition (i) Let ...
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42 views

Self similar process

I am learning long memory process and came cross the definition of self similar. By definition, process $X(t)$ is self similar if $X(at)=_d a^H X(t)$,$a>0$ and $H$ is Hurst exponent. By equality of ...
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18 views

Is it an increasing process?

On a probability space $(\Omega,\mathscr{F},\mathbb{P})$ with filtration generated by Brownian motion, there is a progressivley process $(A_t)_{t\in[0,T]}$. If for any stopping times $0\leq \sigma\leq ...
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83 views

Why Gaussian process is not Ergodic in general?

Can anyone use a simple way to explain this? I heard this in class but I do not know why. By Wiki: a random process is ergodic if its statistical properties can be deduced from a ...
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28 views

How can I know by inspection that a process is WSS?

I have some codes to generate three different Random Sequences: I am getting a 4x100 matrixes where 4 is the number of samples and 100 is the length of the process. I am getting these results: ...
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19 views

Conditions for positive definiteness for a class of matrices induced by a semimetric

Let $X$ be a set, and let $d:X\times X\rightarrow \mathbb{R}$ be a semimetric on that set (i.e. $\forall x,y\in X$, $d(x,y)=d(y,x)\ge 0$, and $d(x,y)=0$ iff $x=y$). I seek conditions on $X$ and $d$ ...
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1answer
60 views

Marginal Probability of Stochastic Process

I have a wide sense stationary stochastic process x(t)=asin(2πf0t)+bcos(2πf0t) where a & b are independent gaussian random variables. How can I find the Marginal probability of x(t)? I am ...
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93 views

Asymmetric simple random walk?

It comes from the book Probability: Theory and Example. I don't understand the part marked with red line. Why it cannot converge to an interior point of $(a,b)$? Can anyone help? Thanks so much!
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202 views

Finding Conditional Expectation and variance E(Y|X=x)

I want to find the conditional Expectation and variance of random function Y for a given value of random function X, i.e. E(Y|X=x). Here X is x(t) and Y is x(t+τ). Also, x(t) is a stationary Gaussian ...
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46 views

Property of submartingale and supermartingle?

Is it true that for a submartingale, $$E(X_n) \le E(X_m)$$ for $n \le m$. And for a supermartingale, $$E(X_n) \ge E(X_m)$$ for $n \le m$. If it is true, then why? I feel confused because the ...
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77 views

Markov property of Brownian motion

There are two statements about Markov property: $B_t $ is Brownian motion and $\mathcal{F}$ is generated by $B$ If $s>0$ and $Y$ is bounded and measuable, then ...
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45 views

Where is the assumption of right continuity used in the following proof?

Lemma:If $X$ be a right-continuous positive local martingale then , $X$ is a generalized super martingale Proof: $\forall s<t$ $$E[X_t\mid F_s]=E[\lim_{n\to\infty} X_{t \wedge\tau_n}\mid F_s] \leq ...
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74 views

Birth immigration process

I'm having some problem with this question. A model for the distribution of the number of goals scored in soccer matches suggests that if n goals have already been scored by time t, then the ...
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51 views

If a stochastic process follows Geometric Brownian Motion, does it imply that it is Log-normally distributed and vice-versa?

This might be a naive question, but it doesn't stop haunting me. Wiki page for GBM writes the SDE for GBM process and shows it follows log-normal distribution. Is it true every time or are there any ...
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192 views

A die whose score cannot be as before (Markov chains)

A die is "fixed" so that each time it is rolled the score cannot be the same as the preceding score, all other scores having probability $1/5$. Given that the first score is 6, what is the probability ...
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115 views

Why do two points never 'arrive at once' in a Poisson point process

In the following, all the measure spaces are endowed with the Borel $\sigma$-algebra corresponding to their topology (we take the usual topology on $[0,\infty)$). Let $E$ be a Polish space and let ...
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83 views

Cellular automata (Random walk)

Here is the context of my question below. I cite from "Some Rigorous Results for the Greenberg-Hastings Model" by Richard Durrett- Consider the following cellular automaton known as the ...
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128 views

Donsker's Theorem for triangular arrays

Assume we have a sequence of smooth i.i.d. random variables $(X_i)_{i=1}^{\infty}$. Given $\alpha>0$, does some sort of Donsker's Theorem hold for $\left(\frac{X_i}{n^{\alpha}}\right)_{i=1}^n$? ...
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87 views

Marginal probability density function of Stochastic process

I was solving the following question and I derived the Auto correlation function and proved that it is a WSS process. However, I am not sure how to go about finding the Marginal probability density ...
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91 views

Usual augmentation filtration? (Sigma algebra generated by a descreasing family of sets)?

My aim is to understand the usual augmentation filtration, for example as given in the appendix in the book Financial Derivatives in Theory and Practice: Revised Edition by P.J. Hunt and J.E. ...
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1answer
766 views

Finding mean and variance of stochastic process

If I'm given a Stochastic Process Xt that satisfies a stochastic diff. equation, let's say fXt, what is the formula to find the mean and variance of Xt? I think it's: $mean = dE(X_t) = dX_0e^t$ ...
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1answer
44 views

Algebra and partions of a set

My book in mathematical finance introduces algebras and partitions of a set, in order to explain how information is modeled to the investor. But there is one thing I don't get. They say that for every ...
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438 views

quadratic variations of Brownian motion squared

I'm trying to refresh my memories about stochastic processes. We know that Brownian motion has as quadratic variation equals to t. What is the quadratic variation of the Brownian motion squared ? ...
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63 views

Recurrence/Transience of random walk with +2/-1 steps

Consider the Markov chain with state space $S=(0,1,2,...)$ and transition probabilities: $p(x,x+2)=p$ , $p(x,x-1)=1-p$, $\forall$ $x>0$. $p(0,2)=p$ , $p(0,0)=1-p$. For which values of $p$ is this ...
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53 views

an exercise about mean and probability

Let $(\Omega, \mathcal{F}, P)$ be a probability space, $X : \Omega \rightarrow \mathbb{R}$ be a discrete random variable and $$\phi : [0, \infty) \rightarrow (0, \infty)$$ an increasing function (so ...
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1answer
37 views

$E[1_{\lbrace P_T-P_{\tau_n}=0\rbrace}\int_{\tau_n}^T h(s)dN_s]=0?$

If $P_t$ is a standard Poisson process, and $N_t=P_t-t$ the associated martingale then $\int_0^t h(s)dN_s$ is a martingale (assuming that h satisfies the neccessary hypothesis). Thus, considering ...
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82 views

Inequality for the expected values of norm of stochastic processes

Let $\underline{X}=(x_1, x_2, x_3), \; x_i \sim \mathcal{N(0,1)}$ i.i.d. For any fixed $t>0$ and $\underline{X}_0$ prove that the following holds ($\Vert\cdot\Vert$ is the Euclidean norm): ...
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67 views

Optional Sampling Theorem Application

Let x, y > 0. Define the first passage time of a Brownian motion $W_t$ as $\tau_a$ = min{t $\ge$ 0: $W_t$ = a}. I need to show that $$E[e^{-u\tau_x}1_{\tau_x < \tau_{-y}}] = ...
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33 views

distribution of $\sup\limits_{0\le t \le 1}|W(t)|$

My prof on class told us that distribution of $S=\sup\limits_{0\le t \le 1}|W(t)|$ has been well studied, where $W$ is a Wiener process, but I need a table to find $c$ such that $P(S < c) = 0.95$. ...
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36 views

Bivariate stopped processes

Take two dependent Levy processes $L_1(t)$ and $L_2(t)$ with law $\mathcal{L}(L_1(1),L_2(1)$. If we stop the first process at a general time $t=s_1$ and stop the second process at another general time ...
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1answer
43 views

Showing that if $B_t$ is a Brownian motion then $t B_{1/t}$ is Gaussian

I want to show that if $B_t$ is a Brownian motion then $t B_{1/t}$ is a Gaussian process, i.e. that it has increments which have the normal distribution. It seems like a trivial fact, since the ...
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1answer
118 views

Stochastic differential of Bessel process [closed]

Let $ \underline{B}_{t}=(B_1(t), \dots, B_d(t))$ be a $d$-dimensional Brownian motion. How to calculate the stochastic differential of $ \Vert{\underline{B}_t}\Vert$? $\Vert . \Vert$ denotes the ...
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1answer
44 views

mean of random variables

Let $(\Omega,\mathcal{F}, \mathbb{P})$ be a probability space, $X : \Omega \rightarrow \mathbb{R}$ a discrete random variable and $g : \mathbb{R} \rightarrow \mathbb{R}$ a random variable. I can't ...
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31 views

Stochastic scheduling to maximize the expected number of customers arrived at the root of a Jackson tree

In a Jackson network, organized as a tree rooted at queue r, several customers are queued at time 0 and there is no new customer arrival. The service time of each customer in queue i is geometrically ...
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113 views

Probability of Wiener process hitting a particular point at an independent stopping time

Assume we have a stopping time $T$ that is independent of a Wiener process $W$. If $T$ were taking discrete values (let's say in $\mathbb{N}_0$), one can easily show (using the independence and the ...
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36 views

Level sets of a Wiener process

Assume we have a Wiener process $W$ starting at $W_0=0$. What can one tell about the Lebesgue measure of "level sets" $A_y = \{t>0; W_t=y\}, y \in \mathbb{R}$? I actually need to estimate these ...
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190 views

Martingale regularization with right continuous filtration

The standard textbook presentation of the Doob Regularization Theorem for a martingale $(X_t, \mathcal{F}_t)$ assumes that the filtration satisfies the usual conditions. It is clear that the ...
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1answer
73 views

Stochastic intensity poisson process

I am wondering if, considering $N_t$- an $\mathcal{F}_t$ poisson process with stochastic intensity $\lambda_t$ on $(\Omega,P)$ and $\tilde{N}_t$- an $\mathcal{G}_t$ poisson process with stochastic ...
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44 views

How to calculate probability of an event in a stochastic setting?

Let $\left(\, B_{t}\,\right)_{t\ \geq\ 0}$ be a Brownian motion. Calculate the probability of the event: $$ E\equiv\left\{\,\exists\ \epsilon > 0 : \forall\ 0 < h < \epsilon, \max_{t\ \in\ ...
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44 views

A question on proving the existence of a martingle which has a deterministic square bracket

Let $g:\mathbb{R^+} \to \mathbb{R^+}$ be a non decreasing and continuous function . Show that there exists a continuous martingale M such that its square bracket $<M>_t=g(t)-g(0)?)$ I have ...
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100 views

On the definition of Markov chains

A Markov chain with discrete time dependence and stationary transition probabilities is defined as follows. Let $S$ be a countable set, $p_{ij}$ be a nonnegative number for each $i,j\in S$ and assume ...
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200 views

Jumps of Lévy process

Let $X:=(X_t)_{t\geq0}$ be a Lévy process with triple $(b,A,\nu)$. Is there any known relation between the "distribution" of its jumps and the Lévy measure $\nu$? E.g. can we express something like ...
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74 views

Applying Ito's formula

This is probably an easy question but I am getting aquanted with Ito's formula and stuck on an exercise in my textbook. Let $X_{t}=W_{t}-a t/2$ where $a$ is a real number and $W_{t}$ is brownian ...
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89 views

How to connect the deterministic and probabilistic descriptions of the SIR model

I am a 17 year old student and I was reading up on epidemic modelling for a math project, specifically the SIR model and I came across this: "This" refers to the assumptions to which the Markov Chain ...