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.

learn more… | top users | synonyms

0
votes
1answer
30 views

multivariate probability generating function

Suppose I have three random variables $X_1$, $X_2$ and $X_3$, with probability generating functions $g_1(z)$, $g_2(z)$ and $g_3(z)$. Now I have a joint-distribution $P(X_1-X_2,X_1-X_3)$, whose ...
0
votes
1answer
23 views

Question about limit of Stochastic Process

Given $\mu_t$ continuous stochastic process that satisfies $\int_0^t \mu_s^2\;ds<\infty$. Define $X_t\equiv \int_0^t \mu_s\;ds$. Let $|\cdot|$ denote floor function. Then where does ...
0
votes
1answer
26 views

Mean time spent in transient states/Markov chain

I dont get this in my book: For transient states $i$ and $j$ , let $s_{ij}$ denote the expected number of time periods that the markov chain is in state $j$ , given that it starts in state $i$. Let ...
0
votes
0answers
10 views

linear system output when input is a Gaussian process?

Rectently, I read a technical book that says:" the linear transform of a Guassian process is also a Guassian process. i.e. for continuous time case: $$ x(t)*h(t)=y(t)$$ the input $x(t)$ is a ...
0
votes
1answer
22 views

$dX_t=-\mu X_tdt + \sigma dW_t$. Prove that $X_t = e^{-\mu t}X_0 + \sigma \int_0^t e^{-\mu(t-u)}dW_u $

So the solution says use Ito-s formula, taking $Y_t:= e^{\mu t}X_t$ to obtain $dY_t = [\mu e^{\mu t}X_t - e^{\mu t}\mu X_t + e^\mu t \sigma dW_t] $. As far as I can see though, Ito's formula says ...
0
votes
1answer
31 views

Coupled stochastic differential equations?

I'm a physics student working on a quantum information project (so please be gentle with me). My work involves stochastic processes and I'm new to the topic, so I'm asking some help about a system of ...
3
votes
0answers
35 views

Markov chains and natural filtration

I have the following problem Consider a homogeneous Markov chain $(X_n)$ with countable state set $E$. Suppose that $A$ is a proper subset of $E$ and consider the stopping times $\tau^0=0 $ and ...
1
vote
2answers
41 views

Is this true about Brownian Motion?

I have the following in my notes and I'm not sure if it's true or not. Any help would be highly appreciated. If $\{W_t\}_{t\geq0}$ is a standard Brownian motion stochastic process, $\Delta>0$ and ...
0
votes
0answers
23 views

Why is chemotaxis considered an emergent behavior?

this is an applied math question. I could have posted this under a biological stackexchange, but the idea of emergent behavior or emergent properties of a system seems more appropriate to an applied ...
2
votes
0answers
33 views

Ito formula for $f(X_t, Y_{t-s})$

I have a situation where I have two stochastic processes (say 2 OU processes) and I have the function $f(X_t, Y_{t-s})=\frac{X_t}{Y_{t-s}}$. How do I apply Ito lemma in this case?(is Ito lemma still ...
1
vote
1answer
56 views

Use Ito's Lemma to show:

I am somewhat unsure how to go about showing this: Use Ito's Lemma to show for any deterministic differentiable function, $f$: $$ \int_0^t f(s) dB(s) = f(t)B(t) - \int_0^t B(s)f'(s)ds $$ Where $B(t)$ ...
0
votes
0answers
11 views

Parameter estimation using characteristic function

Is it possible to do parameter fitting using log-returns data & the characteristic function(CF) in Matlab? I have been trying it on the Variance Gamma Scaled Self-Decompasable (VGSSD) model CF for ...
1
vote
1answer
35 views

Fixed-time Jumps of a Lévy process

If one defines a Levy process as a stochastic process $(X_t)_{t\geq0}$ that has independent and stationary increments with (a.s.) cadlag paths (hence a def. withouth stochastic continuity). How can I ...
0
votes
0answers
21 views

probability of supremum of martingale

Let $Z_{t}$ a continuous nonnegative martingale with $\lim Z_t=0 $ a.s. for every $s\geq0$ and $ b>0$ . show 1/ $ \textbf{P}(\sup_{t>s}Z_t \geq b\mid\textbf{F}_s) =\frac{1}{b} Z_s$ on ${Z_t ...
1
vote
1answer
33 views

supremum and expectation of a martingale

Let $X_{t}$ a right continuous $\textbf{F}_{t}$ martingale and $\textbf{F}_{t}$ satisfying the usual condition Show that $ \sup_{t\geq 0}\textbf{E}(X^{2}_{t})<\infty$. I know that $X^{2}_{t}$ is ...
0
votes
0answers
16 views

Implication of uniform stochastic boundedness?

Let $\theta \in \Theta \subseteq \mathbb{R}^d$ be a parameter vector. Let $Q: \Theta \rightarrow \mathbb{R}$ be a function mapping from the parameter space to the real numbers. Let $Z_T$ be a a ...
0
votes
1answer
30 views

Markov chains for group decision making

I am new to Markov chains since I am doing my own studying on it recently. I was doing some questions and came across this one that got me stuck. Suppose there are four employees and they need to ...
0
votes
0answers
22 views

Girsanov's theorem for OU process

Say that I've got the process $dr_t=a(b-r_t)dt+\sigma dB_t$ and that I want to calculate $E[\exp(-\int_0^T r_s ds)]$ by using Girsanov's theorem. How do I do this? I cannot seem to find an explicit ...
1
vote
1answer
297 views

Deriving SDE for process with two uncorrelated Brownian motions and factor

Derive SDE for the following 2 dimentional process $Y(t) = wX_1(t) + \sqrt{1-w^2}X_2(t)$ where $X_1$ and $X_2$ are brownian motions with drifts and brownian increments $dX_1(t)= \mu_1dt + ...
1
vote
1answer
43 views

Probability of Renewal Processes

Suppose that there are two brands of replacement components, Brand X and Brand Y, and that for political reasons a company buys a replacements of both types. When a Brand X component fails it is ...
1
vote
1answer
27 views

Variance of Martingale Difference Sequence

I'm having trouble understanding part of one of the examples here. This is taken from Hamilton's book Time Series Analysis, p. 194. My question is this. I don't understand why $$ E[X_t^2] = ...
4
votes
0answers
111 views

An exercise from Revuz, Yor; equality in distribution of 2 integrals.

Here is the exercise I have been struggling to solve. It is taken from this book by Revuz and Yor: link. Here is the full text of the problem ( Exercise 3.32, chapter 4). Exercise (3.32). Let $B$ and ...
2
votes
1answer
51 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
0
votes
0answers
12 views

a question which is somhow related to law of large number

suppose that p = [p1, p2, ..., pn]' is a random vector. (' == transpose) and each element of p like pi is a Gaussian random variable with zero mean (E(pi)=0) and variance vi (E(pi^2)=vi). the ...
1
vote
0answers
47 views

probability of a stopping time

Let $T_{y}=\inf\left\{t: M_{t}\geq y\right\}$ , $x<y$ where $M_{t}$ a right-continuous martingale satisfying: $M_{0}=x \in \textbf{R}_{+}$ and $\lim_{t\longrightarrow \infty} M_{t}=0$ a.s Show ...
0
votes
1answer
32 views

a martingale equality

Let $X_{t}$ a positive continuous martingale satisfying: $\lim_{t\longrightarrow \infty}X_{t}=0 $ ps and $X_{0}=a \in {R_{+}}$ Show that $\textbf{P}(\sup_{t\geq0}X_{t} \geq b)=\frac{a}{b}$ , a < ...
1
vote
0answers
22 views

is following model stationary?

I am interested if following model is stationary,model is represented by following formula $$ x(n) = \sum_{p=1}^{P} a_p \cos(2\pi f_pn + \phi_p) + \epsilon(n) $$ $n$ is changing from $1$ to $N$, I ...
0
votes
1answer
42 views

conditional probability poisson process

I am stuck on how to find the conditional probability of a poisson process. I know generally, if you have a poisson process with intensity parameter $\lambda$, then the conditional probability of ...
0
votes
1answer
38 views

Representation of Markov process adapted to given Filtration

Let $X$ be continuous Markov process adapted to a filtration generated by Brownian motion $B$. Does there exist a function $f$ such that $X_t = f(t,B_t)$? My guess is that it should have such ...
0
votes
1answer
32 views

General Poisson process

Define a generalised Poisson process as an arrival process that begins at time 0 and that satisfies: The independence property: the number of arrivals during two non-overlapping intervals ...
2
votes
1answer
16 views

Set of simple predictable processes is a vector space

I have a question, which is probably very easy for you to answer. How can I show that the set of simple predictable processes a vector space is? It's clear that I only have to show that the sum of ...
0
votes
2answers
31 views

Finite in probability implies finite expectation

Let $T_n$ be a random variable with $T_n=X_1+...+X_n$ where the $X_i$'s are iid. Further we set $N(t)=max\{ n: T_n\leq n\}$. If $\Pr(N(t)<\infty)=1$, does this implies ...
2
votes
2answers
143 views

Traversing an array and counting the number of distanct number from the given elements in an array.

You are given an array $A[0 \ldots n-1]$ of $n$ numbers. Let $d$ be the number of \emph{distinct} numbers that occur in this array. For each $i$ with $0 \leq i \leq n-1$, let $N_i$ be the number of ...
0
votes
1answer
34 views

Ornstein-Uhlenbeck processs: Markov, but not martingale?

I'm puzzled about properties of the Ornstein-Uhlenbeck process, given by the Itō integral $$ X_t = x e^{-\lambda t} + \sigma \int_0^t e^{-\lambda(t-s)} d W_s \,. $$ I compute that $\{X_t\}$ is not ...
0
votes
0answers
27 views

Cubed Brownian motion

I have to do the following exercise: Let $(W_t)$ be a Brownian motion. (a) Does X given by $X_t:=W_t^3$ have constant expectation? (b) Is it a martingale? (c) Does it have independent increments? ...
1
vote
0answers
14 views

Considering the backwards motion of a particle described by $X(t+\Delta t)=X(t)+f(X(t),t)\Delta t+g(X(t),t)\sqrt{\Delta t}\zeta$

If we have a particle whose position $X(t)$ is stochastically described by: $$X(t+\Delta t)=X(t)+f(X(t),t)\Delta t+g(X(t),t)\sqrt{\Delta t}\zeta$$ Where $\zeta \sim N(0,1)$. What is we want to ...
2
votes
1answer
35 views

question about martingale

In my lecture notes,I found the following problem: Let $X$ an $F_{t}$ adapted continuous process and $G_{t}\subset F_{t}$. show that $$E\left(\left. \int^{t}_{0}X_{s}ds ...
1
vote
2answers
37 views

Expectation of a parallel system

A system consists of $n$ components in parallel. The lifetimes of the components are i.i.d. exp($\lambda$) random variables. The system functions as long as at least one of the $n$ components is ...
1
vote
1answer
41 views

If $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$ then $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s

Let $X_t$ be an Ito's process where $dX_t=b_tdt+\sigma_tdW_t=\tilde{b}_tdt+\tilde{\sigma}_tdW_t$. Prove $b_t=\tilde{b}_t$ and $\sigma_t=\tilde{\sigma}_t$ a.s Here my solution for ...
0
votes
2answers
30 views

If $u(z),$ where $Z_t=W^1_t-iW^2_t$, is a complex anlyt. fx, show $du(Z_t)=u'(Z_t)dZ_t$

If $u(z)$ is a complex analytical function, where $Z_t=W^1_t-iW^2_t$ is a complex Wiener process, show $du(Z_t)=u'(Z_t)dZ_t$.
0
votes
1answer
30 views

Quadratic Variation for $X_t= \int \sigma_s dW_s$ where $\sigma_s \in S$

Let $\sigma_s \in S$. Setting $X_t=\int^t_0 \sigma_s dW_s$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ such that $d_n=\max_i |t^n_{i+1}-t^n_i| \rightarrow 0$ as $n \rightarrow ...
0
votes
0answers
10 views

Change of Measures for Lévy-Processes

If $X$ is a Lévy-Process on a filtered probability space $(\Omega,\mathcal{F}_t, \mathbf P)$ and $Q$ an equivalent probability measure. Under which circumstances is $X$ also a Lévy-Process under ...
1
vote
1answer
62 views

An application of the strong Markov property in the proof of the connection between Brownian motion and harmonic functions

Let $U$ be an open, connected set in $\mathbb{R}^n$ and let $(B(t))_{t \geq 0}$ be an $n$-dimensional Brownian motion with start at $x \in U$ and let $\overline{B_x(\delta)}$ be the closed ball about ...
1
vote
1answer
27 views

What does this mean in the context of Stochastic Calculus?

I've reading into some Stochastic Calculus books and I've been stumped by two concepts used recurringly in the book. The first is a subscripted 1 which appears in the definition of a simple process ...
0
votes
0answers
22 views

Transition matrix, stationary distribution and expected number

A company wants to operate s identical machines, but they are subject to failure according to a given probability law. To replace them, the company orders new machines at the beginning of each week to ...
0
votes
0answers
41 views

Expected time to reach 6th return time of Markov Chain

I'm having a hard time figuring out this problem from Resnick's Adventures in Stochastic Processes: Harry is negotiating a new tv show and the negotiations follow a discretely indexed Markov chain. ...
2
votes
0answers
31 views

A Lemma in the book “ Mathematical Method for financial markets” (Chapter 5, Section 5.7)

In page 307, Section 5.7, Chapter 5 of the book "mathematical methods for financial markets" by Jeanblanc, Yor and Chesney, Lemma 5.7.1 is given as follows: Lemma 5.7.1.1 Let $W$ be a Brownian ...
7
votes
2answers
172 views

A planar Brownian motion has area zero

I'm looking for proofs of Paul Lévy's theorem that a planar Brownian motion has Lebesgue measure $0$. I know of only two proofs: one is in Lévy's original paper (Théorème 12, p. 532) and the other is ...
0
votes
1answer
40 views

Quadratic Variation for $X_t= \int b_s ds$ where $b_s$ is an $F_t$ adapted process.

Let $b_S$ be an $F_t$ adapted process, Borel measurable in $t$ st $\int |b_s|^2ds < \infty$ (a.s). Setting $X_t=\int^t_0 b_sds$ and partitioning the interval $[0,t]$ i.e. $0=t^n_0<t^n_1... $ ...
1
vote
0answers
8 views

How to calculate the average waiting time from an in-homogeneous Poisson process.

From the in-homogeneous Poisson process $\lambda(t)=10-5cos(2\pi t)dt $, we know $\lambda(t)$ is the arrival rate at time $t$ and $\int_{0}^{t}10-5cos(2\pi t)$ is the average arrivals during time $0$ ...