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

1
vote
0answers
29 views

Brownian motion independent RVs

Let $(W_t)_{t\in\lbrack 0,T\rbrack}$ be a standard Brownian motion. Does there hold that $W_s(W_t-W_s)$ and $W_k(W_l-W_k)$ for $0\leq s<t\leq k<l\leq T$ are independent RVs?
1
vote
0answers
20 views

Transition functions and Markov processes

I am wondering whether there is a one-to-one correspondence between transition functions and homogeneous Markov processes? We say that $(X_t,\mathcal{F}_t)_{t\geq 0}$ is a Markov process if $\mathbb{...
1
vote
0answers
20 views

Renewal Equation without $[0,t]$ on integral

a renewal process $\{N(t),t \ge 0\}$ with continuous inter-renewal time distribution $W$. Let $A(t)=t-S_{N(t)}$ be the age process. For given constant $x>0$ derive its renewal-type equation for $P(...
2
votes
0answers
58 views

Poisson process and Heaviside function

Show that Poisson process $p(t)$ of intensity $\lambda$ can be written as $$p(t)=\sum_{t>t_n}\delta(t-t_n),$$ where function $\delta:\mathbb{R}\rightarrow\mathbb{R}$ is Heaviside's function: $$\...
2
votes
0answers
38 views

Show that $W^2 _t - t$ is a $\mathbb{P}$-martingale.

Claim: $V_t = W^2 _t - t$ is a $\mathbb{P}$-martingale. I have shown via Ito's formula, that $dV_t = 2 W_t \, dW_t$. For reference, I will list this "Proposition": If $X$ is a stochastic process ...
1
vote
1answer
42 views

Log normal simulation.

I want to calculate numerically the expectation of a lognormal random variable $Y=e^X$, where $X$ is normally distributed with mean $m$ and variance $V$. The expectation is known as $e^{m+\frac{1}{2}...
1
vote
1answer
32 views

Find constants such that transformed simple symmetric random walk is martingale

Let $$S_0 :=0, \quad S_n = X_1 + ... + X_n \quad \forall n \in \mathbb{N}$$ be the simple symmetric random walk on $\mathbb{Z}$, i.e. the $X_i$ are i.i.d. with $$P[X_i = +1] = P[X_i = -1] = 1/2.$$ ...
1
vote
0answers
30 views

Is the process Markov or not?

Consider the stochastic process with $X_0=0$ and $$ X_t= \begin{cases} 0 & \text{ for } \ \ t<\tau_1 \\ 1 & \text{ for } \ \ \tau_1\leq t < \tau_1+\tau_2 \\ 2 & \text{ for } \ \ \...
1
vote
0answers
24 views

Simulation of a diffusion on $[0,1]$

I have a diffusion process $X=(X_t)_{t \ge 0}$ with the generator $$Af(x)=\frac{1}{2}(a(1-x)-bx)f'(x)+\frac{1}{4}x(1-x)f''(x),$$ where $a,b >0$ are constants. I want to simulate $X$ to a ...
1
vote
1answer
37 views

Ito formula when g(t,x) is an integral

Suppose we have a stochastic process which is written as an Ito process. $$dX_t=\mu_t\ dt +\sigma_t\ dB_t$$. If $Y_t$ is defined as a stochastic process as a function of $X_t$, then we can find $dY_t$ ...
3
votes
0answers
29 views

A stochastic process $X$ is a martingale $\iff$ $X$ is driftless.

A Collector's Guide to Martingales: If $X$ is a stochastic process with volatility $\sigma _t$ (that is, $dX_t = \sigma _t dW_t + \mu _t dt$) which satisfies $\mathbb{E}[(\int_{0}^{T} \sigma^2 _s \, ...
3
votes
0answers
30 views

Book for learning stochastic processes for a beginner

I am a social science student interested in learning stochastic processes. The book used in my university is Grimmett and Stirzaker. I tried to study discrete time markov process from it and was able ...
0
votes
1answer
30 views

Do Markov generators form a linear space?

Let $G_1$ and $G_2$ be generators for two distinct continuous-time Markov processes $X^{(1)}$ and $X^{(2)}$ on a common probability space $\Omega$ (with Markov semigroups $S^{(1)}$ and $S^{(2)}$) so ...
0
votes
1answer
33 views

About the expected transitions in Markov Chain

The problem is here: The given answer is here: K = $2+ X_1 + X_2$, where $X_1$ and $X_2$ are independent exponential random variables with parameters $2/3$ and $3/5$. $$ E[K] = 2=2+1/p_1 +1/p_2 = ...
1
vote
0answers
51 views

what does this integral stand for?

i would really appreciate some advice concerning a paper i'm reading: http://pages.stern.nyu.edu/~dbackus/GE_asset_pricing/disasters/Leland%20port%20ins%20JF%2080.pdf on page 586, there is a problem ...
3
votes
2answers
84 views

Crossing of Brownian Motion Sample Paths

I would like to ask for a more rigorous statement and proof of Lemma on page 5 of this paper. In essence, it states that two distinct sample paths of a Brownian motion does not strictly cross (meaning ...
3
votes
0answers
73 views

Finding the right $\sigma$-algebra. Question on uncertainty related to the secretary problem.

I'm working on a problem related to the secretary problem. Let me give a short overview on the topic I research: You are supposed to choose the best item presented to you in a row of n items. Any ...
4
votes
0answers
48 views

Radon-Nikodym on a Process wrt to filtration

Given a probability space $(\Omega,\mathcal{F},P)$. Let $(X_t)_{t\geq0}$ be a stochastic process defined on it with cadlag paths, lets say on $(\mathcal{X},\mathcal{B}(X))$. Let be $\mathcal{F}_{t}$ ...
0
votes
0answers
19 views

Indistinguishable Processes under local Lipschitz Condition

Let $a,b, \rho, \sigma$ be locally Lipschitz functions on $\mathbb{R}^d$, G an open subset of $\mathbb{R}^d$ and assume that on $G$ we have the equalities $a=b$ and $\rho=\sigma$. Let $\xi \in G$ and ...
0
votes
2answers
32 views

Why are Optional Stochastic Processes Important?

I understand to some degree why adapted processes, progressive processes, and predictable processes are important. EDIT: I am referring only to the continuous time case, NOT discrete time. But why do ...
1
vote
1answer
29 views

Finding the mean given the probability

I'm doing some work on branching processes and would like to know where the process becomes extinct. If $X$ is the number of offspring of an individual, then the process goes extinct when $\mathbb{E}[...
3
votes
1answer
27 views

Expected Square Distance from Origin of Random Walk in $\mathbb{Z}^2$

I'm trying to find the expected value of the squared distance from the origin of a simple symmetric random walk in $\mathbb{Z}^2$ at time $n$. So far, I have calculated that if $(X,Y)$ is the ...
0
votes
0answers
22 views

System of SDEs and independence [closed]

I am recently reading a paper that seems to claim the following fact without justification: $Y^1_t, \ldots, Y^n_t$ are stochastic processes defined on $\mathbb{R}$. Let $b: \mathbb{R}^2 \...
2
votes
0answers
27 views

How do linear operators acting on paths of Gaussian processes influence the covariance function?

It is well-known that applying a linear transformation $A$ on an $n$-dimensional centered Gaussian distribution with covariance matrix $\Sigma$ results in another centered Gaussian distribution with ...
2
votes
1answer
53 views

Why Are Semimartingales the Largest Possible Class of Stochastic Integrators?

I am trying to understand why semimartingales are the most general possible class of stochastic integrators. (I was hoping that this question would give me my answer, but it didn't.) I thought at ...
2
votes
1answer
37 views

What is the probability that a given $ n $ event trains match the beginning of a Poisson process?

Here is my question with which I'm confusing myself: Assume that some event times $ \{\tau_i\}_{i \in \mathbb{N}} $ are a point process with rate $ \mu $ such that number of events that occurred ...
5
votes
1answer
42 views

Functions of a random walk and martingales

Let $\xi_1,\xi_2,\ldots$ be a sequence of iid random variables, such that $$\mathbb{P}(\xi_i=1)=p\ne \frac{1}{2},\,\mathbb{P}(\xi_i=-1)=q=1-p.$$ Consider the corresponding random walk $X_n=\xi_1+\xi_2+...
3
votes
0answers
44 views

Convergence of a sequence over supremum

Given a cadlag-process $X_{t}$ with stationary independent increments (Levy process) for which $E\left[\sup_{s\in[0,t]}\left|X_s\right|\right]<+\infty$ for all $t>0$. For $n\in \mathbb{N}$ the ...
0
votes
2answers
29 views

Uncorrelated but not independent uniform distribution

Let $X = (X_1, X_2)$ be uniform distributed on $\{(-1,0), (1,0), (0,-1), (0,1)\}$. First of all I want to show that $X_1$ and $X_2$ are uncorrelated but not independent. Secondly I thought about ...
1
vote
1answer
26 views

Do Optional and Progressive Processes Have Counterparts in Discrete Time?

We know that predictable $\implies$ optional $\implies$ progressively measurable. Source Predictable processes have obvious/simple counterparts in discrete time. Do optional processes and ...
0
votes
0answers
26 views

Distribution of Double Stochastic Integral

Assume that $f(s)$ is a $C^\infty$ univariate function and that $\{ (W_{1,t}, W_{2,t})\}_{t \geq 0}$ is a two-dimensional, correlated Wiener process. Then, does the random variable $X_T \equiv \int\...
10
votes
1answer
120 views

Hard Question in Stochastic processes - variance Martingales

I got some hard challenge to solve and I am looking for a small clue/help. My question goes like this: 10 Englishmen are trying to leave a pub in a rainy weather. They do it in the following ...
3
votes
0answers
33 views

$2D$ random walk stopping time

A $2D$ random walk starts at $(X_0, Y_0) = (k, k)$ where $k>0$ is an integer. At each step $(X_{n+1}, Y_{n+1}) = (X_{n}-1, Y_{n})$ or $(X_{n+1}, Y_{n+1}) = (X_{n}, Y_{n}-1)$ with the same ...
2
votes
1answer
46 views

What is the difference between an adapted process and a predictable process?

As the footnote on page 1 of this document mentions, even most experts in the field of stochastic processes don't seem to know rigorously what the difference is. However, since I don't have any idea ...
0
votes
1answer
32 views

Conditional Probability in Poisson Process

Suppose that $N(t), t\ge 0$ is a Poisson process such the $E[N(9)]=6$. (i) Find the mean and variance of $N(8)$. (ii) Find $P(N(2)\le3)$ (iii) Find $P(N(4)\le5|N(2)\le3)$ - How do I solve this? ...
1
vote
1answer
65 views

Indicator Functions - Can someone check my working?

This is a very easy question but since some of my codes aren't coming out properly I thought I should check my theory to see if everything's okay. Say we have two values $K_{1}$ and $K_{2}$ and that ...
0
votes
1answer
17 views

Riesz theorem and $L^p$ norm in expectation

I am reading a paper that uses the following fact, which claims to be from the Riesz's theorem: For a continuous stochastic process $\{ X_t \}$, let $u_t$ be its density function at each time ...
0
votes
1answer
40 views

To test whether a process is a Martingale (Stochastic calculus)?

If $W_t$ is a standard Brownian motion, I was trying to prove $Y_t = \exp (\int_{0}^{t} s\cdot dW_s)$ is a martingale ! First I started finding $dY_t$ using Ito formula. But I am confused how to ...
1
vote
2answers
55 views

How to prove that the stochastic integral process is gaussian?

I would like to prove that for a $C^1$-function f and a Wiener process W, the integral process defined by $$ Y_t:= \int_0^t f (s)dW_s := f (t)W_t -\int_0^t W_s f'(s)ds $$ Is a centered gaussian ...
2
votes
0answers
14 views

Residual life distribution for renewal process after time T

Suppose we have a renewal process with inter-arrival times $\boldsymbol X=\{X_1, X_2, ...\}$, where $X_i$ are i.i.d variables. Assume that the CDF and PDF for $X_i$ is $F(x)$ and $f(x)$. 1) Let $A_t$ ...
1
vote
1answer
18 views

Is equality of processes stable to multiplication with an independent process.

Assume that all processes to be considered are regular (say cadlag). Assume $X^1$ and $X^2$ are stochastic processes such that $X^1_t = X^2_t$, that $Y^1$ and $Y^2$ are processes such that $Y^1_t=Y^...
0
votes
0answers
26 views

Integrating over random boundary

What are some correct stochastic integral notions or theories which make formal sense of the problem of "integrating a function over the boundary of random domain"?
2
votes
1answer
45 views

Conditional independence of stopping times from i.i.d. stochastic processes

My question is somewhat arbitrary but I was thinking about independence of processes and stopping times. Say that we define two processes $X,Y$ on different probability spaces $(\Omega^i,\mathcal{F}^...
1
vote
1answer
28 views

What is the conditional distribution function of the moment when the first event happens when the total number of event is given?

Suppose books are getting missing from a library as a Poisson process $N(t)$ with intensity $\lambda$. Suppose we know that at the moment $t$, $N(t) = n$. What is the conditional distribution ...
0
votes
0answers
22 views

Are sojourn times independent in non homogeneous poisson process?

We know that in homogeneous Poisson process, sojourn times are independent. But are they still independent if $\lambda (t)$ varies? If so, what is the proof? Thanks the problem I have now is that ...
1
vote
0answers
29 views

Information in Filtrations

Is the “information” kept track of by filtrations the same as information-theoretic “information”? If not, is there some way the two concepts can be reconciled?
2
votes
0answers
37 views

Survival probability of a biased random walker

A random walker moves to $+1$ with probability $p$ and moves to $-1$ with probability $q=1-p$. If he starts at point $m$, what is the probability that he doesn't hit the point zero after $k$ steps, ...
0
votes
1answer
36 views

Lévy-process property

I get a problem that comes up in the construction of the Lévy-Itõ decomposition. For a Lévy-process $X$ there is a independently scattered poisson random measure $N$, such that for each t, and for ...
0
votes
1answer
20 views

Intuition of Doob-Meyer decomposition ( case of totally inaccessible jumps)

I try to understand Theorem 10 on page 107 of Protter's Stochastic integration and differential equations. The proof is really long, and for now, I just want to get an intuition. Here is the theorem: ...
1
vote
1answer
84 views

A Stochastic Integral Inequality

Let $B(t)$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)$ are continuous functions, and $$dr(t) = \mu(t,r(t))dt+\sigma(t,r(t))dB(t).$$ Is there a pair $(\mu,\sigma)$ such that $$\infty&...