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

2
votes
0answers
20 views

Changing domain of the CGF of a stochastic process

Given a stochastic cadlag process $(X_t)_{t\geq 0}$. Define $A_{t}$ as the domain of the cumulant generating function by $f_t(s):=\log E(e^{sX_t})$, for which this expression is welldefined. I search ...
4
votes
1answer
46 views

Ito's formula and Taylor expansions for jumps processes.

Consider some model $$ dX_t = \mu d t + \sigma dW_t $$ where $\mu, \sigma$ are some constants. Now let $f \in C^{1,2}$ and consider $$ Y_t = f(t,X_t). $$ Say we (informally) consider a second order ...
3
votes
0answers
27 views

Stochastic process is brownian motion by Levy's characterization

I would like to know if $B_t=W_t-\int_0^t \frac{W_u}{u}du$ is a brownian motion. I know that $W_t$ is a brownian motion. For that i would like to use Levy's characterization, so I have to show that ...
1
vote
1answer
30 views

If M is a transition matrix, 1 is one of its eigenvalues?

If $A$ is an $n\times n$ Markov, Transition or Stochastic matrix (i.e. $A_{ij} \geq 0$ and $\sum_{i=1}^n A_{ij}=1$) one of its eigenvalues is equal to 1?
0
votes
0answers
18 views

Expectation of product of Geometric Brownian Motion

If I have two GBMs where $s < t$ $A_t = A_0 \exp((a-\frac{1}{2}\sigma_A^2)t+\sigma_AW_t^A) $ $B_s = B_0 \exp((a-\frac{1}{2}\sigma_B^2)s+\sigma_BW_t^B) $ How would I calculate $E(A_tB_s)$ My ...
0
votes
0answers
41 views

Reformulate a SPDE parameterized by space and time as an SDE parameterized by time (as it is possible for PDEs)

Let $d\in\mathbb N$ $\mathcal V_t\subseteq\mathbb R^d$ be a bounded domain for $t\ge 0$ $\Phi_t:\mathcal V_0\to\mathcal V_t$ be bijective for $t\ge 0$ with $$\Phi(\;\cdot\;,x_0)\in C^1\left(\mathbb ...
2
votes
2answers
30 views

Markovian Gaussian stationary process with continuous paths

Could you, please, help me figure out the following problem. We call a stationary Gaussian process $\xi_t$ (with continuous paths) an Ornstein-Uhlenbeck process if its correlation function ...
0
votes
1answer
23 views

Probability of exit from compact set

I have a continuous real valued diffusion $\{ X_t \}_{t\ge0 }$ that is contained in a compact set $[a,b] $of $\mathbb{R}$, where $a > 0$ and. Define the stopping times \begin{equation} \tau_c=\inf ...
2
votes
0answers
36 views

Gaussian process via RKHS construction: joint measurability comes for free?

Billingsley's "Probability and Measure" (and other books) show the joint measurability of the Brownian motion using the continuity of paths. Makes me wonder if we can say the joint measurability ...
3
votes
1answer
19 views

Monotone property of transition density of rotational $\alpha$-stable process

For a Brownian motion $B_t$ in $\mathbb R^d$, the transition density of $B_t$ is the normal distribution $$P_x[B_t\in dy]=(2\pi t)^{-d/2}e^{-\frac{|x-y|^2}{2t}}dy$$ and obviously the density is ...
3
votes
1answer
29 views

convergence in distribution of exponential of a brownian motion

If $(B_t)_{t≥0}$ is a standard Brownian motion, show that, as $t \to \infty$, $$ \left(\int_0^t e^{B_s} \, ds\right)^{1/\sqrt{t}} \text{ converges in distribution to} \ e^{M_1}, $$ where $M_1 = ...
3
votes
0answers
60 views

Strong Markov property proof

Let $X$ be a Markov chain with state space $\mathcal{S}$ and denote $\mathbb{N} := \{0,1, \cdots\}$. I need to show that for any stopping time $\tau < \infty$ and any bounded measurable function ...
2
votes
1answer
25 views

Geometric Brownian motion with exponential of sum of iid's

Glasserman's "Monte Carlo Methods in Financial Engineering" on p. 265 states that the geometric Brownian motion can be modelled with : $$S(t_n)=S(0) \exp(\sum_{i=1}^n X_i)$$ where $X_i$ are iid. I ...
1
vote
0answers
20 views

Kolmogorov forward and Backwards equation interpretation

Let $\lambda_i$ be the sojourn rate of state i, $q_{ij}$ be the transition rate form i to j, and $p_{ij}$ be the transition probability from i to j. The Kolmogorov Forward and backwards equation are ...
0
votes
1answer
17 views

Why does Euler-Maruyama method use a square root of the time step

Euler-Maruyama method is supposed to be an extension of the Euler method for ODE, but applied to SDE. This means that if we have an equation: $$ dY_t = Y_t dW_t $$ where $W_t$ is the Wiener process, ...
0
votes
0answers
25 views

Uniqueness of Predictable Quadratic Covariation

In order to prove that $\langle M,N \rangle$ is the only process which is continuous and has bounded variation such that \begin{align} M_tN_t - \langle M,N \rangle_t \end{align} is a continuous ...
0
votes
1answer
40 views

A question about Bernoulli process ( maybe conceptual)?

Question: Al performs an experiment comprising a series of independent trials. On each trial, he simultaneously flips a set of three fair coins. Whenever all three coins land on the same side in any ...
0
votes
1answer
27 views

probability of splitted exponentially distributed random variables

Let $X$ be a exponentially distributed random variable(time interval) with mean of $u$ And $Y$ be a exponentially distributed random variable (time interval)with mean of $\lambda$ And $s$ be a ...
0
votes
2answers
24 views

A relation between first passage time and occupation time

Let's think about a discrete time Markov chain $X_t$ with only one recurrent state. Let $T$ be the random variable that is the number of steps taken from a given state $i$ to the recurrent state (ie. ...
0
votes
2answers
40 views

Simple Random Walk - Why are these two events the same?

Let $S = (S_n)_{n \geq 1}$ be a simple random walk. We denote the hitting time of a point $b$ by $\tau_b = \min \{i \geq 1 : S_i \geq b\}$. My text says that the events $\displaystyle\{\max_{k \leq ...
2
votes
0answers
22 views

Hypothesis Testing on Renewal Processes

We have time $[0,T]$ to observe a renewal point process, where the inter-renewal timings are i.i.d, and then decide whether the observation is according to a renewal process in which the pdf of ...
1
vote
1answer
61 views

Itô-formula proof, remainder term.

I have a question about the proof a a certain version of the Itô-formula. First the author defines an Itô-process and states the formula: My question is in regarding the proof. The proof uses ...
1
vote
1answer
24 views

Are queues CTMC?

The $M/M/1$ queue have all the properties of the countable state continuous time markov chain. Is any general queue also a countable state CTMC?
0
votes
0answers
24 views

Deduce stochastic differential equation

Let $X$ be a stochastic process with $dX_t = \alpha X_t dt + \sigma X_t dW_t$ and $Y$ a stochastic process with $dY_t = \gamma Y_t dt + \delta Y_t dV_t$, where $W$ and $V$ are independent ...
0
votes
0answers
14 views

stochastic experiments like galton board?

do you know some other stochastic/statistic experiments like galton board? I'm looking for something that could be build for learners or people who are interested in mathematics; some sort of ...
3
votes
0answers
16 views

How can an Ornstein-Uhlenbeck process be shown to be continuous/diffusion?

Given the Ornstein-Uhlenbeck transition pdf (where $t_2\geq t_1 \geq 0$ and $x_2 \geq x_1 \geq 0$ and $\gamma >0$): $$p(x_2,t_2;x_1,t_1) = \frac{1}{\sqrt{2\pi(1-e^{-2\gamma(t_2-t_1)})}}\exp \left( ...
2
votes
0answers
22 views

Non-uniform convolution with discrete wavelet transform

I understand that if you have a circular N-dimensional convolution matrix, it can be diagonalized with the Fourier transform of the convolution operator. This makes it easy to calculate the density of ...
0
votes
1answer
34 views

How do you find the probability of a brownian motion?

If $B(t)$ is a brownian motion what do these two questions mean? 1. What is the probability of $B(2)$ 2. What is the probability of $B(2) \gt B(1)$ I know this is also called a Wiener Process and ...
1
vote
1answer
28 views

Convergence of sequence of random variables 2

If I know $\lim\limits_{n \to \infty} \mathbb{P}(X_n<c-\gamma)=0$ for all $\gamma>0$, how can I prove supremum of all reals $\alpha$ for which $\lim\limits_{n \to \infty} \mathbb{P}(X_n\leq ...
0
votes
0answers
12 views

Comparing two hitting times of Bessel process

Suppose $X$ is a Bessel process of dimension $1 < d \le 3$ with $X_0 = 0$. Then $X$ satisfies the SDE $ dX_t = \frac{d - 1}{2X_t} dt + d W_t$ for some Brownian motion $W_t$. Let $a > 0$. Let ...
0
votes
1answer
27 views

Is this martingale constant 0?

I have a martingale X where $X_0 = 0$ a.s. And for each $\omega$, the path $f(t)=X_t(\omega)$ is of bounded variation in the classical sense. That ...
2
votes
2answers
55 views

References on probability theory, stochastic processes, Monte Carlo and convex optimisation, with similar writing style to Terence Tao

I learned a lot from prof Tao's notes and books because unlike many authors, he seems to prefer writing more words, explanations and intuitions rather than just mathematical formulae. His approach is ...
0
votes
1answer
15 views

Can Wiener process be axiomized without normal increments

A common characaterization of Wiener's process is the following which I took directly from Wikipedia: $W_0 = 0$ a.s. $W$ has independent increments: $W_{t+u} - W_t$ is independent of $σ(W_s : s ≤ ...
1
vote
0answers
22 views

Intuitive difference between Laplace functional of Poisson Point Process (PPP) and independently marked PPP

The Laplace functional of the Poisson Point Process (PPP) $\Phi$ with intensity measure $\Lambda$ on $\mathbb{R}^d$ for non-negative function $f(x)$ is: $$ \mathcal{L}_\Phi(f) = ...
0
votes
1answer
29 views

What is the proof of this equation? (Stochastic process)

$$\mathbb{E}[g(X)h(Y)]=\mathbb{E}[h(Y)\,\mathbb{E}[g(X)|Y]]$$ I am reading the book "An Introduction to Stochastic Modeling". This equation appears a lot but I can not see why. Can anyone please ...
1
vote
0answers
19 views

Pure Birth Question. Find the probability that the population at time $t$ is an odd # given it starts at $0$.

Here is the question. Consider a pure birth process $\{X(t) : t ≥ 0\}$ with birth parameters $\lambda_{2n} = α>0$ and $\lambda_{2n+1} =β>0$ for $n∈N$. Compute $Pr\{X(t) \text{ is odd } \mid ...
0
votes
0answers
25 views

Finding the stationary distribution of specific homogeneous Markov chain and determining its uniqueness

I am presented with $P =\begin{bmatrix} 0.5 & \alpha & \beta \\ \alpha & \beta & 0.5 \\ \beta & 0.5 & \alpha \end{bmatrix}$ where $\alpha+\beta=0.5$ and $\alpha,\beta \in ...
2
votes
0answers
45 views

Problems with a Black-Scholes modified equation

I haven't really studied much financial mathematics until about 2 months ago so I'm quite new to this stuff, so I'm sorry if this is a trivial question. At the moment I'm trying to work out what the ...
0
votes
0answers
14 views

Convolution to establish Gaussian process

A Gaussian process $z(s)$ can be established by convolving a gaussian white noise process $x(s)$ with a smoothing kernel $k(s)$ http://ftp.stat.duke.edu/WorkingPapers/01-03.pdf $$\\z(s)=\int_{S}^{} ...
1
vote
0answers
18 views

Problem with compound Poisson process

Let $X_k$ for $k=1,2,...$ be a sequence of i.i.d. random variables with $\mu_k=0$ and $\sigma_k^2=1$ for all $k$. Consider de random process $$S(t)=\sum\limits_{k=1}^{N(t)}X_k $$ where $N(t)$ is a ...
1
vote
2answers
22 views

Function evaluated in Brownian motion vanishes implies that the function itself vanishes?

tl;dr, here's my question: Question. Let $f(t,x)$ be a measurable function such that $f(t,B_t)=0$ almost everywhere on $[0,T]\times\Omega$ for a Brownian motion $B_t$. Does this imply that ...
0
votes
0answers
10 views

Identification of Infinite Dimensional State in Hidden Markov Model

Consider a hidden markov model (HMM) where the state, $X_t(\alpha)$, is a stochastic distribution over $\alpha \in \mathbb{R}_+$ and one observes a signal $Y_t$, which is simply a moment of this ...
0
votes
0answers
17 views

Can you identify this stochastic process?

So I run into this problem the other day and I cannot even think of the keywords I need to use to look it up. For the discrete random variable $X$ we have: $P_{\Delta X(t)} = F\big(X(t-1), ...
1
vote
0answers
11 views

Describe the law of a Bessel process conditioned on hitting $b$ before $0$

We are given the Bessel process SDE $$dX_t=\frac{\delta -1}{2X_t}dt+ dB_t, X_0>0.$$ Where $B_t$ is a standard Brownian motion, at least until $X_0=0$. We need to solve four problems: Show that ...
0
votes
0answers
34 views

Question regarding regular stochastic matrix

We say that a stochastic matrix is regular iff $\exists n\in \mathbb N$ such that $p_{ij}(n)>0$ for all states $i,j$ How many powers of a matrix do we need to compute at most in order to verify ...
1
vote
0answers
8 views

Distribution of an autoregressive process

Say that we are given a AR process. Also, lets assume that the residuals of the process come form a distribution $P_R$ which, while known to us, is not necessarily normal. Can I derive the ...
0
votes
1answer
32 views

What is the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$?

I am trying to find the expectation of $\int_0^t \sqrt{s+B_s^2}dB_s$, but am unable to use Ito's Formula because of the nasty integral. Is there another solution I am missing? Thanks!
4
votes
0answers
33 views

Stochastic domination

Suppose we have two probability measures on a space $X$, $\mu$ and $\nu$, such that $\nu$ stochastically dominates $\mu$, i.e.there exist a coupling of $\mu$ and $\nu$ on the product space $X \times ...
0
votes
0answers
56 views

Why do we always consider real-valued $f$ in the Itō formula to find an expression for $f(t,X_t)$

The Itō formula (see Da Prato, Theorem 4.32) yields an expression for $f(t,X_t)$ where $${\rm d}X_t=\phi\;{\rm d}t+\Phi\;{\rm d}W_t\;,\;\;\;X_0=\xi\;.\tag 1$$ Even when $X$ takes values in a Hilbert ...
1
vote
0answers
9 views

Existence of first passage time density for time-inhomogeneous diffusion

Let $X$ be a time-inhomogeneous diffusion process in $\mathbb{R}^d$: $$dX_t=b(t,X_t)dt+\Sigma(t,X_t)dB_t,$$ where $\Sigma_{d\times d}$ is uniformly elliptic, and coefficients are such that the above ...