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
33 views

A problem about the supremum of countable stopping times

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with a countable filtration $F=\left\{ \mathcal{F}_0,\mathcal{F}_1,\cdots,\mathcal{F}_n,\cdots \right\}$. $\left\{ T_n \...
2
votes
1answer
45 views

a.s. convergence and conditional expectation

I have a stochastic process $X(t,\omega)$ which is a martingale. It is showed that there exists a r.v. $X(\infty)$ such that in $L^1(\Omega)$, $\lim_{t \rightarrow \infty}X(t) = X(\infty)$. In my ...
1
vote
0answers
64 views

Ito's Formula applied to a weird equation…

EDIT: One thing I forgot to mention before is that this is all under the $\mathbb{Q}$ measure in case that changes anything I was just wondering if someone could explain how to solve this problem. I ...
3
votes
0answers
53 views

Hitting times in two-dimensional case: expectation of Brownian motion at a hitting time

Consider two Brownian motions $$X_{1t}=\mu t+\sigma_1B_{1t}$$ and $$X_{2t}=\mu t+\sigma_2B_{2t}.$$ Here $B_{1t}$ and $B_{2t}$ are uncorrelated. Let $\tau_1$ and $\tau_2$ be the stopping times: \begin{...
1
vote
0answers
16 views

The finite-dimensional distribution of a stochastic process

Let $K(s,t)$ be a real function over $T\times T$, where $T$ is arbitrary. $K$ has two properties: $K$ is symmetric ($K(s,t)=K(t,s)$). $K$ is nonnegative-definite ($\sum_{i,j=1}^k K(t_i,t_j)x_ix_j\...
5
votes
1answer
82 views

Modified gambler's ruin problem: quit when going bankruptcy or losing $k$ dollars in all

In each round, the gambler either wins and earns 1 dollar, or loses 1 dollar. The winning probability in each round is $p<1/2$. The gambler initially has $a$ dollars. He quits the game when he has ...
-1
votes
1answer
84 views

A proposition about a stochastically continuous process with independent increments.

Let $\left(\Omega\ ,\mathcal{F}\ ,\mathbb{P} \right)$ be a probability space with filtration $F=\left\{ \mathcal{F_t} \right\}_{t\geqslant0}$ , and $X=\left\{ X_t \right\}_{t\geqslant0}$ is an ...
0
votes
0answers
21 views

Modified Bernoulli trials

Consider the modified version of i.i.d. Bernoulli trials where the first success in a success run is converted into a failure, e.g. 'FFSSSSFFSSS' $\rightarrow$ 'FFFSSSFFFSS'. Let the original success ...
1
vote
1answer
39 views

Solve Kolmogorov differential equations for birth-death process with constant rates

I need to solve the Kolmogorov forward equations for a birth-death process whose birth/death rates $\lambda_k,k=0,\ldots$ and $\mu_k,k=1,\ldots $ are constant, i.e., $\lambda_k=\lambda$ and $\mu_k=\mu$...
2
votes
0answers
29 views

Applying Ito formula to Ito process

I would like to simplify the expression $\left(\phi(s_{1})\cdot(X_{s_{1}}-X_{s_{2}})+\phi(s_{2})\cdot(X_{s_{2}}-X_{s_{3}})+\ldots+\phi(s_{n-1})\cdot(X_{s_{n-1}}-X_{s_{n}})\right)^{2}$ where $X_{t}$ ...
1
vote
0answers
14 views

“Return probability” to origin of a variant of the random walk.

Let $\{\epsilon_t\}_{t\ge0}$ be an iid sequence of random variables and let $\lambda>1$. I am interested in the following process: Let $X_0 = 0$ and $$ X_{t+1} = \lambda(X_t+\epsilon_t). $$ This ...
1
vote
0answers
19 views

M/G/1 queuing system with two arrivals

I have a queuing system with two independent Poisson arrivals with rates $\lambda_1$ and $\lambda_2$. But, the service time for each arrival is different. Suppose f_1(s) and f_2(s) are the pdf of ...
1
vote
0answers
32 views

ergodic theorem for expectation of positive recurrent diffusion

Suppose $X_t$ is a positive recurrent diffusion on $\mathbb{R}$ with invariant probability measure $\mu$. There is an ergodic theorem (see V.53. in Rogers & Williams volume II) that states $$\lim_{...
3
votes
2answers
42 views

Nonuniqueness of Stochastic Differential Equation

Let $B_t$ be the standard Brownian motion, $\mu(t,x)$ and $\sigma(t,x)\ne 0$ are real valued continuous functions where $|\mu(t,x)|+|\sigma(t,x)|$ is NOT Lipschitz continuous, and $$dX_t = \mu(t,X(t)...
0
votes
0answers
21 views

Data transmission process PDF

Given the quasi-defined data transmission random process: $X(t) =\sum_{n=-\infty}^{+\infty} a_n \pi_T(t - nT)$ where $a_n$ are statistically independent RVs that can either assume the value 0 or 1 ...
3
votes
1answer
40 views

Minimal value of probability according to the difference of a Levy-process

Can we conclude for a Levy-Process, that for all $\epsilon>0$ it holds that $\min_{s\in [0,t]} \mathbb P\left(\left|X_t-X_s\right|\leq \epsilon\right)>0$? Stochastic continuity doesn't seem to ...
3
votes
1answer
118 views

Distributional equality

Let $(W_t)_{t\geq0}$ be a standard Brownian motion. I have to show that the following equality holds in distribution. Does someone has a good hint to show this? $\sup_{t \geq 0}( |W_t| -t) = \sup_{t \...
2
votes
1answer
29 views

Showing a “signed Markov transition density” will lead to a trivial measure on path space.

Let for all $t>0$, $x\mapsto p(t,x)$ be a Schwartz function on $\mathbb R$, satisfying $\int_{\mathbb R}p(t,x)\mathrm dx=1$ and $\int_\mathbb{R}|p(t,x)|\mathrm dx\equiv C>1$ for all $t>0$ (so ...
2
votes
0answers
29 views

Gaussian volterra process. Conditional distribution?

Asssuming a probability space $(\Omega,(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ such that $(\mathcal{F}_t)_{t\geq 0}$ is generated by a Brownian motion $W_t$. We assume that $s>0$ is fixed and $t\in[...
0
votes
0answers
19 views

If $(W_t)_{t\ge 0}$ is a $L^2(D)$-valued Wiener process, then $W_t(x)$ is normally distributed

Let $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $D\subseteq\mathbb R^d$ be a domain $U:=L^2(D)$ and $\langle\;\cdot\;,\;\cdot\;\rangle:=\langle\;\cdot\;,\;\cdot\;\rangle_U$...
0
votes
1answer
22 views

Relationship Between $\mathbb{E}$(time) and $\mathbb{E}$(Repetition)

Consider aa Stochastic Process with Expected value of time of occurring =T (less than infinity). Can we deduce that Expected value of number of occurrences until time T is equal to 1?? If not, in ...
1
vote
0answers
12 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
1
vote
0answers
26 views

Return probability of a SRW in an even number of steps

I am looking for some references for the following problem. Consider a graph $G$ and a simple continuous time random walk $(X_t)_{t\geqslant 0}$ on this graph. Consider the family of events $(e_t)...
-2
votes
1answer
57 views

Find the variance of W when given $W = x + 2y + 3z$. [closed]

x,y,z are random numbers given w = x + 2y + 3z. also given that the mean of x,y,z= 1,8,0 respectively. what is the mean of the random number w ? Assuming the Standard deviation of the random numbers x,...
2
votes
1answer
120 views

Calculation of $\ln\left( \frac{S_{1}(t)}{S_{2}(t)}\right)$ where $S$ are stocks

Assume we have a probability space $(\Omega,\mathcal{F},\mathbb{P})$ where $\mathcal{F} =(\mathcal{F}_t)_{0 \leq t \leq \tau}$ is a Filtration of an incomplete finance market with stocks $S_j(t)$ for $...
3
votes
0answers
23 views

Doob Meyer decomposition in an exercize

I have to find the Doob Meyer decomposition for the following process: $Y_t=e^{(1+B_t^2)}$ I think that the method is to derive with the Ito's formula the process and I've obtained: $dY_t=2B_te^{(...
1
vote
1answer
49 views

prove that Doléans-Dade exponential is a local martingale

I want to prove that $Z_t$ the Doléans-Dade exponential is a local martingale i.e. that there exists a stopping time $\tau_n$ tending to infinity such that the stopping process $\mathbb{1}_{\tau_n>...
0
votes
1answer
24 views

Explain the orderliness of Poisson process

For an Orderly Poisson Process, events occur at distinct points and not simultaneously. However, the reverse is not necessarily true, i.e, even if the events occur at distinct points, the process may ...
1
vote
0answers
41 views

Understanding an equation

I am trying to understand an equation from the paper "Dynamic Model for generating Synthetic ECG signal" (http://web.mit.edu/~gari/www/papers/ieeetbe50p289.pdf). The equation is: $$S(f) = \frac{\...
1
vote
0answers
34 views

Differences between additive processes and Lévy processes

A real valued stochastic process $\left\{ X_{t}:\ t\in\mathbb{R}^{+}\right\} $ is termed additive if $\forall n\in\mathbb{N}$, $0\leq t_{0}<t_{1}<...<t_{n}<+\infty$, $X_{t_{0}},X_{t_{1}}-...
0
votes
1answer
91 views

Weak $L_1$ convergence

Given a sequence $Y_{un}$, where $Y_{1n},Y_{2n},\ldots$ have the same domain. Assume for every $u\in \mathbb{N}$ we have $e^{itY_{un}}\rightarrow \mathbb{E}[e^{it M}]$ weakly in $L_1$ as $n\rightarrow ...
0
votes
1answer
21 views

Optimal average utility of the processing network needed

In "Utility Optimal Scheduling in Processing Networks" by Michael J. Neely et al an example of processing network is provided. There are three queues ($q_1,q_2,q_3$) in the network and two processors (...
1
vote
0answers
23 views

Role of alpha-stability for subordinators

A Lévy process $\left\{ X_{t}\right\} $with values in $\mathbb{R}^{+}$ is termed a subordinator if it is a.s. increasing as a function of $t$, i.e. the map $t\mapsto X_{t}(\omega)$ is increasing for ...
1
vote
1answer
51 views

What is the uncertainty of a discrete sum given the uncertainty of an individual element?

I have a measurement $$X=\sum_{i=1}^nX_i,$$ and I am interested to know standard deviation $\sigma_X^2$ of measurement $X$, assuming I know $\sigma_i^2$, the standard deviation of all measurements $...
1
vote
1answer
27 views

For a one-dimensional Brownian motion $B_t$ $Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}.$

A one-dimensional Brownian motion $B_t$ has exponential moments of all orders, i.e. $$Ee^{\zeta B_t}=e^{t\zeta ^2/2}\; \text{for all} \; \zeta \in \mathbb{C}. (2.6)$$ This is given as a corollary to ...
1
vote
0answers
42 views

Approximating Geometric Brownian Motion numerically

I am trying to generate a numerical solution to the SDE for Geometric Brownian Motion. The stochastic process is given by $S_t = \exp(\sigma W_t + \mu t)$, and by Ito's lemma, we have that the SDE is ...
1
vote
1answer
54 views

Stationary solution of a Fokker-Planck equation

I have a question that's driving me crazy. I have a Fokker-Planck equation $$\frac{\partial P}{\partial t}=x\frac{\partial P}{\partial x}+D\frac{\partial^2 P}{\partial x^2}$$ I look for the ...
1
vote
1answer
21 views

Deriving the Power Spectral Density of a Maximum Entropy Process

In Elements of Information Theory, Chapter 12, Section 6 Burg's Theorem is derived: Presented with the first $p$ values of the autocovariance function $R(k) = E[X_i X_{i+k}]$ a stochastic process ...
2
votes
0answers
38 views

Fractional powers of Markov generators

Let $H$ be the generator of a symmetric Markov semigroup on $L^2(\mathbb{R}^n).$ Why the fractional power $H^\alpha$ (defined on a proper domain) with $0 < \alpha < 1$ turn out to be the ...
2
votes
0answers
73 views
+100

How can we prove that $\langle\int_0^t\Phi_s{\rm d}W_s,x\rangle_H=\sum_{n\in\mathbb N}\int_0^t\langle\sqrt{λ_n}\Phi_se_n,x\rangle_H{\rm d}B_s^{(n)}$?

Let$^1$ $U$ and $H$ be separable $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U,H)$ be nonnegative and symmetric operator on $U$ with finite trace $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $...
2
votes
1answer
77 views

The jumping times of a càdlàg process are stopping times.

Protter first proves this theorem: Let $X$ be an adapted càdlàg stochastic process, and let $A$ be a closed set. Then the random variable: $T(\omega)=\inf\{t: X_t(\omega)\in A \text{ or } ...
1
vote
0answers
17 views

First time of passage, discrete random walk with disjoint absorbing regions

I have a sum $T^i$ of zero/one Bern$(p)$ random variables $T_i$ and multiple disjoint absorbing regions, i.e. the absorbing region is a union of disjoint, closed sets: $$T^i \in \bigcup_{u \in \...
2
votes
1answer
24 views

A question about Poisson process?( Merged process)

Question: Alice shows up at time zero and spends her time exclusively in typing emails. The times that her emails are sent are a Poisson process with rate $λ_A$ per hour. And Bob just finished ...
3
votes
1answer
32 views

Show that a cadlag adapted martingale is a local martingale (help using DCT to show uniform integrability)

EDIT 2: With the correct definition, I think I have a proof. Want to show $\lim_{M\to\infty} \sup_t E[|X_{t\wedge n}|; |X_{t\wedge n}|\ge M]=0$. Fix $n$. Note that $\sup_t E[|X_{t\wedge n}; |X_{t\...
0
votes
1answer
64 views

Law of large numbers limit dependend on a second variable. What happens when both limits are taken at once?

The question I have is as follows. I have a i.i.d. sequence of random variables $(X^\alpha_n)_{n \in \mathbb{N}}$ with a expectation $\mathbb{E}X^\alpha$ which depend on a Markov process with a scaled ...
2
votes
0answers
74 views

Radon-Nikodym with respect to Stochastic Measure?

Question This question is now concerning stochastic processes. Let $(X_t)_{t\geq0}$ be defined on the probability space $(\Omega,\mathcal{F},P)$ with $\mathcal{F}_t=\sigma(X_s:s\leq t)$. Assume that ...
2
votes
0answers
117 views

Can Local Martingales be characterized only using their FV process and BM?

Prove or Disprove: A process $(X_t)_{t \ge 0}$ is a (continuous) local martingale if and only if it can be represented in the form: $$\int_0^t \xi dB = \large B_{\int_0^t \xi_s^2 ds} $$ where the ...
0
votes
0answers
16 views

Coupling a “partially” stationary process?

Take the stationary process $X$ on $\{0,1\}$ with distribution $\pi=(\pi_0,\pi_1).$ Then introduce the rates: $$ \begin{aligned} 0\rightarrow2 & \quad \text{ at rate } \quad \gamma_{02} \\ 1\...
2
votes
2answers
51 views

Simulating a Stochastic Integral of OU process

The stochastic integral I want to simulate is $$\int_{0}^{1}J_c(s)dJ_c(s)$$ where $J_c(s) = \int_{0}^{s}e^{-c(s-r)}dB(r)$, is an OU process. I simulate the data using Matlab and the sample codes are ...
0
votes
0answers
10 views

Composition of limits of functions | Switching limits of function

I have a question which I am having some trouble with. I have a double indexed sequence of stochastic processes (martingales in fact), denoted $X_{m,n}(t)$. Now I can prove that $\underset{m \...