Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [stochastic-processes]

For questions about stochastic processes, for example random walks and Brownian motion.

A stochastic process is a collection of random variables representing the evolution of a system of random variables over time. A typical example is a .

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.

16128 questions
2
votes
0 answers

Convergence of the sum of two stochastic processes

I've one question regarding the convergence of the sum of two stochastic processes. Let $(X^n_{t})_t \rightarrow (X_t)_t$ and $(Y^n_{t})_t \rightarrow (Y_t)_t$ for $n \to \infty$ where $\rightarrow$ denotes convergence in distribution in the uniform…
pierreparis1
  • 125
2
votes
2 answers

Question about the Poisson process

A service center consists of two servers, each working at an exponential rate of two services per hour. If customers arrive at a Poisson rate of three per hour, then, assuming a system capacity of at most three customers, What fraction of…
bringingdownthegauss
  • 530
2
votes
1 answer

What is Transition intensity?

What do we eaxctly mean by the term transition intensity and how is it different from transition probability? Transition Intensity = lim dt-0 d/dt (dtQx+t/dt) where dtQx+t= P(person in the dead state at age x+t+dt/given in the alive state at age…
Aman Sanganeria
  • 23
2
votes
1 answer

Higher powers of transition matrix of the Markov chain $X_n =\max\{Z_1, ..., Z_n\}$

I have a question I am working on: Suppose $\{Z_n, n ≥ 1\}$ are iid outcomes of successive throws of a fair die. Then, let $X_n = \max\{Z_1, ..., Z_n\}$. It is easy to show that $X_n$ is Markov. I am trying to find the higher powers of $P$,…
user123276
  • 3,445
2
votes
1 answer

Locally FV processes are FV processes

Does anyone see why any process which is of Locally (see definition 1) Finite Variation has to be of Finite Variation ? Best regards
TheBridge
  • 5,721
2
votes
0 answers

Question about Lundberg Theorem

Hope that my English is readable after all... In the textbook it is written: Consider the process $U_t=u+ct-\sum\limits_{i=1}^{N(t)}y_i$ where $y_i\geq 0\,$ iid, $(N(t))_{t\geq 0}$ is a Poisson proces with intensity $\lambda$ independent from…
7O'clock
  • 90
2
votes
1 answer

Can we compute the probability that a Brownian motion hits a quadratic?

Let $A$ be the event that there is some $t$ such that $B_t=1+t^2$, where $B$ is a Brownian mation. Is there any way to compute the probability of $A$, or to approximate it well? I ask, because we can calculate the probability that $B$ hits a line…
Ben Derrett
  • 4,592
2
votes
1 answer

limiting distribution of $Y_t$ in the mean-reverting Ornstein-Uhlenbeck process

The mean reverting Ornstein-Uhlenbeck process is of the equation: $$dX_t=(a-cX_t) \, dt+\sigma \, dW_t$$ If we are told that both $a$ and $c$ are larger than $0$, what then is the limiting distribution of $X_t$ as $t$ tends to infinity? Obviously…
user146378
  • 23
2
votes
3 answers

Throwing dice a randomly determined number of times

Let $X$ be the number that shows up when rolling a die. Now throw another dice $X$ times $(Y_1, ..., Y_X)$ and calculate the sum $Z = \sum_{k=1}^X Y_k$. What kind of Stochastic Process is this? How do you calculate mean value and variance of $Z$?
wnrph
  • 238
2
votes
2 answers

what's the relationship between a.s. continuous and m.s. continuous?

suppose that X(t) is a s.p. on T with $EX(t)^2<+\infty$. we give two kinds of continuity of X(t). X(t) is continuous a.s. X(t) is m.s. continuous, i.e. $\lim\limits_{\triangle t \rightarrow 0}E(X(t+\triangle t)-X(t))^2=0$. Then, what's the…
Jim
  • 505
2
votes
1 answer

Number of crossings in a two-dimensional random walk

Given the standard two-dimensional random walk (up, down, left, or right 1 unit with equal probability), what is the expected number of crossings of the origin after $x$ steps? It strikes me as slightly unnatural to count each crossing separately…
Charles
  • 32,122
2
votes
0 answers

$N$-point distribution functions of Brownian local time

What is the most explicit formula for $N$-point distribution functions of the local (or occupation, sojourn) time of Brownian motion (Wiener process) with exponentially distributed time duration?
Vaclav
  • 21
2
votes
1 answer

Variance of a Gaussian process

I want to prove that if $Y_t$, $0\leq t\leq 1$ is a zero mean Gaussian process such that there exist $a,b$ with $$\operatorname{Var}(Y_t-Y_s) \leq a|t-s|^b, \;\; s,t\in[0,1]$$ then there exists a version of $Y$ with continuous paths on $[0,1]$. It…
Nagato
  • 21
2
votes
3 answers

Stochastic in finance

I need of a undergraduate guide level to study stochastic process with finances. Starting from a review of probability theory. Eg books, papers or posts. I'll apreciate some help.
cyb3rpunk
  • 21
2
votes
1 answer

Limit of a left continuous process

Suppose we are given a left continuous process $X=(X_t)_{t\ge 0}$ and define $$Y^n_t=n\int^t_{t-\frac{1}{n}}\mathbf1_{\{|X_{s\vee 0}|\le n\}}X_{s\vee 0}ds$$ Why does it hold that $\lim_nY^n\to X$? It should follow from the left-continuity. Clearly…
math
  • 4,347
1 2 3
…
46 47