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
1
vote
1 answer

1-dimensional diffusion process

Let $(X_t, t ≥ 0)$ be a 1-dimensional diffusion process with generator $Af(x) =\frac{1}{2}a(x)f''(x)+b(x)f'(x), \mathcal{D}(A)=C^2({\mathbb{R}})$ where $b$ and $a=\sigma^2$ are continuous functions of $x$ and $\sigma(x)> 0$ for all $x$. Let…
user9641
  • 41
1
vote
1 answer

Name of a stochastic process

Suppose we have $n>1$ cells that are arranged in a row. Each cell contains a coin. We label the coins uniformly by integers ranging from $1$ to $k$, where $k$ is chosen such that $\ell k = n$ for some parameter $\ell\geq 1$. The initial order of…
Syd Amerikaner
  • 447
1
vote
1 answer

backward stochastic diff. eq.

Let $W_t$ be a standard Brownian motion. Let $T$ be the terminal date, $X_T=x$, and $$ dX_t=f_tdt+B_tdW_t $$ where $f_t$ and $B_t$ (yet to be determined) have to be adapted to the filtration generated by $W$. Assume $x$ is a constant. One possible…
tsm
  • 564
1
vote
0 answers

Can fractional Brownian motion with a Hurst exponent < 0.5 be equivalent to an ornstein-uhlenbeck process?

When a fractional brownian motion has a Hurst exponent < 0.5, it corresponds to a mean reverting process. Are there values of the parameters of a fractional brownian motion and an Ornstein-Uhlenbeck process for which they correspond to the exact…
Simon Ouellette
  • 11
  • 1
1
vote
0 answers

existence of martingale having a given terminal law and deterministic quadratic variation

I wonder whether there exists a continuous-time martingale satisfying the following: let $\mu$ be a centered probability measure whose support is compact (one can assume, if necessary, that support of $\mu$ is a finite set contained in an…
LDS
  • 81
1
vote
1 answer

"strong stationary process" equivalent to a process with "indentically distributed" random variables?

I want to know if a strong stationary process is equivalent to a process with identically distributed RVs. In another words : if A is the set of s.s.s processes and B is the set of processes with identically distributed RVs, is it true that A = B ?…
Afaf
  • 11
1
vote
1 answer

How to show a function belongs to $H^2$

How do you show that $\{\exp(B_t(\omega))\}_{0 \le t \le T} \in H^2$ where $B_t$ is a standard wiener process $H^2=\{f\in L^2(P\times m):f~~\text{adapted}\}$ and $P\times m : {\cal F} \times {\cal B}[0,T] \to \cal R $. I'm mostly just confused on…
ActuariallyImpaired
  • 321
1
vote
0 answers

What is the probablistic distribution of exponential convolution of a Gaussian white noise process?

Say $x(t)$ is a Gaussian white noise process with $\sigma^2$ as variance. Now what is the probabilistic distribution of $Y(t)$? $$ Y(t) = \int_{0}^t x(s) e^{a(t-s)} ds $$ say $a \in \mathbb{R}$. I want to know the PDF. What I know when $a = 0$,…
ArtificiallyIntelligent
  • 733
1
vote
1 answer

Homogeneous Markov Chain

If $\{X_n\}$ is a homogeneous Markov chain, is it true that ${X_{n^2}}$ ($n$ is of the power $2$ not $X_n$) is also a homogeneous Markov chain? And why?
Jennyfer Lahoud
  • 13
1
vote
2 answers

modified stochastic process

Is there any study of stochastic processes where the probability matrix (for a finite state process) is time dependent? For example, probability I go from school to home is higher at night as compared to in the morning where it is lower.
picakhu
  • 4,906
1
vote
1 answer

Wheel of Fortune - Optimal Stopping Time

I am interested in the solution of the following exercise: One spins a wheel with fields marked 1 to 50. One has $N \in\mathbb{N}$ turns. After every turn on can either stop and receive the amount shown in the field or spin once more. The…
Lük
  • 188
1
vote
0 answers

There is no measure such that $\mu(\lbrace f : (f(t_1), ... , f(t_n)) \in A\rbrace) = \mu_{(t_1,...,t_n)}(A)$

Let $C[0,\infty)$ denote the space of continuous functions $f : [0,\infty) \rightarrow \mathbb{R}$. Let $\mathcal{C} [0,\infty)$ denote the $\sigma$-algebra of subsets of $C[0, \infty)$ generated by sets of the form $$ \lbrace f ∈ C[0,\infty) : f(t)…
Rocco van Vreumingen
  • 321
1
vote
1 answer

Birth Process Question

The Yule process is a pure birth process with parameter $\lambda_n = n\beta$. If $X(0) = 1$, then find the probability there are no births during the time interval $(5,8]$. I was thinking of conditioning on $X(5)$ but I was unsure on how to proceed……
icobes
  • 1,109
1
vote
0 answers

How can I find the hazard rate of a time-dependent accelerated failure time model

Suppose I have the following accelerated failure time model: $t = \exp(-\beta t)u$ where the c.d.f of $u$, $F_u(x) = \dfrac{x}{1+x}$. I tried the following way: $u = t\exp(\beta t) = \Psi(t)$, or $t = \Psi^{-1}(u)$ $F_t(x)=Pr(t \leq x) =…
skyindeer
  • 123
1 2 3
…
46 47