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
0
votes
1 answer

Why is there "markov property" in proving the renewal equation for a renewal process?

When proving the renewal equation for a renewal process in Wikipedia The renewal function satisfies $$ m(t) = F_S(t) + \int_0^t m(t-s) f_S(s)\, ds $$ where $F_S$ is the cumulative distribution function of $S_1$ and $f_S$ is the…
Tim
  • 47,382
0
votes
1 answer

Differences between a Markov jump process and a continuous-time discrete-state Markov process?

What are the difference and relation between a Markov jump process and a continuous-time discrete-state Markov process? By "a continuous-time discrete-state Markov process", I understand it same as a continuous-time Markov chain. Is a "jump process"…
Tim
  • 47,382
0
votes
4 answers

Using integration by parts to solve

This question has been asked in a similar format, but not that I've found exactly the same as mine. It may be that they are the same, but I don't understand a fundamental part so would appreciate some guidance. I am trying to solve…
MathsIsFun
  • 183
0
votes
1 answer

Extinction probability not starting at 1

Consider a branching process where $X_n$ represents the number of individuals in generation $n$. Suppose that you know the extinction probability when $X_0 =1$: $$\alpha=P(extinction|X_0=1)$$ Could you please help me prove that…
Raphaël
  • 159
0
votes
0 answers

An inequality regarding Poisson process

Let $N$ be a Poisson process with intensity $\lambda$, for $0<\sigma<\tau$, we have $$E[\sup_{\sigma\le t\le \tau}(N_t/t-\lambda)^2]\le 4\tau\lambda/\sigma^2$$ I think this is by Doob’s inequality for a martingale, but it works only for a positive…
Apocalypse
  • 888
0
votes
1 answer

Difference of two Poisson processes with same parameter

If I have two Poisson processes, $X$ and $Y$, each with rate $\lambda$, then what is the rate of $Z$ where $Z=X-Y$. Is it $2 \lambda$? and would this differ if $X$ and $Y$ had different rates? Thank you.
Kabamaro
  • 31
0
votes
1 answer

Expected value of a markov history chain

What is $E[X_{n+1}|X_0 = i_0, . . . , X_{n−1} = x_{n−1}, X_n = i]??$ Given that $\sum_j^n jp_{i,j}=i$.
confused doughnut
  • 57
0
votes
2 answers

Markov Chain initial state conditional probability

Suppose ${X_0, X_1, . . . , }$ forms a Markov chain with state space S. For any n ≥ 1 and $i_0, i_1, . . . , ∈ S$, which conditional probability, $P(X_0 = i_0|X_1 = i_1)$ or $P(X_0 = i_0|X_n = i_n)$, is equal to $P(X_0 = i_0|X_1 = i_1, . . . , X_n…
confused doughnut
  • 57
0
votes
1 answer

Covariance of Brownian Motion

What is the covariance function for $U(t)$ if $U(t) = e^{-t}B(e^{2t})$ for $t \geq 0$ where $B(t)$ is standard Brownian motion? Any help would be great
Brent
  • 1
0
votes
1 answer

Expectation of $X_t\cdot X_s$

I have a following task: Let $X_0=1$ and then in every $t>0$, $X_t$ is $-1$ or $1$. The number of "changes" from $-1$ to $1$ in interval $\Delta$ is described by Poisson distribution with expectation $\lambda\Delta$. What is the expectation of…
mwrooo
  • 327
0
votes
0 answers

Proving zeros for Ito process

Suppose that Ito process $$X_t=\int_0^tK_sds+\int_0^tH_sdB_s=0, \,t\geq0.$$ Then $$K_t=H_t=0 \text{ a.s.},\,t\geq0.$$ To prove this, it is suggested to apply Ito formula to the process $Y_t=e^{-X^2_t}$, but I fail to see the connection.
TheGrandDuke
  • 351
0
votes
0 answers

When is the composition of two Gaussian Processes a Gaussian Process?

Say I have a probability space $(\Omega, \Sigma, P)$ and two Gaussian Processes over this space $X_1, X_2$ such that: \begin{align*} X_1: \mathbb{R} \times \Omega \rightarrow \mathbb{R} \\ X_2: \mathbb{R} \times \Omega \rightarrow…
gigalord
  • 321
0
votes
1 answer

Can a right-continuous stochastic process be predictable?

I know that a stochastic process is said to be predictable if it's measurable with respect to the predictable $\sigma$-field $\mathcal P$, namely the $\sigma$-field generated by all left-continuous adapted processes. I furthermore know that if $X$…
Chaos
  • 3,327
0
votes
1 answer

Showing that $\text{Var}(X_t) = t$ if $(X_t)_{t > 0}$ is a gaussian process

Let $(X_t)_{t > 0}$ be a stochastic process such that $(X_t)_{t > 0}$ is a gaussian process, it has continuous trajectories, $\mathbb{E}(X_t) = 0$, $\text{Cov}(X_t, X_s) = \min \{t, s \}$. I would like to show that $(X_t)_{t > 0}$ is a Brownian…
Hendrra
  • 2,868
  • 1
  • 18
  • 35
0
votes
1 answer

Distribution Function in Unit interval

Let $Z=(Z_1,Z_2,Z_3)$ be a triple coin toss with $p=1/4$. Let $Y:=\frac{1}{2}Z_1 + \left(\frac{1}{2}\right)^2Z_2+\left(\frac{1}{2}\right)^3Z_3$ be a random variable $\in [0,1]$. Calculate the value F (b) for the distribution function of Y for i) b =…
Marceline
  • 1
1 2 3
…
46 47