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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.

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Canonical processes of a stochastic process

From a old handwritten note without references cited, the first canonical process of a stochastic process $\Omega \times T \to S$ is defined as the identity mapping on $S^T$. I was wondering if there are concepts such as the second canonical…
Tim
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Find expectation of brownian motion

How would I do the following question. I know how to do it with two variables (just B(U) and B(U + V) but I do not know how to figure this out with 4 (or even 3) terms) Thanks for the help. E [B(U)B(U+V)B(U+V+W)B(U+V+W+x)] where U + V + W > U + V >…
icobes
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Transition Matrix problem

We have a biased coin which, if tossed, shows heads (H) with probability 1/3 and tails (T) with probability 2/3. First, we will toss the coin 3 times. We will model this triple of tosses as well as further tossing of the coin by an MC with states…
Saria Ahmad
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Strong Markov property explained.

I have got 2 theorems, Theorem 1 The increment $ (N_{t+u} - N_t)_{u\geq 0} $ of a Poisson process rate $\lambda$ is again a Poisson process rate $\lambda$ and is independent of $(N_s)_{0\leq s \leq t}$ Proof $ P(N_{t+u} - N_t)= k|N_t=n,…
Rosie
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definition of a sample path of a stochastic process

Why do we fix $\omega$ in the definition of a sample path of a stochastic process? If $f:T \to \Omega$ is an arbitrary function, can't we define sample path to be equal to $\{X(t, f(t)): t \in T\}$?
Denisof
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How to compute conditional probability for a homogeneous Poisson process?

Let $N$ be a homogeneous Poisson process with intensity $\lambda$. How do I compute the following probability: $$P[N(5)=2 \, | \, N(2)=1,N(10)=3]?$$
Faheem
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Is a stochastic process being Markovian or Martingale completely determined by its law?

Suppose a stochastic process is Markovian. Let $L$ be its law on its sample path space (note that here I assume its initial distribution is known, not just conditional distributions). If there is another stochastic process with the same law $L$,…
Tim
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Verify that a random variable is a stopping time

Let $\lbrace X_{n}\rbrace$ be a stochastic process adapted to filtration $\lbrace \mathcal{F}_{n}\rbrace$. Let $B\subset \mathbb{R}$ be closed. Then $$\tau(\omega):=\mathtt{inf}\lbrace n\in\mathbb{N}:X_{n}(\omega)\in B\rbrace $$ is a stopping time.…
czachur
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Ways for characterizing stochastic processes

I was wondering what are some general approaches to characterize a stochastic process? Sorry for the vagueness that my questions may have, but let me try to make clear as much as I could that question I have been having for a long time. First of…
Tim
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Death process (stochastics)

From what I understand, the question is asking me to find P(X(t) = n | X(0) = N). I know that with a linear death rate this probability is (N choose n) * [e^(-alpha*t)]^n * [1 - e^(-alpha*t)]^N-n but I don't think this is true for a constant death…
icobes
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problem to understand branching process

Here the definition of wikipedia of Branching process. Let $z_n$ the size of the generation $n$. So, $$Z_{n+1}=\sum_{k=1}^{Z_n}X_{n,i}$$ where $X_{n,i}$ is the number of offspring of the $i-$th individual. We suppose that $Z_0=1$. My problem is…
user386627
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Statistical self-similar process

Can someone please explain the following: If the process $z={z(t)}$ for $t>0$ is self similar, then the finite dimensional distributions of $z$ on the positive real line are completely determined by those on any interval of finite length.
prayag gowgi
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Brownian Motion Expectation-Like Integral

How much is $$\int_0^T tB_t \, dt$$ where $B_t$ is Brownian motion and $T$ an universal constant?
Troy McClure
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what is exactly a predictable process?

Let $(\Omega ,\mathcal F,\mathbb P)$ a probability space. A stochastic process is predictable with refer to the filtration $(\mathcal F_n)_n$ if $X_{n+1}\in \mathcal F_n$. Could someone tel me what it mean exactly ? (except the fact that $X_{n+1}$…
user380364
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How to calculate the expectation of a trajectory dependent random variable

Now, I have a stochastic differential equation, $dx=f(x)dt+h(x)dB$ with intitial condition $x_0$, and a random varaible $g\triangleq \exp\{\int_0^{t} F(x_s) ds\} $. Here, $f(\cdot)$, $h(\cdot)$ and $F(\cdot)$ are arbitrary functions. My question…
Zhou Fang
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