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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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the relationship of the counting process and the renewal process?

The textbook wrote like the following: A natural generalization is to consider a counting process for which the interarrival times are independent and identically distributed with an arbitrary distribution. Such a counting process is called a…
Danny_Kim
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The expected value of a stochastic process

I am asked to calculate the expected value of the maximum of the stochastic process $X(t)=At+B,\,0\le t \le 1$, where both $A$ and $B$ are independent, normally distributed random variables with mean $0$ and variance $\sigma^2$. Any ideas will be…
RLP
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Stopping time wiener process

Suppose that $T=\inf\{t\geq 0: W_{t} \notin [-a,b]\}$ where $a,b>0$ and $W_{t}$ denotes a Wiener process. Now I'm wondering if this is a stopping time but don't know how to work this out.
Roos Jansen
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The variance of wiener process at different points in time

Given that I have $W_t$ and $W_s$ for some $t>0$ and $s>0$ and $W_n = \sum_{i=1}^n R_i$ I do not understand why $\text{Var}[W_t-W_s]=|t-s|$. I understand why $\text{Var}[W_n]=n$. This is because $\text{Var}[W_n]=\text{E}[(\sum_{i=1}^n…
Naz
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Poisson Process. Three independent processes

David is fishing together with two friends. Each of them catches fish independently of the others according to a Possion process with rate 2 per hour. What is the expected time until everyone has caught at least one fish?
user299219
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Birth and Death Rate Processes

I am stuck on trying to figure out how to get started? Can someone maybe push me in the right direction? A bank that is planning to install an ATM must choose between buying one Zippytel machine or two Klunkytel machines. Although one Zippy costs…
daultongray8
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Wrong Solution to Geometric Brownian Equation

I attempted to treat SDE as ODE to solve the Geometric Brownian Equation and obviously got the wrong answer. My question is, where did it go wrong? $dX_t=\mu X_tdt+\sigma X_tdW_t$ as given. $\dfrac{dX_t}{X_t}=\mu dt+\sigma d W_t$ dividing $X_t$ on…
Chris L
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Equivalence classes and abuse of notation, a question concerning Stochastic processes

On Karatzas and Shreve - Brownian motion and Stochastic Processes pg 146 one reads I don't understand how saying that $\mathcal{P}^* \subset \mathcal{P}$ is an abuse of notation. $X \sim Y$ if and only if $\int_0^T (X_u - Y_u)\, d\langle M…
Conrado Costa
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Expected number of arrival in a poisson process which are not followed by arrival in next time $\delta$ .

Let $a,\lambda,\delta > 0$ . Compute the expected number of arrivals in a Poisson process with intensity function $\alpha(t)=ae^{-\lambda t}$, which are not followed by another arrival with time interval of length $\delta$. Suppose the number of…
Subhasish Basak
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If $N(t)$ is a Poisson process with parameter $\lambda(t)$ then is $N'(t)=N(t+2)-N(2)$ a poisson process?

If $N(t)$ is a Poisson process with parameter $\lambda(t)$ then is $N'(t)=N(t+2)-N(2)$ a poisson process? I think it should be poisson process as it is like observing a poisson process after time $2$ but how to prove it.
Subhasish Basak
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Question about a step in the proof of the bracket process

Let's first state the theorem $\forall M$ continuous local martingale, there exists a unique increasing continuous process $\langle M\rangle $ zero at $t=0$ and such that $M^2-\langle M \rangle $ is again a continuous local martingale. Further for…
user20869
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Death process - median time to die out

Question: A pure death process $\{X(t); t \ge 0\}$, where $X(t)$ denotes the number of individuals alive at time $t$, starts with $X(0) = 8$. The lifetime of each of these individuals is exponential with mean $\frac 1 \upsilon$. Solve for the…
Seow Fan Chong
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Distribution and continuous time markov chains

Let be a probability distribution on the nonnegative integers such that $\pi_i > 0$ for all i. Write down the transition matrix of an irreducible, aperiodic, recurrent Markov chain on the nonnegative integers that has as its stationary probability…
mary
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Find a family of measures that satisfies the requirements for measurable selection

In chap 12 of Stoock and Varadhan Multidimensional diffusion processes in section 12.2 markov selections page 290 one reads I couldn't find an example that fit the conditions (a)-(d). One would guess that when a there is a unique strong solution …
Conrado Costa
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When does an uncountable collection of random variables define a stochastic process?

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $( \mathcal{X}, \mathcal{B})$ be a measurable space. Let $\{X_t\}_{t\in [0,1]}$ be an uncountable collection of random variables such that $$X_t:(\Omega, \mathcal{F}, \mathbb{P}) \to…
user2808118
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