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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Is aperiodicity necessary for a irreducible Markov chain with finite state space to exclude positive probability of infinite hitting time?

I encountered a Lemma: For any irreducible aperiodic Markov chain $(X_0, X_1, \ldots)$with state space $S =\{s_1,\ldots, s_k \}$ and transition matrix $P$, we have for any two states $s_i,s_j \in ...
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Galton-Watson Branching process, most recent common ancestor

Consider the Galton-Watson branching process such that $X_n =$ # of individuals in generation $n$, starting with $X_0=1$. Assume that the family size distribution is the same for all individuals and ...
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Brownian motion hitting probability

Let $B_t$ be a brownian motion and $g(t)$ a function of the time $t$. $B_0=0$. Let $\Phi$ be the c.d.f. of a normal distribution. At time $t$, the probability that $B_t > g(t)$ equals ...
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Wiener Process $dB^2=dt$

Why is $dB^2=dt$? Every online source I've come across lists this as an exercise or just states it, but why isn't this ever explicitly proved? I know that $dB=\sqrt{dt}Z$, but I don't know what ...
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Question regarding Ito Process

I am new to Ito Process, so I have a following question. Consider a standard Ito Process, $$X_t=X_0+\int_0^t\mu_sds+\int_0^t\sigma_sdW_s$$ where W is the m-dimentional Brownian motion and X is a ...
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probability (waiting time = infinity) for a poisson process

I am new here so if I violate any rule, please inform. Consider the stochastic process given by $\{ N(t) : t \geq 0 \}$ which is time homogeneous poisson process with arrival rate $\lambda$. Let ...
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Is filtration necessary for continuous random variables?

Define $(\mathscr{F}_t)_{t\geq 0}$ being the natural filtration induced by the Brownian motion $(B_t)_{t\geq 0}$. That is $$\mathscr{F}_t=\sigma(B_s\mid 0\leq s\leq t), \forall t\geq 0,$$ i.e. ...
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stochastic cash flow [on hold]

I need Finance data with stochastic cash flows for my thesis. please introduce how could i access to these data? or introduce a site.
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Stochastic process that is Martingale but not Markov? [duplicate]

Can you please help me by giving an example of a stochastic process that is Martingale but not Markov process for discrete case?
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A question concerning the joint probability distribution

Here is the original question: Given a stochastic process $X(t)=Y_1+tY_2$, where $Y_1,Y_2$ are i.i.d satisfying $Y_1 \sim N(0,1)$. Derive the joint probability distribution for $(X(t),X(s))$ where ...
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Using characteristic function to deduce convergence of Bernoulli random variables

Let $Y_1, Y_2,...$ be a sequence of independent Bernoulli(0.5) random variables and $X_n = \sum_{i=1}^{n} Y_i 2^{-i}$ I need to use the characteristic function to deduce that $X_n$ converges in ...
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How to prove Brownian motion is Gaussian Process?

I'm reading Bernt Oksendal's "Stochastic Differential Equations" and this is one of the proof that I'm totally lost. This is from Ch2.2, page 12-13 (sixth edition). First, Brownian motion is ...
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Given particle undergoing Geometric Brownian Motion, want to find formula for probability that max-min > z after n days

Consider a particle undergoing geometric brownian motion with drift $\mu$ and volatility $\sigma$ e.g. as in here. Let $W_t$ denote this geometric brownian motion with drift at time $t$. I am looking ...
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probability terminology for parameter in a Markov process

Suppose $$P(\text{feature present at time} \ t \ \text{and} \ t+\Delta t) = \beta^{2}+\beta(1-\beta) \exp(\Delta t/\tau)$$ where $\tau = 1/(\pi_{01}+\pi_{10})$. What is $\tau$?
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poisson convergence for a counting process

Given N(t) is a counting process, I am trying to prove it is poisson (by applying the poisson convergence: unsure how to apply this). To do this the proof says divide the interval into $n$ equal ...

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