Tagged Questions
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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4 views
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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10 views
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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15 views
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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301 views
+50
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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26 views
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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2answers
139 views
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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16 views
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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14 views
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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51 views
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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2answers
30 views
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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23 views
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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46 views
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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1answer
379 views
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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2answers
230 views
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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18 views
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 ...