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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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measurable process stopped at random time

Let $\big( X_t , t\geq 0 \big)$ be a measurable process, that is, $$\big( t, \omega \big) \in \mathbb{R}_+\times\Omega\longmapsto X_t(\omega)\in\mathbb{R} \quad\text{is $\mathscr{B}\big( \mathbb{R}_+ \big)\otimes\mathscr{F}…
Chival
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About stochastic continuity

One of the definitions of stochastic continuity is: $\forall a > 0$ and $\forall s \geq 0$ $$\lim_{t\rightarrow s}\;P(|X(t)-X(s)|>a) = 0.$$ What does it mean intuitively? I know that it implies the jumps but am not sure how and why.
Vaolter
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Is a local martingale which is nonnegative at a deterministic time, nonnegative.

Assume $M$ is a continuous, local martingale s.t. for a single given $T$ we have $M(T)\geq 0$ and $P(M(T)>0)>0$. Can we then deduce $M(t)\geq 0$ for $t\leq T$? I'm trying to use the good old Fatou trick showing a nonnegative local martingale is a…
user121006
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Characterizing previsible processes

I'm having trouble understanding the concept of a previsible process in continuous time, so I'm asking this question to get a better idea of what it means for a process to be previsible. (In what follows, suppose we're working with respect to a…
Ben Derrett
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Stochastic stability and convergence

Consider a Markov process $X$ on $\mathbb R$. Suppose that $X^2$ is $\mathsf P_x$-supermartingale for any $x\in \mathbb R$. If we want that for some neighborhood $U_0$ of $x=0$ holds: for each $x\in U_0$ a condition $X_0 = x$…
SBF
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exponential stochastic process

Let $T$ be a random exponentially distributed time. $P \left(T > t \right)=e^{-t}$. Define $M$ via $M_t = 1$ if $t-T \in Q^+$, $M_t = 0$ otherwise. Where $Q^+$ being positive rationals. let $\{F_t\}$ be a filtration generated by the process $M$.…
babu
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Do uniform convergence and pointwise convergence to normality imply convergence to a gaussian process?

A function $\widehat{f}(x) \rightarrow f(x)$ uniformly over a compact interval $[\underline{x},\overline{x}]$, in addition, for some sequence ${a_{n}}$, $\forall x\in [\underline{x},\overline{x}]$, $a_{n}(\widehat{f}(x)-f(x))\rightarrow_{d}…
Jie Wei
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Collecting coupons which arrive as a Poisson process

I'm trying to answer the following question: "A person collects coupons one at a time, at jump times of a Poisson process $(N_t)_{t\geq 0}$ of rate $\lambda$. There are m types of coupons, and each time a coupon of type j is obtained with…
Ben Derrett
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Stopping times and $\sigma$-algebras

We have the usual $(\Omega, \mathcal{F}, P)$ stochastic basis. Let $\rho, \tau: \Omega \to T \cup \{+\infty\}$ be stopping times and $\mathcal{F}_{\rho}, \mathcal{F}_{\tau}$ their respective $\sigma$-algebras. Prove that $[\rho \le \tau ] \in…
shimee
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Stochastic process, stochastic differential equation

A stochastic process $\{X_t , t ≥ 0\}$ satisfies stochastic differential equation $$\frac{dX_t}{X_t} = 3\mu\ dt + 2\sigma dB_t.$$ where $-\infty<\mu<\infty$ and $\sigma>0$ are given constants, and $\{B_t , t \geq 0\}$ is the standard Brownian…
Mathsstack
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Determining the balance equations for a Poisson Process

I'm trying to do an exercise (not homework) and I fail to understand the solution the reader is giving me. Consider a gas station with one gas pump. Cars arrive at the gas station according to a Poisson process with an arrival rate of 20 cars per…
Stijn
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stochastic process question and strong law of large numbers

It is given that $\{X_i\}_{i = 1}^{\infty}$ is a sequence of i.i.d. random variables distributed as $$ \Pr(X_i=2)=1/2, \Pr(X_i=1/2)=1/4, \Pr(X_i =1/4)=1/4$$ Also let $\displaystyle Z_n = z_0\prod_{i=1}^n X_i $. If I define $W_i =\log_{2}(X_i)$,…
strahd
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Poisson arrivals followed by locking

following is my problem: Pulses arrive at a processor according to a Poisson process of rate λ. Suppose each arriving pulse that is processed by the processor locks the processor for a fixed time T, so that pulses arriving during the locked period…
Tim
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Limiting distribution of alternating renewal process

Consider an alternating renewal system that can be in one of two states: on or off. Initially it is on and it remains on for a time $Z_1$, it then goes off and remains off for a time $Y_1$, it then goes on for a time $Z_2$, then off for a time $Y_2$…
Tim
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optimized upper bound; stochastic

Let $I_t=\int_0^t f_tdB_t,$ where $(f_t,t\ge 0)$ is a bounded process, $|f_t|\leq M$ almost surely for all $t \ge 0.$ Show that $$\mathcal{P}\left[\sup_{0\leq t\leq T}|I_t|>\lambda\right]\leq \exp\left(-\frac{\lambda^2}{2M^2T}\right).$$ Hint:…
user9641
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