Mathematics Stack Exchange
Mathematics Stack Exchange

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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Book for learning stochastic processes for a beginner

I am a social science student interested in learning stochastic processes. The book used in my university is Grimmett and Stirzaker. I tried to study discrete time markov process from it and was able to understand the proofs. However, I probably…
canseeker
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How can an Ornstein-Uhlenbeck process be shown to be continuous/diffusion?

Given the Ornstein-Uhlenbeck transition pdf (where $t_2\geq t_1 \geq 0$ and $x_2 \geq x_1 \geq 0$ and $\gamma >0$): $$p(x_2,t_2;x_1,t_1) = \frac{1}{\sqrt{2\pi(1-e^{-2\gamma(t_2-t_1)})}}\exp \left( -\frac{(x_2-x_1e^{-\gamma…
Naz
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Time until x successes, given p(failure)?

I hope this is the right place for help on this question! I expect this should be easy for this audience (and, no, this isn't homework). I have a task that takes $X$ seconds to complete (say, moving a rock up a hill). However, sometimes the task is…
redtuna
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Black-Scholes formula with non-constant volatility (function of time)

Let's have the following stochastic process: $$dS_t = r S_t dt + σ(t) St dW_t$$ where $W_t$ is the Brownian motion, r the drift and $σ(t)$ the volatility, a deterministic function of the time. Applying Ito's lemma, I have reached that : $$S_t =…
Edin_91
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Variation of a function

What exactly is the variation of a function ? Is it a distace or an element of some space The total of a real valued function $f\colon [0,t] \mapsto \Re $ is as below say $\pi = \{0=t_0,t_1,\cdots , t_n=t\} $ then the $p^{th}$ variation of $f$ is…
almost_sure
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Does there exist a continuous-time i.i.d. process on a standard probability space?

Is there are standard probability space $(\Omega, \mathcal{A}, P)$ and a process $X_t : \Omega \to \{ -1, 1 \}$, $t \in [0,1]$ such that $X_t$ is uniformly distributed on $\{ -1, 1 \}$ and all the $X_t$ are independent (or uncorrelated)? By…
yada
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Master equation to Fokker Planck equation

How can we convert a given master equation to a Fokker Planck equation.Is there any general method for this transformation?
Anu
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Aggregate arrivals from a renewal process

This is a follow-up question of the question "Aggregate arrivals from a Poisson Process". The inter-arrival time of a renewal process, t, conforms to a general distribution, denoted by PDF $f(t)$. Next we aggregate the requests according to the…
Bloodmoon
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Are $L$-diffusions unique in law?

I've been trying to understand diffusions. We can show they exist by noting they solve particular SDEs, but are they unique? More precisely: Fix a filtered probability space satisfying the usual conditions and locally bounded measurable functions …
Ben Derrett
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Weak convergence of an integral of an exponential of a Wiener process

EDIT: corrected the problem thesis. Suppose $(W_s)_{s \geq 0}$ is a Wiener process. Define $$ V_t := \frac{1}{\sqrt{t}}\log \left[\int_{0}^{t}\exp(W_s)ds\right] $$ Show that $$ V_t \xrightarrow{t \rightarrow \infty} \sup_{s \in[0, 1]}W_s $$ in…
tosi3k
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Local time for reflected random walk

Say I have a process starting from 0, and last for 100 steps, each step either moves up or down by one unit, within the boundary -10 and 10. My understanding is that since this is a martingale, the expected hitting time would be 100, but how to…
butterbetter
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How to prove that a probability density has second moment?

The setting of my question is the following. We have a diffusion process $$dS(t) = \mu S(t) \; dt + v(t,S(t)) \; dW(t)$$ where $W$ is a standard Brownian motion under an equivalent martingale measure $Q$ and $v$ satisfies all necessary regularity…
DoubleTrouble
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Expected value of integrals of a gaussian process

I have limited knowledge of the theory of stochastic processes. While working on a problem I've stumbled upon some expected values of time integrals of Gaussian stochastic processes. Before starting to delve into the literature on stochastic…
Pincopallino
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Brownian motion and Bessel process

Can you help me with the following: Prove that a geometric Brownian motion can be represented as a time-changed Bessel process $$ \exp(B_t+vt)=R_{A_t} $$ where $A_t= \int_{0}^t \exp(2(B_s+vs)) ds$ and $(R_t)$ is a Bessel process of parameter…
nick
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proof T is a stopping time

let $X(t) $ is a stochastic process and is cadlag and adapted, let $T = \inf\{t:|X(t)| \ge c\}$, proof T is a stopping time. i.e.$\{T\le t\} \in F_t$
annimal
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