# Questions tagged [stochastic-processes]

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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### integration by parts formula

Below is from Liptser, Shiryaev "Theory of martingales", page 200: I have a question: How from eq. 3.5 and 3.8 they got the eq. 3.9? Since $G$ is of finite variation $\mathcal{E}^{-1}(G)$ is of ...
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### Infinitesimal generator of diffusion process in one dimension

I consider a diffusion process $$dX_t = b(X_t) dt + \sigma(X_t) dB_t.$$ From the general theory, we know that if $f\in C^2(\mathbb{R})$ and has a compact support, then the infinitesimal generator ...
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### Poisson process with time exponentially distributed

While studying Poisson processes, I have found a problem I can't solve: Two people, A and B, arrive at a bank and wait to be attended. A and B are in the waiting queue and A is before B. The ...
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### Multi-folds of Generating Functions

Stochastic Process Generating Function Practice I don't understand how $G_{N}(s) = G_{M}(G_{Y_i}(s))$. $N = Y_1 + Y_2 + Y_3 + ... +Y_M$, which to my understanding the main generating function should ...
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### Decomposition of a cadlag and of local bounded variation function $Z:\mathbb{R}_+\rightarrow\mathbb{R}$

Let $Z:\mathbb{R}_+\rightarrow\mathbb{R}$ be cadlag and of local bounded variation with $Z(0)=0$ and $V_Z(t)$ denotes the value of the total variation of $Z$ on $[0,t]$ for all $t\in\mathbb{R}_+$. I ...
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### Is the sum of positive jumps from a jump process adapted again?

Let $(\Omega,\mathcal F,\{\mathcal F_t\}_{t\in [0;T]},\mathbb P)$ be a filtered probability space satisfying the usual conditions and let $\{X_t\}_{t\in[0;T]}$ be an adapted stochastic process, whose ...
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### Why is $Y_n$ not a bernoulli process?

for $n \in \mathbb{N}$, let $X_n$ be a Bernoulli process with parameter $p = \frac12$, let $N = \min \{n \geq 2: X_1 \neq X_n \}$ for $n \in \mathbb{N}$, let $Y_n = X_{N +n -2}$. in a question it ...
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### Stochastic process vs regression for modelling data

why one would model something (anything?) with a stochastic process, such as Ornstein–Uhlenbeck (OU), rather than a regression, like $y(t) = \beta_0 + \beta_1e^{( - rt)}$? With regression, you can use ...
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### Stochastic Optimization and Monte Carlo

Assume we are in a Brownian filtration where I denote $W$ the Brownian motion. My problem is to numerically compute $$\min_X E (\int^1_0 X^2_tdt),\ \ \ \ (*)$$ where $X$ is adapted to the filtration ...
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### Stopped uniformly integrable process in discrete time is uniformly integrable?

I'm studying for work the book: "Stochastic Calculus and Application" by Choen and Elliot 2 ed. In section $4.2$ (pg. 91) it states the discrete version of the Optional Stopping Theorem for bounded ...
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### Is stochastic integral the same as Lebesgue integral of Banach-valued function?

I'm reading a book about stochastic processes and I've come to "stochastic integration." In this section stochastic processes are basically functions X:[0,\infty)\rightarrow L^2(\Omega, \mathbb{C})....
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### Stochastic Proof Piece

As a part of the proof of recurrence criterion, (Page 22 here: http://web.math.ku.dk/noter/filer/stoknoter.pdf), it is shown that $(P^n)_{i,j} = \sum_{m=1}^n (P^{n-m})_{j,j}f_{ij}^{(m)}$. This makes ...