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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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About cylindrical $\sigma$ fields and countably many random variables

I am reading this article and in the second page I am not being able to understand the argument. In essence we have two random process $(X_t)_{t > 0}$, $(Y_t)_{t > 0}$, such that both have the same finite dimensional distributions. We are interested…
Barreto
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Correlation Coefficient can be Greater Than 1??

I'm in a stochastics class this semester, and during our last class, a classmate casually said that "correlation coefficients of random processes can be greater than 1 in the case of exponential growth series." I thought this was wrong, but the…
Camellia99
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From $\lim_{s\to t}E e^{i \lambda\left(\zeta_{t}-\zeta_{s}\right)}=1$ prove $\zeta_t$ stochastically continuous at $t$.

From $$ \lim_{s\to t}E e^{i \lambda\left(\zeta_{t}-\zeta_{s}\right)}=1 $$ prove $\zeta_t$ stochastically continuous at $t$. I don't know why it can be proven from such a condition. If you need additional conditions, then $\zeta_t$ can assume to be…
Knt
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stopping time and martingale

If $h(x_{t\wedge \tau})$ is a submartingale such that $h(0)=1$ and $\tau$ is a stopping time. Why is it that $h(x_\tau)=1_{{\tau<\infty}}$?. Also $h(x)$ is decreasing.
Vaolter
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Natural filtration of a one-jump counting process

Since a one-jump counting process $N(t) := I(T \leq t)$ only attains values in $\{0, 1\}$, am I correct in thinking that its natural filtration $\mathcal{N}_t := \sigma(\{N(s): s \leq t)\}$ does not change in time? I.e. $\mathcal{N}_s =…
harisf
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Every local martingale (Yn) can be represented as a martingale transform (Cn)•(Xn) of some martingale

Every local martingale (Yn) can be represented as a martingale transform (Cn)•(Xn) of some martingale This is from the following lecture notes https://qmplus.qmul.ac.uk/pluginfile.php/2479408/mod_resource/content/3/L3-2020.pdf Theorem 3.12. Is…
jk001
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On the sample path right continuity and hitting times

I have a question on the sample path right-continuity of a stochastic process. Let $X=\{X_t\}_{t \ge 0}$ be a stochastic process on a locally compact separable metric space $E$. We assume that $X$ starts from $x \in E$ and the sample path of $X$ is…
sharpe
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How to derive the dynamics of the perpetual bond price

I have a perpetual bond price dynamics under a martingale measure $Q$: $$ dp(t,T)=p(t,T)r(t)dt+p(t,T)v(t,T)dW(t) $$ where $W$ is a vector-valued $Q$-Wiener process. I also have a bond pricing formula: $$ C(t) = \int_t^\infty p(t,s)ds $$ How can I…
Cyan
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When is a family of densities a solution to the Fokker-Planck equation?

Given a family of densities $\{p(x, t)\}_t, t \in [0, \infty),$ with (tractable) stationary density $\lim_{t \to \infty} p(x,t) = p_\infty(x)$ under what conditions is the family a solution to the Fokker-Planck equation? That is, under what…
tdockhorn
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On finite dimensional distributions

I have a question on distributions for some stoachastic processes. Let $(\Omega_i,\mathcal{F_i},P_i)$, $i=1,2$, be probability spaces. Let $X=(X_t)_{t \in [0,\infty)}$ be an $\mathbb{R}^d$-valued stochastictic process defined on the probability…
sharpe
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Notation clarification: Kolmogorov's Extension Theorm in Øksendal

In Økensdal's Chapter 2 (6th ed. of Introduction to Stochastic Differential Equations), the author states the Kolmogorov's Extension Theorem as follows: For $k$ Borel sets $F_1, \cdots, F_k \in \mathbb{R}^n$, for all $t_1, \cdots, t_k \in T$, $k…
Apoorv
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Galton-Watson process : branching property

I can’t find a definition of the ‚branching property‘ of a Galton Watson process on the internet. Can someone help me, how it is defined or give me an example. I have a branching processes book, but there is the ‚branching property‘ only for ‚lines‘…
toni_iva
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Add a random variable to a random process

I have a statistical model, $Y_i=Bf_i+W_i$, where $i=1,\ldots,N$ $B\thicksim\mathcal{N}(0,\sigma_B^2)$ $W_i\thicksim\mathcal{N}(0,\sigma_w^2)$ $B,W_0,\ldots,W_N$ are i.i.d. and $f_i$ is a known deterministic signal. I need to write the likelihood…
HVW
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Poisson counting process

Let $N_t$ be number of customers that arrived at shop until moment $t$. Let's say that shop opens at 9:00. $N_t$, $t\geq0$ is Poisson process with $\lambda=1$ per hour. What is the probability that, there will be at least 2 arrivals between 10:00…
atomoutside
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Two processes that are modification and have continuous trajectories are indistinguishables

I'm having problems to solve this problem If X and Y are processes, Y is a modification of X and both have continuous trajectories a.s. Then X and Y are indistinguishable. I know that if they are indistinguishable then are modifications of each…
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