Stochastic geometry is an area of mathematics that investigates random spatial patterns and the probabilistic foundations of geometric structures. A key focus within this field is the study of random ...
A stochastic process is a colection of random variables defined on the same probability space. Please explain further what parts of this definition are escaping you.
What's the difference between stochastic and random? There is an anecdote about the notion of stochastic processes. They say that when Khinchin wrote his seminal paper "Correlation theory for stationary stochastic processes", this did not go well with Soviet authorities. The reason is that the notion of random process used by Khinchin contradicted dialectical materialism. In diamat, all ...
Stochastic Calculus for Finance I: Binomial asset pricing model and Stochastic Calculus for Finance II: tochastic Calculus for Finance II: Continuous-Time Models. These two books are very good if you want to apply the theory to price derivatives. Stochastic Differential Equations: An Introduction with Applications Bernt Oksanda.
An intuitive logical consequence of that interpretation is that the "law" or "underlying mechanism" that determines the stochastic process must be time-invariant. On the other hand, my understanding of the time homogeneous condition is that it explicitly states the time-invariance of the "law" or "underlying mechanism" of the stochastic process.
Now a "stochastic process" is simply a collection of many such variables, usually labeled by non-negative real numbers $t$. So $X_t$ is a random variable, and $X_t (\omega)$ is an actual number.
Complex behaviour in many systems arises from the stochastic interactions of spatially distributed particles or agents. Stochastic reaction–diffusion processes are widely used to model such behaviour ...