The study of stochastic differential equations (SDEs) has long been a cornerstone in the modelling of complex systems affected by randomness. In recent years, the extension to G-Brownian motion has ...
The story of Brownian motion began with experimental confusion and philosophical debate, before Einstein, in one of his least well-known contributions to physics, laid the theoretical groundwork for ...
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.
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 ...
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.