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.
With stochastic process, the likelihood or probability of any particular outcome can be specified and not all outcomes are equally likely of occurring. For example, an ornithologist may assign a greater probability that a bird will select a nesting location based on how far it is from the edge of the refuge or whether the location is shielded ...
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.
What you need is a good foundation in probability, an understanding of stochastic processes (basic ones [markov chains, queues, renewals], what they are, what they look like, applications, markov properties), calculus 2-3 (Taylor expansions are the key) and basic differential equations. Some people here are trying to scare you away.
18 I have experience in Abstract algebra (up to Galois theory), Real Analysis (baby Rudin except for the measure integral) and probability theory up to Brownian motion (non-rigorous treatment). Is there a suggested direction I can take in order to begin studying stochastic calculus and stochastic differential equations?
When studying stochastic processes/stochastic calculus/statistics you certainly need to know PT- so I would say this is the primary course here. Jonas has mentioned measure theory - and it is indeed essential for the proper understanding of probability (thanks to Kolmogorov's axiomatization), so in some universities, students learn measure ...
Stochastic processes are often used in modeling time series data- we assume that the time series we have was produced by a stochastic process, find the parameters of a stochastic process that would be likely to produce that time series, and then use that stochastic process as a model in predicting future values of the time series.
Stochastic analysis is looking at the interplay between analysis & probability. Examples of research topics include linear & nonlinear SPDEs, forward-backward SDEs, rough path theory, asymptotic behaviour of stochastic processes, filtering, sequential monte carlo methods, particle approximations, & statistical methods for stochastic processes.
Can someone please elaborate on what the drift of a stochastic process for eg. a Markov process mean? And what role does it play with respect to establishing the stability of that process ?
