How many stochastic processes are there really? I know there is the Wiener process and Brownian motion process but what about otherst.trying to learn math
>>16258271It's been some years since I last dealt with this. I'll try to give a broad overview.First of all, Wiener process is just another name for Brownian motion. To give a very short answer: There are uncountable infinitely many stochastic processes. Any consistent assignment of finite-dimensional distributions defines a stochastic process. Luckily most of them are too insignificant or unphysical to be relevant. For all practical purposes there are two big classes of stochastic processes based on their properties: Markov processes and martingales. Intuitively speaking, Markov processes are processes where the distribution of a future event only depends on the current state and not on the entire history. This includes discrete Markov chains, random walks, Brownian motion. Martingales are, loosely speaking, processes whose expectations cannot be manipulated by knowing their history, such as fair gambling games or financial markets under ideal conditions. There are some deep connections between martingales and Markov processes.THE most fundamental and most important stochastic process is indeed Brownian motion, both a martingale and a Markov process. The entire theory of Ito integration and stochastic differential equations is based upon BM. Via martingale representation theorems many martingales can be expressed in terms of BM.So far this was about temporal stochastic processes. Of course another generalization are spatial or spacetime stochastic processes. The generalization of BM here are Gaussian random fields allowing for a rich theory of stochastic differential geometry. The other large group of spatial stochastic processes are point processes stemming from the fundamental Poisson process and their generalizations to random measures.
>>16258271Hey, I saw you posting this gif on the /pol/ thread this morning. What's going on there?
>>16258271none, they’re all perfectly predictable.
>>16258350You ready for this?
>>16258519No, that wasn't me. I took that image from him and made this thread.
>>16258350Damn. I don't know if I understood all of that. When I see random shit I just think it is random shit and it's course in time is the will of God.
>>16258271>>>/sci/psg
>>16258760God is not predictable.
>>16258350What should I read if I want to understand this shit? I just finished taos Analysis 1
>>16258350Kind of sounds like set theory to be honest.
>>16258889You need a good understanding of basic measure theory, so I hope your analysis book covers this. Unfortunately I can't give you recommendations for that since I learned measure theory only in my native language. For stochastic processes the most comprehensive introduction is >Kallenberg - foundations of modern probability It's a very big book though. If you're only interested in Markov processes you may want to read>Norris - Markov chains>Liggett - continuous time Markov processesFor a beginner friendly intro to Brownian motion and Ito calculus there is >Calin - an informal introduction to stochastic calculus with applicationsFor the very advanced study of random fields and spatial stochastic processes I only know of >Adler and Taylor - random fields and geometry >Kallenberg - random measures But surely there are other knowledgeable anons here who have more book recommendations.
