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 processes in nature are characterized by deterministic development, transformation etc, so the phrase "random process" itself sounded paradoxically. Therefore, Khinchin had to change the name. After some search, he came up with the term stochastic, from στοχαστικὴ τέχνη, the Greek title of Ars conjectandi. Being popularized later by Feller and Doob, this became a standard notion in English and German literature.
Funny enough, in Russian literature the term "stochastic processes" did not live for long. The 1956 Russian translation of Doob's monograph by this name was already entitled Вероятностные процессы (probabilistic processes), and now the standard name is случайный процесс (random process).
Neither word by itself has a commonly accepted formal definition in mathematics, so one cannot really ask about "the difference" between them.
They are used in phrases such as "random variable," "random walk," "stochastic process," "stochastically complete," etc, which have accepted definitions of their own. In all cases both words tend to refer to an element of chance or unpredictability. But they are generally not interchangeable; if you talk about a "stochastic walk" people will be confused.
Random process and stochastic process are completely interchangeable (at least in many books on the subject). Although once upon a time "stochastic" (process) generally meant things that are randomly changing over time (and not space). See relevant citations:
In English the word "stochastic" is technical and most English speakers wouldn't know it, whereas, from my experience, many German speakers are more familiar with the word "Stochastik", which they use in school when studying probability.
The word "stochastic" ultimately comes from Greek, but it first gained its current sense, meaning "random", in German starting in 1917, when Sergei Bortkiewicz used it. Bortkiewicz had drawn inspiration from the book on probability by Jakob (or Jacques) Bernoulli, Ars Conjectandi. In the book, published 1713, Bernoulli used the phrase "Ars Conjectandi sive Stochastice", meaning the art of conjecturing. After being used in German, the word "stochastic" was later adopted into English by Joseph Doob in the 1930s, who cited a paper on stochastic procsses written in German by Aleksandr Khinchin.
The use of the term "random process" pre-dates that of "stochastic process" by four or so decades.
Although in English the word "random" does come from French, I strongly doubt it ever meant random in French. In fact, it originally was a noun in English meaning something like "great speed". It's related to the French word "randonée" (meaning hike or trek), which is still used today. To describe a random variable, French uses the word "aléatoire", stemming from the Latin word for dice (which features in a famous quote "Alea iacta est." by Julius Caesar). The English equivalent "aleatory" is not commonly used (at least in my random circles).
The term stochastic in Hydrology science refers to a process which periodically and apparently-independently happens but a kind of dependency exists. For example, if the flow of a river in last (say) 2 weeks has been low, it will probably be low in the next weeks too. So, the flow of a river is not a complete random variable but stochastic.
The terms "stochastic variable" and "random variable" both occur in the literature and are synonymous. The latter is seen more often. Similarly "stochastic process" and "random process", but the former is seen more often.
Some mathematicians seem to use "random" when they mean uniformly distributed, but probabilists and statisticians don't. I suspect those who do that haven't thought about it much.
A random process is unpredictable such as the movement of the tip of a feather In wind. If we assume that the movement of a roller coaster is deterministic, then a stochastic process would be the movement of the tip of a feather attached to a moving roller coaster. That is to say, stochastic processes have components that are both deterministic AND random; e.g. Martingales.
I will quote from Robert Gray Gallager's MIT OpenCourseWare notes for "Discrete Stochastic Processes" (1):
"Stochastic and random are synonyms, but random has become more popular for rv’s (random variable) and stochastic for stochastic processes. The reason for the author’s choice is that the common-sense intuition associated with randomness appears more important than mathematical precision in reasoning about rv’s, whereas for stochastic processes, common-sense intuition causes confusion much more frequently than with rv’s. The less familiar word stochastic warns the reader to be more careful." (Chapter 1: Introduction and review of probability, page 15, fn 15).
I believe there is a difference between random and stochastic. Random has no preciptating or a priori cause i.e acausal. Random action stands alone - not within any system. Stochastic is random, but within a probablistic system. In other words an act of God is random, but a hurricane hitting the east coast of the US is stochastic event. Any individual hurricane may be random but it also exists as a mathmatical probability within a system of many hurricanes that hit or do not hit the east coast every year. The later is therefore stochastic. A coin flip has an interesting difference than a hurricane since each individual flip of a coin is already stochastically limited to the 50% statistical probability of two possible results.
So. There are Random ODEs and Random PDEs that are not synonymous to SODEs and SPDEs -- although wide classes of RODEs can be converted to SODEs (usually replacing random normal forcings for Ornstein-Uhlenbeck dynamics) through the Imkelller-Schmallflusss correspondence.
I don't know that much about the theory because my work is in simulating/numerically solving these. But they're a natural formulation for systems such as earthquakes, tumor growth, etc.
A nice wide-ranging textbook aimed at scientists (spends a third of the book introducing rigorous probability before entering the subject) is
NECKEL, Tobias; RUPP, Florian. Random Differential Equations in Scientific Computing. Walter de Gruyter, 2013.
A textbook of numerics, a little narrower in scope is
Han, Xiaoying, and Peter E. Kloeden. "Random Ordinary Differential Equations and Their Numerical Solution." (2017).
Kloeden is well known for his textbook on numerical SDEs, of course. If you plug these on Google Scholar you can browse more recent papers that cite them for hours.
EDIT: I found this slideset from a talk by Neckel [PDF link]. It defines RODEs and explains the intuition for the Imkeller-Schmallfluss correspondence.
I would make a distinction: for example in a queueing system the arrival times (or interval times) might be modelled by a Poisson process which would be time independent and would not be bound my initial conditions. This would be an example of a random process which outputs random variables. Service time in the queue would be dependent on the previous state(s) of the system and possibly initial conditions. This would be an example of a stochastic process which also outputs random variables. I am not sure that the term ‘stochastic variable’ has any real meaning except possibly to indicate how the variable was produced.
There is absolutely a difference between stochastic process and randomness. For example, if I take one step then let's suppose my friend takes two steps. Now my friend's steps are not random, those are dependent on my steps. That means my friend's steps has a process which is the number of steps I take. But it is random because my friend doesn't know how many steps I will take. So the steps I take is a random walk.