## What is a Deterministic Function?

Deterministic means the opposite of randomness, giving the same results every time. So in a sense,

*all*mathematical functions are deterministic, because they give the same results every time; The output of the “usual” function is only determined by its inputs, without any random elements; There are exceptions in stochastic calculus.

The opposite of a deterministic function is a **nondeterministic function** (sometimes called a* random function*). Broadly speaking, those terms don’t actually refer to a “function” in the true sense of the word but rather a random *process*, like predicting weather events, Brownian motion, or other phenomena with elements of randomness. These random processes always have a deterministic part, which might be an average value or expected pattern. In other words, the process isn’t truly random, but is expected to produce outputs within a certain range. The “nondeterminisitc” part is the random element (which is why it’s so hard for weathermen to get forecasts right!).

## Deterministic Functions in Stochastic Calculus

In calculus, we usually aren’t concerned with whether functions are deterministic or not, as it’s assumed all functions are deterministic. However, there is a branch of calculus called **stochastic calculus**, that deals with functions that have random elements.

In stochastic calculus, functions are evaluated to account for random elements. For example, stochastic integrals are generalized ordinary integrals where the integrator is not the usual “dx” but rather a stochastic process with random elements. This allows you to evaluate processes that cannot be evaluated with ordinary calculus methods, like Brownian motion (The Wiener process) [1].

The integrand (the function you want to integrate) can also be nondeterministic. These require completely different definitions for integrals and include the Ito Integral and Stratonovich Integral.

## References

[1] The Azimuth Project. (2015). The Stochastic Integral. Retrieved April 14, 2021 from: https://www.azimuthproject.org/azimuth/show/Stochastic+integral

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