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Stochastic Calculus: Overview

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Stochastic calculus deals with stochastic (random) processes or equations that involve statistical noise. It’s a blend of probability and calculus which allows us to apply the principles of calculus to events with random elements. Problems in the field are defined by algebraic, integral, or differential equations with random coefficients or inputs [1].


Key Elements of Stochastic Calculus

A key part of stochastic calculus is Brownian motion, named after Robert Brown’s observation of the random walk pollen particles take when suspended in water. Brownian motion is actually a blend of several random and non-random elements: it is part Gaussian, part martingale and part Markov process (a random process where the future is independent of the past). The paths of Brownian motion are irregular and nowhere differentiable [2], making them impossible to study in “regular” calculus. Later on, Norbert Weiner made mathematical sense of the apparently random motion; This led to the development of a limiting stochastic process called the Weiner process[3], which is now used to model Brownian motion.

Another element of stochastic calculus is the Ito integral, defined as:
ito integral stochastic calculus

Where:

  • H = a locally bounded predictable process,
  • X = a semimartingale.

This new type of integral was developed because Brownian motion is not of bounded variation— a requirement of the usual Riemann integral.

The stochastic differential equation (SDE) also plays a key role. It is defined as:

dXt = b(Xt)dt + σ(Xt) dBt
Where:

  • Bt = one-dimensional Brownian motion.

One way to think of this is as the differential equation dXt/dt = b(Xt) perturbed by noise. The most well-known use of the stochastic differential equation is in modeling stock prices [4].

References

[1] Grigoriu, M. (2002). Stochastic Calculus: Applications in Science and Engineering. Birkhauser.
[2] Stochastic Calculus. Retrieved April 18, 2021 from: https://www.math.purdue.edu/~stindel/teaching/stoch-calc/stoch-calc.html
[3] Van Handel, R. (2007). Stochastic Calculus, Filtering, and
Stochastic Control. Retrieved April 18, 2021 from: https://web.math.princeton.edu/~rvan/acm217/ACM217.pdf
[4] Cohen, S. & Elliot, R. (2015). Stochastic Calculus and Applications. Birkhauser.

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Stephanie Glen. "Stochastic Calculus: Overview" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/stochastic-calculus/
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