Drew Williamson

2022-10-09

Derivatives and Integrals of Stochastic Processes?
I am trying to wrap my head around this idea and am trying to understand why this might be useful. For instance, suppose I have some stochastic process like the Brownian Motion. Why might I be interested in knowing "how quickly the Brownian Motion might change" at some point (i.e. the derivative) and the "area that the Brownian Motion might cover over two time intervals" (i.e. the integral)?
I understand this is more complicated than evaluating derivatives and integrals on deterministic functions. In a deterministic function, you only need to take one derivative and one integral to answer your question. On the other hand, each time I simulate a Stochastic Process on a computer, each realization of this Stochastic Process will look different. Thus, it is likely more challenging to take the integral and derivative of a function that can take many forms, compared to a function that can only take a single form.
But this point aside - why is it useful to know the derivative and the integral of a stochastic process? What do I gain from knowing this - how might I be able to apply this information to some real world application? For example, in financial models such as the Black-Scholes Model, are we using Stochastic Calculus to infer the amount of "risk" or "volatility" (i.e. the area under the stochastic process) that the stochastic process corresponds to over some period of time?

Haylie Campbell

Step 1
We can give a somewhat geometric interpretation for the stochastic integral but in probability this is not what is important. Besides, this intepretation is lost in some sense when we take the ${L}^{2}$-limit used in the definition of the integral.
Note also that for stochastic processes the derivative is not the usual one: you need Ito's lemma to compute the derivative because, as it turns out, we have an additional term taking into account the fact that we have an additional contribution to the drift given by "$\left(d{W}_{t}{\right)}^{2}=dt$". Another problem you have is that Browninan Motion is nowhere differentiable and is of infinite first variation and so defining a concept analogous to the classic one is not feasible. So much so, that even the common writing $dX=\mu dt+\sigma d{W}_{t}$ is just a short-writing and has to be understood in the integral sense:
${X}_{t}={X}_{0}+\int \mu dt+\int \sigma d{W}_{t}$
with all the necessary hypothesis.
Step 2
Stochastic integration was introduced to study SDEs (the analogous of ODEs) and these have countless applications: finance, physics, biology, etc.
Taking your example of BS equation: we use SDE and calculus to model the dynamics of risky assets that are driven by a random component given as a Browninan Motion. For instance in the BS framework the risky assets is said to follow a Geometric Brownian Motion which is the process that is the solution to the following SDE: $d{S}_{t}=\mu {S}_{t}dt+\sigma {S}_{t}d{W}_{t}$. Why? Because it has been seen that the stock price resamble this process. But other choices are possible and each choice corresponds to a different SDE.

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