Do you need to know stochastic processes to learn stochastic calculus?

Question

I have been researching topics that I can study next year for an independent reading course, and stochastic calculus seems very interesting. i have not studied stochastic processes at all, and from my research online I have found that the two subjects have very little overlap. I don't know how true that is, and I was wondering if its possible to study stochastic calculus without knowing stochastic processes?

Answer 1

I think it depends in what context you want to study stochastic calculus.

At the core of it, it's impossible to study stochastic calculus without having some intuition for stochastic processes. After all, stochastic calculus is used to evaluate differentials of stochastic processes.

However, it is possible to memorize Ito's lemma and "derive" it from intuition based on squared variation and Taylor expansion. No knowledge of stochastic processes is required to understand that if:

$$dX = \mu dt + \sigma dZ$$

Then:

$$dF \sim \frac{\partial F}{\partial t}dt + \frac{1}{2}\frac{\partial F}{\partial x}dX + \frac{\partial^2 F}{\partial X^2}(dX)^2$$ $$= \frac{\partial F}{\partial t}dt + \frac{\partial F}{\partial x}(\mu dt + \sigma dZ) + \frac{1}{2}\frac{\partial^2 F}{\partial X^2} (\mu^2 dt^2 + 2 \sigma\mu dtdZ + \sigma^2 dZ^2)$$

Now, here's where Ito comes in:

$dt^2 = 0$, $dZdt = 0$, $dZ^2 = dt$ And all higher order terms go to $0$

so,

$$dF = \frac{\partial F}{\partial t}dt + \frac{\partial F}{\partial x}(\mu dt + \sigma dZ) + \frac{1}{2}\frac{\partial^2 F}{\partial X^2} (\sigma^2 dt)$$

That's it! No stochastic processes needed. You can go about solving your problem by subbing in the above expression for $dF$ instead of the standard:

$$dF = \frac{\partial F}{\partial t}dt + \frac{\partial F}{\partial Z}dZ$$

But, what kind of understanding does that build? For example, why does $E[dZ^2] = dt^2$?

It has to do with the fact that a random walk, of which brownian motion is a limiting case, has variance $\sim N$ after $N$ steps.

Stochastic calculus also makes heavy use of martingales. A martingale is a stochastic process where:

$E[X_{t} | X_s] = X_s$ for $s<t$.

$E[X^2 - t^2]$ is a martingale for brownian motion. That often makes solving problems a lot easier.

The Feynman Kac formula is another example of a concept in stochastic calculus rooted in stochastic processes. You can tilt your head to the side and say it looks like a green's function, but it's derivation is straight forward if you understand random walks and diffusion.

In summary, Stochastic differential equations describe stochastic processes that occur in continuous time. The two subjects have tremendous overlap. Diffusion is a stochastic process; jumps follow a poisson process. Moreover, if you want to actually use stochastic calculus, you'll have to have some intuition for stochastic processes. It's impossible to formulate a stochastic optimization problem or predict a most likely path if you can't describe the underlying process.

In order to get something out of your independent study, you'll have to at least familiarize yourself with probability, random walks, markov processes, and poisson processes. You don't need measure theory, though it is helpful. If you choose to start without it, as you go deeper into stochastic calculus, you'll get a significant amount of measure theory through probability theory and intuition for its role in stochastic processes. I'd recommend getting familiar with the above — it won't take more than a few weeks if you're at a high enough level — and then diving in. Learning a little about stochastic processes will payoff a hundred fold in the end and will help you in other subjects should you choose to pursue them.

Great resources for stochastic processes:

1. http://www.columbia.edu/~ww2040/3106F13/lectures3106.html
2. Anything by Sheldon Ross
3. https://www.amazon.com/Elementary-Probability-David-Stirzaker/dp/0521534283