Integration by parts
We begin our journey by talking about a trick that could be seen as the reverse for the product rule for derivatives. We will use it to develop an entirely new way to solve problems by minimizing a function of functions. FInally, this will follow Wikipedia contributors (2023), which has the classical derivation, with some pieces added by me.
Sketch for deriving the integration-by-parts formula
We will use the
Now, let’s integrate on both sides:
If we rename
Finally, imagine that the functions vanish on the extremes
Calculus of variations
Malliavin Calculus is sometimes called an extension of calculus of variations to stochastic processes. What does that even means? Let’s deal with the first part.
Derivation and application
I’ll assume that you know the basics of calculus. That means you can solve problems like this: let’s say we have a function
To solve it, we take the derivative, make it equal to zero, and obtain the arguments that make it so.
Which means that
This approach works great when you try to minimize a function value over the reals,
But let’s say we want to find the function
From now on, it’s autopilot from any course (I’ll be using the Wikipedia article as the baseline). The first step is to define a “function of functions” as the starting point for minimization, and that function is essentially the integral like this:
To keep things grounded on reality, we will use the classical example of finding the function that represents the smallest distance between two points,
As little changes become infinitesimal, and with some extreme abuse of notation, we switch to an integral and obtain the expression for
Now, we will introduce a generic function
The trick here is that now, instead of minimizing something that depends on unknown functions, which we can’t even start to comprehend, we minimize something that depends on a number,
We take the total derivative of
This looks meaningless, but now comes the magic:
We start from
Whatever the solution function
Shortest Path Example
Why is it helpful? Let’s return to the shortest path between two points, and I’m using
We apply the Euler-Lagrange equation and we see where it leads us. What makes this example super clean is that there’s no explicit
If the derivative is zero, then we can integrate that and the result is an unknown constant
The function with the shortest path between two points is a straight line, and the only thing we needed to know is how the distance between two points is calculated to get a closed-form, analytic solution.