Category Archives: Mathematics

MathJax

This site uses MathJax as implemented in the wordpress plugin MathJax-LaTeX.

Here is an example of how MathJax renders mathematics:

Given \((\mu,\sigma)\) and a setup \((\Omega,\mathcal{F},P,F,W)\), a solution on \((\Omega,\mathcal{F},P,F,W)\) of the SDE \((\mu,\sigma)\) is an \(N\) dimensional Itô process \(X\) on \((\Omega,\mathcal{F},P,F,W)\) with a potentially random initial value \(X(0)\), such that the process \(\mu(X,t)\) belongs to \(\mathcal{L}^{1}\), the process \(\sigma(X,t)\) belongs to \(\mathcal{L}^{2}\), and for all \(t \in [0,\infty)\),
$$X(t) = X(0) + \int_{0}^{t} \mu(X,s) \, ds + \int_{0}^{t} \sigma(X,s) \, dW$$

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Disavowing Lebesgue

No, you cannot disavow Lebesgue.

Lebesgue integration (and measure theory) are indispensable in stochastic calculus, especially for finance applications.

Take for example the martingale representation theorem, which is central to the theory of dynamic hedging and replication. This version is essentially Rogers and Williams (1987), Theorem 36.5:

Theorem
The martingale representation theorem
Let \(W\) be a standard Brownian motion of dimension \(K\). If \(X\) is a martingale with respect to the augmented filtration generated by \(W\), then there exists a process \(b \in \mathcal{L}^{2}\) such that
$$X(t) -X(0) = \int_{0}^{t} b \, dW$$ (in the sense that the two processes are indistinguishable).

But what does it mean to say that the process \(b\) is in \(\mathcal{L}^{2}\)? It means that \(b\) is measurable and adapted and pathwise square integrable on bounded intervals. In other words,
$$\int_{0}^{t} \|b\|^{2} \, ds < \infty $$ with probability one, for all \(t \in [0,\infty)\). Or, to write it out in detail, if the underlying probability space is \((\Omega,\mathcal{F},P)\), then the requirement is that
$$\int_{0}^{t} \|b(\omega,s)\|^{2} \, ds < \infty $$ for \(P\)-almost all \(\omega \in \Omega\), for all \(t \in [0,\infty)\).

For each \(\omega\), the time integral is a Lebesgue integral. The martingale representation theorem does not in any way guarantee that the function \(\|b\|^{2}\) is pathwise Riemann integrable. How could it?

So the statement of the martingale representation theorem requires Lebesgue integration and measure theory.

Mathematical Genealogy

My Ph.D. is in economics, not in mathematics, but I did write a master’s thesis in differential topology, some of which was published in: Transversality and the Inverse Image of a Submanifold with Corners. Mathematica Scandinavica 49 (1981), 211-221.

My thesis advisor was Vagn Lundsgaard Hansen, then at the University of Copenhagen, now at the Technical University of Denmark. Through him, I trace my mathematical ancestry back to Birkhoff, and possibly all the way back to Poisson, Lagrange, Euler, Johan and Jacob Bernoulli, and, finally, Leibnitz.

Apart from being my advisor, Lundsgaard Hansen was also my teacher in first-year undergraduate mathematical analysis. The course started with an unforgettably vivid exposition of the topology of metric spaces — pulling back open and closed sets and mapping compact sets forward and so on. Later on I took his course in algebraic topology, where the main agenda was to use category theory to translate topological questions into easier algebraic ones.

Category theory was apparently invented by Samuel Eilenberg together with Saunders Mac Lane.

Samuel Eilenberg

Samuel Eilenberg

This bust of Samuel Eilenberg stands in the corner of the Faculty Lounge in the Columbia Mathematics Department. Eilenberg was a professor in the department from 1947 and until his retirement in 1982.