Stochastic Differential Equations, Market Impact and Tutte polynomial

Post date: Dec 12, 2015 9:54:40 AM

Recently it came out a third paper of a series of investigations of the differential equation:

(1)

which appears in the study of optimal leverage solutions in financial markets with (linear) market impact:

Optimal leverage paper:

http://arxiv.org/abs/1510.05123

(with Sindoni, Caccioli and Ududec).

One of the first things to note is that in order to evaluate these optimal solutions, one can use the perturbation theory of the solution of equation (1), which turns out being a logistic differential equation in which one needs to solve for the moments of the integrated exponential Brownian motion:

(2)

and which in turn happen to be "easy" (it can be solved with bunch of trickeries), and which appeared somehow before the first paper:

Moments of Exponential Brownian Motion:

http://arxiv.org/abs/1509.05980

(with Mansour, Sindoni, Severini).

More recently, we understood that the mathematical structure occurring in the evaluation of eqn. (3) can be casted into the formalism

of Tutte polynomials. We thus introduced the mathematical framework to study these polynomials on generic graphs:

generalized Tutte polynomials:

http://arxiv.org/abs/1512.02278

(with Ben Geloun).

More to come? We'll see where this brings us.