Research

Applied Mathematics

Mathematical questions, explicit assumptions and careful reasoning.

Overview

Stochastic calculus and numerical methods are the shared mathematical foundation the other research areas build on — a foundation, not a separate line of business, and one where the validity of every downstream result depends on it.

Core questions

Under what conditions does a stochastic differential equation have a well-defined solution?
How does a numerical scheme's error behave as its time step shrinks, and when does that behavior fail?

Mathematical formulation

Itô's lemma

df(X_t) = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma\frac{\partial f}{\partial x}\,dW_t

For a process dXₜ = μ dt + σ dWₜ and a twice-differentiable function f, Itô's lemma gives the differential of f(Xₜ) — the chain rule of stochastic calculus, and the basis of most continuous-time models used elsewhere in this research, from option pricing to optimal execution.

Methods we use

Stochastic integration
Itô, K. (1944). Stochastic integral. Proceedings of the Imperial Academy, 20(8), 519–524.
Self-exciting processes
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika, 58(1), 83–90.
Rough volatility
Gatheral, J., Jaisson, T., & Rosenbaum, M. (2018). Volatility is rough. Quantitative Finance, 18(6), 933–949.

Open problems

When do classical Itô-calculus results need revision for processes with rough, non-semimartingale paths?
How should numerical schemes be adapted for processes with jumps alongside continuous diffusion?

This page describes the field's established methods, not DaraHoosh's own results, parameters or current use of them.