Mathematical finance and market microstructure

How do the trades of many small agents aggregate into a market price? A limit order book records the standing buy and sell orders at every price level; arbitrage between venues forces the aggregate of two limit order books to behave like a single, larger one. Inspired by thermodynamics, this aggregation procedure carries a market-dynamical entropy: a quantitative measure of the loss of liquidity caused by combining venues. With Moritz Voss we developed this picture and worked it out explicitly for Uniswap v2-style automated market makers used in decentralized exchanges.

A complementary question is how a single execution moves the price. In linear propagator models, a trade at time \(s\) shifts the mid-price at time \(t\) by \(K(t-s)\) for some impact kernel \(K\). With Tingwei Meng, Moritz Voss, Nils Detering, Giulio Farolfi, and Stanley Osher we use in-context operator learning — a transformer architecture trained on simulated price-impact trajectories — to recover the kernel from a few observed trades and then optimize execution against it.

I am interested in the rigorous side of market aggregation, in extending propagator-style impact models to settings with cross-asset interaction and nonlinear impact, and in data-driven operator learning that relaxes the linearity and stationarity assumptions standard in the optimal-execution literature.

Further reading.

• Aggregation of financial markets, with M. Voss. arXiv:2309.04116. Frontiers of Mathematical Finance 7 (2025), 30–77.
• In-Context Operator Learning for Linear Propagator Models, with T. Meng, M. Voss, N. Detering, G. Farolfi, S. Osher. arXiv:2501.15106.