Qijin Shi

Abstract

Rough-path calculus provides an alternative way to make sense of the trading gains in a market, yielding pathwise interpretations for classical topics in financial mathematics, including pricing, gamma-hedging, and portfolio theory. Remarkably, it generalises the class of analytically tractable market noises far beyond the classical Itô-semimartingale paradigm. On the other hand, a fundamental market modelling constraint is No-Arbitrage. This raises a key question: does the rich family of rough path calculi support an arbitrage-free market beyond the classical equivalent-local-martingale paradigms?

In this talk, we provide a negative answer to this question within our framework. We formulate „No Controlled Free Lunch“ (NCFL) and develop a Kreps-Yan-type theorem, linking NCFL to the unbiasedness of the underlying price rough path lift, as a rough integrator. We then classify such (1-dim) unbiased rough integrators with respect to two natural classes of test integrands/portfolios. The upshot is a strong rigidity statement: once the admissible trading strategies are reasonably rich, the only random rough paths that can support an NCFL market—under our assumptions—are (extremely close to) the Itô lift of a standard Brownian motion, up to a time change and an equivalent change of measure. Our framework covers continuous non-geometric rough paths in the tensor algebra for α∈ (0,1] arbitrarily small.

Our speaker

Qijin Shi is a third-year PhD student in the Department of Statistics and Applied Probability at the University of California, Santa Barbara, advised by Prof. Tomoyuki Ichiba. He received an M.Sc. in Mathematics from the University of Bonn and a B.Sc. from the University of Heidelberg. His current research lies at the intersection of rough paths, stochastic analysis, and mathematical finance. Beyond that, he is also interested in algebraic renormalisation, signature methods, and operations research.

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