December 2025 CDT Workshop
14:00 – 14:45: Mie Gluckstad, Oxford
Title: An introduction to mass erasure and invariance principles for Galton-Watson forests
Abstract: In 2002, Duquesne and Le Gall established an invariance principle for discrete Galton-Watson forests, characterizing their scaling limits as a class of continuum random trees called Lévy forests, in an analogous manner to how continuous-state branching processes are obtained as scaling limits of discrete Galton-Watson processes. Their invariance principle, however, relied on two assumptions: i) that the Galton-Watson forests are subcritical, and ii) that the so-called Grey’s condition is satisfied in the limit. The invariance principle has since been extended in work by Duquesne and Winkel (2019, 2025+) leaving open only the case where both assumptions i) and ii) fail. In this case the limiting trees may be both unbounded and not locally compact, and as such the classical Gromov-type topologies used for studying convergence of R-trees are not suitable. We develop instead a weaker notion of convergence by extending the technique of mass erasure to R-trees equipped with a suitable class of boundedly finite measures.
The talk is based on ongoing work, and its aim will be to introduce the invariance principles, the notion of mass erasure and some of the main ideas and results of the project. No prior knowledge is assumed, other than standard probability- and measure theory, and the first half of the talk will contain a general introduction to the topic.
14:45 – 15.30: Sturmius Tuschmann, Imperial
Title: A Fredholm Approach to Nonlinear Propagator Models
Abstract: We formulate and solve an optimal trading problem with alpha signals, where transactions induce nonlinear transient price impact described by a general propagator model. In particular, our formulation integrates the well-known square-root law of price impact with the empirical observation that impact decays according to a power law. Using a variational approach, we demonstrate that the optimal trading strategy satisfies a nonlinear stochastic Fredholm equation with both forward and backward coefficients. We prove the existence and uniqueness of the solution under a monotonicity condition reflecting the nonlinearity of the price impact. Moreover, we derive an existence result for the optimal strategy beyond this condition when the underlying probability space is countable. In addition, we introduce a novel iterative scheme and establish its convergence to the optimal trading strategy. Finally, we provide a numerical implementation of the scheme that illustrates its convergence, stability, and the effects of concavity on optimal execution under exponential and power-law decay. This is joint work with Eduardo Abi Jaber, Alessandro Bondi, Nathan De Carvalho, and Eyal Neuman.
15:30 – 16.00: Tea and coffee
16.00 – 16.45: Dr Purba Das, Kings College London
Title: Understanding roughness
Abstract: We study how to construct a stochastic process on a finite interval with given `roughness' (in terms of both Holder and p-th variation). We first extend Ciesielski's isomorphism along a general sequence of partitions and provide a characterization of Hölder regularity of a function in terms of its Schauder coefficients. Using this characterization, we provide a better (path wise) estimator of Hölder exponent. Furthermore, we study the concept of (generalized) p-th variation of a real-valued continuous function along a sequence of partitions. We show that the finiteness of the p-th variation of a given function is closely related to the finiteness of ℓp-norm of the coefficients along a Schauder basis. These characterise allow us to construct paths (not necessarily Gaussian) with a given Hölder and variation index. As an additional application, we construct fake (fractional) Brownian motions with some path properties, and the finite moments of marginal distributions are the same as (fractional) Brownian motions. These belong to non-Gaussian families of stochastic processes which are statistically difficult to distinguish from real (fractional) Brownian motions.
16.45 – 17.30: Prof Jingjie Zhang, China School of Banking and Finance at the University of International Business and Economics (UIBE)
Title: Stackelberg stopping games (Joint work with Zhou Zhou)
Abstract: We study a Stackelberg variant of the classical Dynkin game in discrete time, where the two players are no longer on equal footing. Player 1 (the leader) announces her stopping strategy first, and Player 2 (the follower) responds optimally. This Stackelberg stopping game can be viewed as an optimal control problem for the leader. Our primary focus is on the time-inconsistency that arises from the leader-follower game structure. We begin by using a finite-horizon example to clarify key concepts, including precommitment and equilibrium strategies in the Stackelberg setting, as well as the Nash equilibrium in the standard Dynkin game. We then turn to the infinite-horizon case and study randomized precommitment and equilibrium strategies. We provide a characterization for the leader's value induced by precommitment strategies and show that it may fail to attain the supremum. Moreover, we construct a counterexample to demonstrate that a randomized equilibrium strategy may not exist. Then we introduce an entropy-regularized Stackelberg stopping game, in which the follower's optimization is regularized with an entropy term. This modification yields a continuous best response and ensures the existence of a regular randomized equilibrium strategy, which can be viewed as an approximation of the exact equilibrium.
