arXiv subjects

Graeme Baker

Publications and source records attributed to Graeme Baker.

8 recordsLinked to original sources

arXiv2026

Traveling Waves in Equity Markets with Rank-Based Entry and Exit

We model equity markets using geometric Brownian particles entering and exiting at rank-dependent intensities. In the many-firm limit, the capital distribution converges to the solution of a reaction-diffusion equation with reaction term built from the intensities. Calibrated on CRSP data, the reaction term is bistable, and the long-run distribution is a traveling wave: we prove existence, uniqueness, and, for constant coefficients, exponential relaxation. Turnover, not drift, stabilizes the calibrated market. With measured volatility, the wave tracks the empirical capital distribution in every decade, determines the capitalization growth of diversity-weighted portfolios, and places the market just inside the boundary of the diverse phase. Turnover reclaims most of what rebalancing gains.

math.PR↗

arXiv2026

Multi-Credit Calibration via Elastically Stopped L\'{e}vy Processes

We calibrate credit default swaps and index tranches with elastically stopped L\'evy processes: each firm defaults when the running supremum of a latent, spectrally positive distress process crosses an independent exponential barrier. This yields a Cox construction with totally inaccessible default times, while retaining the interpretability and explicit formulas of a structural approach. Adding a single common compound Poisson jump factor to every firm's latent driver gives a parsimonious multi-credit model with simultaneous defaults, which is priced by an exact Wiener--Hopf Monte Carlo scheme. Its tractability rests on a single-name result we prove: a finite partial-fraction formula for the Laplace transform of the default probability under phase-type jumps. On daily CDX North American High-Yield and Investment-Grade panels, our drivers attain the lowest out-of-sample errors in a six-model field and reproduce the inverted spread curves of names heading into default, which a L\'evy subordinator provably does not. At the index level, the two-parameter dependence structure closes $73\%$ to $89\%$ of the tranche pricing gap left by independent marginals with the dependence parameters frozen, and up to $95\%$ once re-marked to tranche quotes; our framework dominates a single-factor Gaussian copula and the affine intensity benchmark of Duffie--G\^arleanu on both indices.

q-fin.MF↗

arXiv2025

Minimal Solutions to the Skorokhod Reflection Problem Driven by Jump Processes and an Application to Reinsurance

We consider a reflected process in the positive orthant driven by an exogenous jump process. For a given input process, we show that there exists a unique minimal strong solution to the given particle system up until a certain maximal stopping time, which is stated explicitly in terms of the dual formulation of a linear programming problem associated with the state of the system. We apply this model to study the ruin time of interconnected insurance firms, where the stopping time can be interpreted as the failure time of a reinsurance agreement between the firms. Our work extends the analysis of the particle system in Baker, Hambly, and Jettkant (2025) to the case of jump driving processes, and the existence result of Reiman (1984) beyond the case of sub-stochastic reflection matrices.

math.PR↗

arXiv2025

Data-Driven Dynamic Factor Modeling via Manifold Learning

We introduce a data-driven dynamic factor framework for modeling the joint evolution of high-dimensional covariates and responses without parametric assumptions. Standard factor models applied to covariates alone often lose explanatory power for responses. Our approach uses anisotropic diffusion maps, a manifold learning technique, to learn low-dimensional embeddings that preserve both the intrinsic geometry of the covariates and the predictive relationship with responses. For time series arising from Langevin diffusions in Euclidean space, we show that the associated graph Laplacian converges to the generator of the underlying diffusion. We further establish a bound on the approximation error between the diffusion map coordinates and linear diffusion processes, and we show that ergodic averages in the embedding space converge under standard spectral assumptions. These results justify using Kalman filtering in diffusion-map coordinates for predicting joint covariate-response evolution. We apply this methodology to equity-portfolio stress testing using macroeconomic and financial variables from Federal Reserve supervisory scenarios, achieving mean absolute error improvements of up to 55% over classical scenario analysis and 39% over principal component analysis benchmarks.

stat.ML↗

arXiv2025

Particle Systems and McKean--Vlasov Dynamics with Singular Interaction through Local Times

We study a system of reflected Brownian motions on the positive half-line in which each particle has a drift toward the origin determined by the local times at the origin of all the particles. If this local time drift is too strong, such systems exhibit a breakdown in their solutions in that there is a time beyond which the system cannot be extended. In the finite particle case we give a complete characterisation of this finite time breakdown, relying on a novel dynamic graph structure. We consider the mean-field limit of the system in the symmetric setting, which admits a McKean--Vlasov representation, and establish propagation of chaos. In the absence of breakdowns, the McKean--Vlasov equation exhibits multiple stationary and unique self-similar solutions and we prove convergence to these profiles. This work is motivated by models for liquidity in financial markets, the supercooled Stefan problem, and a toy model for cell polarisation.

math.PR↗

arXiv2023

Sensitivity to Initial Data for Physical and Minimal Solutions of the Supercooled Stefan Problem

We address the problem of well-posedness for physical and minimal solutions to a probabilistic reformulation of the supercooled Stefan problem by investigating the sensitivity of these solutions to changes in the initial data. We show that the solution map for physical solutions is continuous under perturbations to the initial condition $X_{0-}$ (in the weak sense of probability measures), provided that $X_{0-}$ admits a unique physical solution. Furthermore, we show continuous dependency of the solution map for minimal solutions when the data is shifted to the right; however, continuity of this solution map is shown to fail at $X_{0-}$ for shifts to the left unless uniqueness of physical solutions holds. As a result, we show that the question of whether the solution map for minimal solutions is continuous at $X_{0-}$ is equivalent to the question of whether the given data $X_{0-}$ admits a unique physical solution.

math.PR↗

arXiv2022

A singular two-phase Stefan problem and particles interacting through their hitting times

We consider a probabilistic formulation of a singular two-phase Stefan problem in one space dimension, which amounts to a coupled system of two McKean-Vlasov stochastic differential equations. In the financial context of systemic risk, this system models two competing regions with a large number of interconnected banks or firms at risk of default. Our main result shows the existence of a solution whose discontinuities obey the natural physicality condition for the problem at hand. Thus, this work extends the recent series of existence results for singular one-phase Stefan problems in one space dimension that can be found in [DIRT15a], [NS19a], [HLS18], [CRS20]. As therein, our existence result is obtained via a large system limit of a finite particle system approximation in the Skorokhod M1 topology. But, unlike for the previously studied one-phase case, the free boundary herein is not monotone, so that the large system limit is obtained by a novel argument.

math.PR↗

arXiv2020

Zero kinetic undercooling limit in the supercooled Stefan problem

We study the solutions of the one-phase supercooled Stefan problem with kinetic undercooling, which describes the freezing of a supercooled liquid, in one spatial dimension. Assuming that the initial temperature lies between the equilibrium freezing point and the characteristic invariant temperature throughout the liquid our main theorem shows that, as the kinetic undercooling parameter tends to zero, the free boundary converges to the (possibly irregular) free boundary in the supercooled Stefan problem without kinetic undercooling, whose uniqueness has been recently established in [DNS19], [LS18b]. The key tools in the proof are a Feynman-Kac formula, which expresses the free boundary in the problem with kinetic undercooling through a local time of a reflected process, and a resulting comparison principle for the free boundaries with different kinetic undercooling parameters.

math.PR↗