arXiv subjects

Andrew L. Allan

Publications and source records attributed to Andrew L. Allan.

11 recordsLinked to original sources

arXiv2026

Rough SDEs and Robust Filtering for Jump-Diffusions

We investigate the existence of a robust, i.e., continuous, representation of the conditional distribution in a stochastic filtering model for multidimensional correlated jump-diffusions. Even in the absence of jumps, it is known that in general such a representation can only be continuous with respect to rough path topologies, leading us naturally to express the conditional dynamics as a rough stochastic differential equation with jumps. Via the analysis of such equations, including exponential moments, Skorokhod continuity, and randomisation of the rough path, we establish several novel robustness results for stochastic filters.

math.PR↗

arXiv2026

Rough Stochastic Analysis with Jumps

We present a new version of the stochastic sewing lemma, capable of handling multiple discontinuous control functions. This is then used to develop a theory of rough stochastic analysis in a càdlàg setting. In particular, we define rough stochastic integration and establish solutions to rough stochastic differential equations with jump discontinuities in both sources of noise, along with an Itô formula for stochastic controlled paths with jumps.

math.PR↗

arXiv2025

Pathwise analysis of log-optimal portfolios

Based on the theory of càdlàg rough paths, we develop a pathwise approach to analyze stability and approximation properties of portfolios along individual price trajectories generated by standard models of financial markets. As a prototypical example from portfolio theory, we study the log-optimal portfolio in a classical investment-consumption optimization problem on a frictionless financial market modelled by an Itô diffusion process. We identify a fully deterministic framework that enables a pathwise construction of the log-optimal portfolio, for which we then establish pathwise stability estimates with respect to the underlying model parameters. We also derive pathwise error estimates arising from the time-discretization of the log-optimal portfolio and its associated capital process.

q-fin.MF↗

arXiv2025

Pathwise convergence of the Euler scheme for rough and stochastic differential equations

The convergence of the first order Euler scheme and an approximative variant thereof, along with convergence rates, are established for rough differential equations driven by càdlàg paths satisfying a suitable criterion, namely the so-called Property (RIE), along time discretizations with vanishing mesh size. This property is then verified for almost all sample paths of Brownian motion, Itô processes, Lévy processes and general càdlàg semimartingales, as well as the driving signals of both mixed and rough stochastic differential equations, relative to various time discretizations. Consequently, we obtain pathwise convergence in p-variation of the Euler--Maruyama scheme for stochastic differential equations driven by these processes.

math.PR↗

arXiv2023

A Càdlàg Rough Path Foundation for Robust Finance

Using rough path theory, we provide a pathwise foundation for stochastic Itô integration, which covers most commonly applied trading strategies and mathematical models of financial markets, including those under Knightian uncertainty. To this end, we introduce the so-called Property (RIE) for càdlàg paths, which is shown to imply the existence of a càdlàg rough path and of quadratic variation in the sense of Föllmer. We prove that the corresponding rough integrals exist as limits of left-point Riemann sums along a suitable sequence of partitions. This allows one to treat integrands of non-gradient type, and gives access to the powerful stability estimates of rough path theory. Additionally, we verify that (path-dependent) functionally generated trading strategies and Cover's universal portfolio are admissible integrands, and that Property (RIE) is satisfied by both (Young) semimartingales and typical price paths.

math.PR↗

arXiv2022

Model-free Portfolio Theory: A Rough Path Approach

Based on a rough path foundation, we develop a model-free approach to stochastic portfolio theory (SPT). Our approach allows to handle significantly more general portfolios compared to previous model-free approaches based on F{ö}llmer integration. Without the assumption of any underlying probabilistic model, we prove a pathwise formula for the relative wealth process which reduces in the special case of functionally generated portfolios to a pathwise version of the so-called master formula of classical SPT. We show that the appropriately scaled asymptotic growth rate of a far reaching generalization of Cover's universal portfolio based on controlled paths coincides with that of the best retrospectively chosen portfolio within this class. We provide several novel results concerning rough integration, and highlight the advantages of the rough path approach by showing that (non-functionally generated) log-optimal portfolios in an ergodic It{ô} diffusion setting have the same asymptotic growth rate as Cover's universal portfolio and the best retrospectively chosen one.

math.PR↗

arXiv2021

Càdlàg Rough Differential Equations with Reflecting Barriers

We investigate rough differential equations with a time-dependent reflecting lower barrier, where both the driving (rough) path and the barrier itself may have jumps. Assuming the driving signals allow for Young integration, we provide existence, uniqueness and stability results. When the driving signal is a càdlàg $p$-rough path for $p \in [2,3)$, we establish existence to general reflected rough differential equations, as well as uniqueness in the one-dimensional case.

math.PR↗

arXiv2021

Robust Filtering and Propagation of Uncertainty in Hidden Markov Models

We consider the filtering of continuous-time finite-state hidden Markov models, where the rate and observation matrices depend on unknown time-dependent parameters, for which no prior or stochastic model is available. We quantify and analyze how the induced uncertainty may be propagated through time as we collect new observations, and used to simultaneously provide robust estimates of the hidden signal and to learn the unknown parameters, via techniques based on pathwise filtering and new results on the optimal control of rough differential equations.

math.PR↗

arXiv2019

Pathwise Stochastic Control with Applications to Robust Filtering

We study the problem of pathwise stochastic optimal control, where the optimization is performed for each fixed realisation of the driving noise, by phrasing the problem in terms of the optimal control of rough differential equations. We investigate the degeneracy phenomenon induced by directly controlling the coefficient of the noise term, and propose a simple procedure to resolve this degeneracy whilst retaining dynamic programming. As an application, we use pathwise stochastic control in the context of stochastic filtering to construct filters which are robust to parameter uncertainty, demonstrating an original application of rough path theory to statistics.

math.PR↗

arXiv2018

Parameter Uncertainty in the Kalman-Bucy Filter

In standard treatments of stochastic filtering one first has to estimate the values of the parameters of the model. Simply running the filter without considering the reliability of this estimate does not take into account this additional source of statistical uncertainty. We propose an approach to address this problem when working with the continuous-time Kalman-Bucy filter. We show how our approach may be reformulated as an optimal control problem, and proceed to analyse the corresponding value function in some detail. In particular we present a novel uniqueness result for the associated Hamilton-Jacobi-Bellman equation.

math.PR↗

arXiv2015

Ergodic Backward Stochastic Difference Equations

We consider ergodic backward stochastic differential equations in a discrete time setting, where noise is generated by a finite state Markov chain. We show existence and uniqueness of solutions, along with a comparison theorem. To obtain this result, we use a Nummelin splitting argument to obtain ergodicity estimates for a discrete time Markov chain which hold uniformly under suitable perturbations of its transition matrix. We conclude with an application of this theory to a treatment of an ergodic control problem.

math.PR↗