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Andrew Allan


Last Name

Allan

First Name

Andrew

ORCID

0000-0003-3324-6237

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2020 - 20216
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Publications 1 - 6 of 6
  • Càdlàg Rough Differential Equations with Reflecting Barriers
    Item type: Working Paper
    Allan, Andrew; Liu, Chong; Prömel, David J. (2020)
    arXiv
    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∈[2,3), we establish existence to general reflected rough differential equations, as well as uniqueness in the one-dimensional case.
  • Model-free Portfolio Theory: A Rough Path Approach
    Item type: Working Paper
    Allan, Andrew; Cuchiero, Christa; Liu, Chong; et al. (2021)
    arXiv
    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 Föllmer integration. Without the assumption of any underlying probabilistic model, we prove pathwise Master formulae analogous to those of classical SPT, describing the growth of wealth processes associated to functionally generated portfolios relative to the market portfolio. 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 considering (non-functionally generated) log-optimal portfolios in an ergodic Itô diffusion setting.
  • Robust filtering and propagation of uncertainty in hidden Markov models
    Item type: Journal Article
    Allan, Andrew (2021)
    Electronic Journal of Probability
    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.
  • Càdlàg rough differential equations with reflecting barriers
    Item type: Journal Article
    Allan, Andrew; Liu, Chong; Prömel, David J. (2021)
    Stochastic Processes and their Applications
    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∈[2,3), we establish existence to general reflected rough differential equations, as well as uniqueness in the one-dimensional case.
  • Robust Filtering and Propagation of Uncertainty in Hidden Markov Models
    Item type: Working Paper
    Allan, Andrew (2020)
    arXiv
    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.
  • A Càdlàg Rough Path Foundation for Robust Finance
    Item type: Working Paper
    Allan, Andrew; Liu, Chong; Prömel, David J. (2021)
    arXiv
    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.
Publications 1 - 6 of 6