# Geometric methods for dimensionality reductions of stochastic (partial) differential equations with applications to signal processing and finance

Lead Research Organisation:
Imperial College London

Department Name: Dept of Mathematics

### Abstract

The project aims to develop and generalize the classic dimensionality reduction theory for the filtering problem in signal processing and more generally for Stochastic (Partial) Differential Equations (S(P)DEs) developed initially in Brigo et al (1998, 1999) and Armstrong and Brigo (2016a, 2016b, 2016c). The methodology is based on projection of the equation coefficients on the tangent space of a chosen manifold or parametrization in a way that is optimal for a chosen metric, typically in mean square. Potential applications might allow to bypass the curse of dimensionality in many fields of engineering and include optimal approximation of SDEs, PDEs including the Fokker Planck Kolmogorov and heat equations, and SPDEs such as the above mentioned filtering problem. A classic example is attitude filtering in navigation, but many applications from medicine are appearing in the information geometry literature. The project also aims at formulating a general pathwise framework for dimensionality reduction based on rough paths theory. The project is at the intersection of three areas of mathematics: geometry of S(P)DEs, the variational approach to the Fokker Planck equation with information geometry, and rough paths theory. In the second phase of the project we will seek active cooperation with industry to obtain data for the implementation of the filtering algorithms. The project is in the EPSRC grow area of applied probability and statistics.

## People |
## ORCID iD |

Damiano Brigo (Primary Supervisor) | |

Claudio Bellani (Student) |

### Studentship Projects

Project Reference | Relationship | Related To | Start | End | Student Name |
---|---|---|---|---|---|

EP/N509486/1 | 01/10/2016 | 31/03/2022 | |||

1943803 | Studentship | EP/N509486/1 | 01/10/2017 | 31/03/2021 | Claudio Bellani |

Description | In my doctoral research, I focused on two tasks: option pricing and trade execution. These two are likely to bisect my thesis in two parts, the first presenting my work ``Option pricing models from a pathwise perspective'', the second presenting my works ``Mechanics of good trade execution in the framework of linear temporary market impact'' and ``Sectional cost of trade execution in the Glosten-Milgrom-Rosenbaum limit order book''. Corresponding chapters will articulate the thesis and their prospective contents are introduced below. In Chapter 1, I will describe how pricing and hedging European options can be formulated within a mathematical apparatus that refrains from using probability. I will encode volatility in an enhancement of the underlying price trajectory and I will give pathwise presentations of the fundamental equations of Mathematical Finance. In particular, this will allow me to assess model misspecification, generalising the so-called fundamental theorem of derivative trading. Moreover, pathwise integrals and equations will exhibit the role of Greeks beyond the leading-order Delta, in particular making explicit the role of Gamma sensitivities. In Chapter 2, I will define the concept of good trade execution and I will construct explicit adapted good trade execution strategies in the framework of linear temporary market impact. Good trade execution strategies are dynamic, in the sense that they react to the actual realisation of the traded asset price path over the trading period; this is paramount in volatile regimes, where price trajectories can considerably deviate from their expected value. Remarkably however, the implementation of these strategies does not require the full specification of an SDE evolution for the traded asset price, making them robust across different models. Moreover, rather than optimising the expected trading cost, good trade execution strategies optimise trading costs in a pathwise sense, a point of view not yet considered in the literature. The mathematical apparatus for such a pathwise optimisation hinges on certain random Young differential equations that correspond to the Euler-Lagrange equations of the classical Calculus of Variations. These Young differential equations characterise our good trade execution strategies in terms of an initial value problem that allows for easy implementations. In Chapter 3, I will define the execution cost of child market orders of a large parent order. This execution cost is sectional, in that it does not presuppose a time extension for the variables at play. Secondly, I will propose a model for the time evolution of limit order books based on the equilibrium configuration in Huang et al. 2019. Combining the sectional cost of execution and the dynamical LOB model allows on the one hand to design inventory trajectories without assuming a market impact function; on the other hand, it allows a proof of concept of the classical Almgren and Chriss inventory trajectory. The proof of concept refers to the idea of diluting the trade execution in time, and I will measure the gains that such a dilution yields by calibrating my models to the order books of AAPL and of INTC. Chapters 1 and 2 share the common theme of a pathwise assessment of model specifications, in the attempt to understand where probability plays a role and to which extent its usage can be reduced. This was based on my study of a probability-free approach to stochastic differential equations that can assist and complement the classical probabilistic one. The specification of a model is always affected by the technical mathematical apparatus that the modeller can use, so that expanding such an apparatus can unlock modelling possibilities. In particular, with the development of Rough Path Analysis by T. Lyons, M. Hairer, M. Gubinelli and P. Friz among others, the modeller's quiver is endowed with the new arrow of pathwise integration of unbounded variation signals, which I tried to employ in my models for option pricing and for trade execution. Chapter 3 instead will represent the most data-driven part of the thesis, the technical aspect of it consisting of Python implementations. The question about model specification that I intend to address in this chapter is on the significance of market impact functions employed in the models of optimal trade execution. The extrapolation of market impact functions from real world-data is problematic and contended in the literature, to the point that their own existence was questioned in Capponi et al. 2019. My approach to this conundrum was to take a step back from the investigation of their precise shape and, rather, to assess their role in the models of trade execution. In these models, their role is to give rise to optimal inventory trajectories that dilute the execution of a large market order over time. What I proposed to assess was the advantage of such a dilution, in a way that refrained from invoking market impact functions to justify the dilution. Such an assessment involved shifting the modelling from that of price evolution and impact function (the classical model formulation in optimal trade execution) to that of the limit order book where the execution takes place, hence increasing the granularity of the model. |

Exploitation Route | The models introduced in our work can be implemented in real trading, relying directly on the algorithms that we described. |

Sectors | Digital/Communication/Information Technologies (including Software),Financial Services, and Management Consultancy |

URL | https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3458454 |