# Analysis and control of path-dependent random systems

Lead Research Organisation:
Imperial College London

Department Name: Mathematics

### Abstract

The study of path-dependent functionals of random processes -quantities which depend on the path of a random process- is an important topic in probability and arises in various applications of probability, in mathematical physics, optimal control theory, population dynamics and mathematical finance. On the other hand, the vast majority of theoretical tools available for the simulation and analysis of stochastic systems focus on "Markovian systems" -random systems with no memory of the past- which are easier to analyse and simulate and for which a wide array of analytical results and tools are available.

The ambition of this project is to capitalize on recent developments in stochastic analysis -in particular the 'Functional Ito Calculus' developed by Rama Cont and collaborators in the recent years- to develop a mathematical framework for sensitivity analysis and optimal control of random systems with path-dependent features.

A key ingredient of this approach is a mathematical framework known as the "non-anticipative functional calculus", developed by Rama Cont and collaborators based on pioneering early work of Föllmer, which extends the classical Ito calculus to a large class of path-dependent functionals. This pathwise calculus, and its probabilistic counterpart, the Functional Ito Calculus, are currently the focus of a burgeoning literature in stochastic analysis, which is starting to be explored and which points to fruitful links with previously established mathematical concepts such as Rough Path theory, Forward-Backward Stochastic Differential Equations and Malliavin Calculus.

The Functional Ito calculus extends well-known relations between Markov processes and parabolic partial differential operators to the case of non-Markovian processes with path-dependent features, and leads to a new class of functional partial differential equations on the space of continuous functions, called 'path-dependent Kolmogorov equations'. The goal of this proposal is to undertake a systematic study of the Functional Ito Calculus and path-dependent Kolmogorov equations, their ramifications in analysis and probability theory and their applications to the sensitivity analysis and optimal control of systems with path-dependent features. We will explore the non-anticipative pathwise calculus and its relation with rough path theory, functional Ito calculus and its relation with the Malliavin calculus, functionals of discontinuous processes, various notions of solutions for path-dependent Kolmogorov equations, applications to non-Markovian stochastic control and Backward Stochastic Differential Equations, and application to Martingale Representation.

A key ambition of the proposed research is to extend the links between stochastic processes and partial differential equations beyond the classical Markovian setting, making a range of analytical tools, formerly restricted to the modelling of Markovian phenomena applicable to a wide range of (non-Markovian) stochastic processes which may arise in various types of application in physics, engineering and the life sciences. By expanding the applicability of these analytical tools to a wider array of problems, our research aims to make them available to a wider array of scientists, engineers and end-users for the purpose of simulation, sensitivity analysis, and optimal control of systems with path-dependent features.

These questions have numerous applications to the analysis, simulation and optimal control of random path-dependent systems in physics, mathematical finance and population dynamics. We intend in particular to explore in detail the computational aspects of the theory and develop open-source software tools for the simulation and analysis of stochastic systems with path-dependent features. Scientists and engineers working on optimal control of systems with memory, models of population dynamics, and mathematical finance are likely to be interested in/to be benefited from the research.

The ambition of this project is to capitalize on recent developments in stochastic analysis -in particular the 'Functional Ito Calculus' developed by Rama Cont and collaborators in the recent years- to develop a mathematical framework for sensitivity analysis and optimal control of random systems with path-dependent features.

A key ingredient of this approach is a mathematical framework known as the "non-anticipative functional calculus", developed by Rama Cont and collaborators based on pioneering early work of Föllmer, which extends the classical Ito calculus to a large class of path-dependent functionals. This pathwise calculus, and its probabilistic counterpart, the Functional Ito Calculus, are currently the focus of a burgeoning literature in stochastic analysis, which is starting to be explored and which points to fruitful links with previously established mathematical concepts such as Rough Path theory, Forward-Backward Stochastic Differential Equations and Malliavin Calculus.

The Functional Ito calculus extends well-known relations between Markov processes and parabolic partial differential operators to the case of non-Markovian processes with path-dependent features, and leads to a new class of functional partial differential equations on the space of continuous functions, called 'path-dependent Kolmogorov equations'. The goal of this proposal is to undertake a systematic study of the Functional Ito Calculus and path-dependent Kolmogorov equations, their ramifications in analysis and probability theory and their applications to the sensitivity analysis and optimal control of systems with path-dependent features. We will explore the non-anticipative pathwise calculus and its relation with rough path theory, functional Ito calculus and its relation with the Malliavin calculus, functionals of discontinuous processes, various notions of solutions for path-dependent Kolmogorov equations, applications to non-Markovian stochastic control and Backward Stochastic Differential Equations, and application to Martingale Representation.

A key ambition of the proposed research is to extend the links between stochastic processes and partial differential equations beyond the classical Markovian setting, making a range of analytical tools, formerly restricted to the modelling of Markovian phenomena applicable to a wide range of (non-Markovian) stochastic processes which may arise in various types of application in physics, engineering and the life sciences. By expanding the applicability of these analytical tools to a wider array of problems, our research aims to make them available to a wider array of scientists, engineers and end-users for the purpose of simulation, sensitivity analysis, and optimal control of systems with path-dependent features.

These questions have numerous applications to the analysis, simulation and optimal control of random path-dependent systems in physics, mathematical finance and population dynamics. We intend in particular to explore in detail the computational aspects of the theory and develop open-source software tools for the simulation and analysis of stochastic systems with path-dependent features. Scientists and engineers working on optimal control of systems with memory, models of population dynamics, and mathematical finance are likely to be interested in/to be benefited from the research.

### Organisations

### Publications

Chiu H
(2021)

*A Model-Free Approach to Continuous-Time Finance*in SSRN Electronic Journal
Chiu H
(2022)

*Causal functional calculus*in Transactions of the London Mathematical Society
Chiu H
(2023)

*A model-free approach to continuous-time finance*in Mathematical Finance
Chiu H
(2022)

*A model-free approach to continuous-time finance*
Chiu H
(2018)

*On pathwise quadratic variation for càdlàg functions*in Electronic Communications in Probability### Studentship Projects

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

EP/N509486/1 | 30/09/2016 | 30/03/2022 | |||

1824430 | Studentship | EP/N509486/1 | 30/09/2016 | 31/12/2020 | Henry Chiu |

Description | A non-probabilistic, generic framework for the analysis of path-dependent (causal) random systems in continuous-time. This framework may be applied to study non-Markovian control problems in physics, engineering, economics and finance. |

Exploitation Route | Better (not necessarily Markovian) solutions to control problems in physics, engineering, economics and finance. |

Sectors | Aerospace Defence and Marine Communities and Social Services/Policy Digital/Communication/Information Technologies (including Software) Education Electronics Energy Environment Financial Services and Management Consultancy Government Democracy and Justice Manufacturing including Industrial Biotechology Retail Security and Diplomacy Transport Other |

Description | 11th World Congress of Bachelier Finance Society |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Professional Practitioners |

Results and Impact | Present latest research. Outcome (publication) has been attributed to this grant. |

Year(s) Of Engagement Activity | 2022 |

URL | http://www.bacheliercongress.com/2022/~bfs2020/index.html |