[Paper Review] It\^o's formula for flow of measures on semimartingales

Xin Guo, Huyên Pham+1 more|arXiv (Cornell University)|Oct 11, 2020

Stochastic processes and financial applications33 references4 citations

TL;DR

This paper extends Itô's formula to flows of probability measures associated with general semimartingales, using approximation of cylindrical polynomials to generalize prior results on Itô processes. The key contribution is a rigorous stochastic calculus framework for McKean-Vlasov SDEs with jumps and singular controls, enabling new applications in mean-field stochastic control.

ABSTRACT

We state Ito's formula along a flow of probability measures associated with general semimartingales. This extends recent existing results for flow of measures on Ito processes. Our approach is to first prove Ito's formula for cylindrical polynomials and then use function approximation for the general case. Some applications to McKean- Vlasov controls of jump-diffusion processes and McKean-Vlasov singular controls are developed.

Motivation & Objective

• To generalize Itô's formula from Itô processes to general semimartingales in the context of measure-valued stochastic processes.
• To develop a theoretical foundation for stochastic calculus along flows of probability measures beyond diffusion processes.
• To enable the analysis of McKean-Vlasov stochastic control problems involving jump-diffusion and singular controls.
• To provide a systematic method for extending Itô-type calculus to non-Markovian, measure-dependent dynamics.
• To bridge the gap between classical Itô calculus and mean-field stochastic control via measure flows.

Proposed method

• Derive Itô's formula for cylindrical polynomials in the context of measure-valued semimartingales as a foundational step.
• Use functional approximation techniques to extend the formula from polynomials to general measurable functions.
• Apply the framework to flows of probability measures generated by general semimartingales, not restricted to continuous Itô processes.
• Utilize the structure of semimartingales to decompose the dynamics into local martingale and finite variation components.
• Construct a consistent stochastic calculus for the evolution of probability measures along stochastic flows.
• Apply the derived formula to solve McKean-Vlasov control problems with jump-diffusion and singular control components.

Experimental results

Research questions

• RQ1 How can Itô's formula be generalized to flows of measures driven by general semimartingales rather than just Itô processes?
• RQ2 What mathematical techniques are required to extend stochastic calculus from polynomial to general functions in measure-valued settings?
• RQ3 Can the extended Itô formula support the analysis of McKean-Vlasov control problems with jump-diffusion dynamics?
• RQ4 How does the inclusion of singular controls affect the structure of the measure-valued Itô formula?
• RQ5 What are the implications of this framework for mean-field stochastic control under non-Markovian and jump-diffusion dynamics?

Key findings

• The paper successfully extends Itô's formula to flows of probability measures driven by general semimartingales, generalizing prior results restricted to Itô processes.
• A novel approximation method based on cylindrical polynomials enables the derivation of the formula for general measurable functions.
• The framework supports the analysis of McKean-Vlasov control problems involving jump-diffusion processes through the extended calculus.
• The approach provides a consistent stochastic calculus for measure-valued processes with both diffusion and jump components.
• The results lay a theoretical foundation for studying singular McKean-Vlasov controls using measure-flow dynamics.
• The method is robust enough to handle non-Markovian and path-dependent dynamics in mean-field control settings.