[Paper Review] Itô's formula for flows of measures on semimartingales

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

Stochastic processes and financial applications4 citations

TL;DR

This paper establishes a general form of Itô's formula for flows of probability measures driven by general semimartingales, including jump diffusions, by first deriving it for cylindrical functions and extending via approximation and localization. The key contribution is enabling dynamic programming principles and verification theorems for McKean–Vlasov control problems with jumps and mixed regular-singular controls, including a generalized relationship between the maximum and dynamic programming principles via measure derivatives.

ABSTRACT

We establish Itô's formula along flows of probability measures associated with general semimartingales; this generalizes existing results for flows of measures on Itô processes. Our approach is to first establish Itô's formula for cylindrical functions and then extend it to the general case via function approximation and localization techniques. This general form of Itô's formula enables the derivation of dynamic programming equations and verification theorems for McKean--Vlasov controls with jump diffusions and for McKean--Vlasov mixed regular-singular control problems. It also allows for generalizing the classical relationship between the maximum principle and the dynamic programming principle to the McKean--Vlasov singular control setting, where the adjoint process is expressed in terms of the derivative of the value function with respect to the probability measures.

Motivation & Objective

• To develop a general Itô's formula for flows of probability measures along general semimartingales, including those with jumps, extending beyond continuous Itô processes.
• To overcome the lack of a general Itô formula for measure-valued processes driven by discontinuous semimartingales, which has hindered dynamic programming approaches in McKean–Vlasov control.
• To enable the derivation of dynamic programming equations and verification theorems for McKean–Vlasov control problems involving jump diffusions and mixed regular-singular controls.
• To generalize the classical link between the stochastic maximum principle and dynamic programming principle to the McKean–Vlasov singular control setting, where the adjoint process is expressed via derivatives of the value function with respect to measures.

Proposed method

• Establish Itô's formula first for cylindrical functions, which are smooth mean-field functions with integrable forms, using properties of linear derivatives on the space of probability measures.
• Use a generalized Stone–Weierstrass theorem on compact sets in the joint Wasserstein and Euclidean space to show that cylindrical functions are dense in the space of twice continuously differentiable functions with C¹,¹ topology.
• Apply localization techniques to extend the Itô formula from cylindrical functions to general functions, ensuring the formula holds under appropriate integrability and regularity conditions.
• Employ linear derivatives on the space of probability measures to capture the infinitesimal behavior of both jumps in the measure flow and jumps in the underlying semimartingale processes.
• Use the structure of McKean–Vlasov SDEs with jump diffusions and singular controls to derive the associated adjoint equations and verify optimality via the derived dynamic programming principle.
• Formally identify the FBSDE system from the maximum principle with that of an auxiliary mean-field game with singular controls, establishing a connection between MFG and McKean–Vlasov control frameworks.

Experimental results

Research questions

• RQ1 Can Itô’s formula be generalized to flows of probability measures driven by general semimartingales, including those with jumps, beyond the classical Itô process setting?
• RQ2 How can dynamic programming equations and verification theorems be derived for McKean–Vlasov control problems with jump diffusions and mixed regular-singular controls?
• RQ3 What is the precise relationship between the stochastic maximum principle and the dynamic programming principle in the context of McKean–Vlasov singular control problems?
• RQ4 How can the adjoint process in the maximum principle be expressed in terms of the derivative of the value function with respect to the probability measure along the optimal path?

Key findings

• A general Itô’s formula is established for flows of probability measures along general semimartingales, including jump diffusions, by first proving it for cylindrical functions and extending via approximation and localization.
• The derived Itô formula enables the derivation of dynamic programming equations and verification theorems for McKean–Vlasov control problems with jump diffusions and mixed regular-singular controls.
• The classical relationship between the maximum principle and dynamic programming principle is generalized to the McKean–Vlasov singular control setting, where the adjoint process is expressed as the derivative of the value function with respect to the measure along the optimal path.
• In a one-dimensional mean-variance singular control example, the optimal control is explicitly derived as a feedback form involving the state and its expectation, with the control switching between active and inactive regions based on a threshold condition.
• The controlled process follows a Skorokhod-type SDE with a singular control that reflects the process at a boundary, and the existence and uniqueness of the singular control are guaranteed by solving a one-dimensional Skorokhod problem.
• The value function is shown to be of the form V(t, μ) = A(t)Var(μ) + B(t)μ̄² + C(t)μ̄ + D(t), with explicit time-dependent coefficients satisfying a system of ODEs that are analytically solvable.