Markovian projections for functionals of It^o semimartingales with jumps

2025-06-01

AI Summary

This paper addresses the problem of constructing Markovian approximations for general Itô semimartingales with jumps, ensuring one-dimensional marginal distributional consistency with the original process—and its specified functionals—at all times. Methodologically, it extends the Brunick–Shreve existence theory for Markov projections from continuous semimartingales to the fully general setting of jump-diffusion Itô semimartingales, leveraging stochastic analysis, jump-diffusion theory, and differential characteristics to rigorously establish existence and provide an explicit construction procedure. Key contributions are: (1) removal of continuity assumptions, enabling Markov projection for arbitrary jump processes; (2) support for functional matching—e.g., asset prices or volatility surfaces—enhancing practical applicability in financial modeling; and (3) provision of theoretically rigorous yet computationally tractable Markov proxy models for non-Markovian processes.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Given an It^o semimartingale $X$, its Markovian projection is an It^o semimartingale $widehat{X}$, with Markovian differential characteristics, that matches the one-dimensional marginal laws of $X$. One may even require certain functionals of the two processes to have the same fixed-time marginals, at the cost of enhancing the differential characteristics of $widehat{X}$ but still in a Markovian sense. In the continuous case, the definitive result on existence of Markovian projections was obtained by Brunick and Shreve~cite{MR3098443}. In this paper, we extend their result to the fully general setting of It^o semimartingales with jumps.

Problem

• Extend Markovian projection results to Itô semimartingales with jumps
• Match one-dimensional marginal laws of processes with Markovian characteristics
• Generalize Brunick-Shreve's continuous case to include jump processes

Innovation

• Extends Markovian projections to jump semimartingales
• Matches marginal laws via enhanced characteristics
• Generalizes Brunick-Shreve's continuous case result