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Arcade Processes for Informed Martingale Interpolation

Georges Kassis, Andrea Macrina

TL;DR

This work introduces arcade processes (APs) and their randomized extensions (RAPs) to construct continuous-time martingale interpolants that match a finite sequence of convexly ordered targets almost surely. Filtration-based filtered arcade martingales (FAMs) are then built as $M_t = \mathbb{E}[X_n \mid \mathcal{F}_t^I]$, with Bayesian updating enabled when RAPs are conditionally Markov, yielding tractable SDEs for $M_t$ and its innovations process $W_t$. The framework extends to reverse-time interpolants (FARMs) and to multi-arc settings (n-arc FAMs), providing explicit SDEs under Gaussian and standard RAP assumptions. The paper connects these constructions to stochastic filtering, and outlines a path to information-based martingale optimal transport (IB-MOT), where the pathwise interpolation informs coupling selection under noise, with potential applications across finance, insurance, biology, and climate studies.

Abstract

Arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between zeros at fixed pre-specified times. Their additive randomisation allows one to match any finite sequence of target random variables, indexed by the given fixed dates, on the whole probability space. The randomised arcade processes (RAPs) can thus be interpreted as a generalisation of anticipative stochastic bridges. The filtrations generated by these processes are utilised to construct a class of martingales that interpolate between the given target random variables. These so-called filtered arcade martingales (FAMs) are almost-sure solutions to the martingale interpolation problem and reveal an underlying stochastic filtering structure. In the special case of conditionally Markov randomised arcade processes, the dynamics of FAMs are informed by Bayesian updating. The same ideas are applied to filtered arcade reverse-martingales, which are constructed in a similar fashion, using reverse-filtrations of RAPs, instead. Several explicit examples for RAPs and FAMs are provided and simulated. This paper concludes with an outlook on potential connections between FAMs and martingale optimal transport, and related applications.

Arcade Processes for Informed Martingale Interpolation

TL;DR

This work introduces arcade processes (APs) and their randomized extensions (RAPs) to construct continuous-time martingale interpolants that match a finite sequence of convexly ordered targets almost surely. Filtration-based filtered arcade martingales (FAMs) are then built as , with Bayesian updating enabled when RAPs are conditionally Markov, yielding tractable SDEs for and its innovations process . The framework extends to reverse-time interpolants (FARMs) and to multi-arc settings (n-arc FAMs), providing explicit SDEs under Gaussian and standard RAP assumptions. The paper connects these constructions to stochastic filtering, and outlines a path to information-based martingale optimal transport (IB-MOT), where the pathwise interpolation informs coupling selection under noise, with potential applications across finance, insurance, biology, and climate studies.

Abstract

Arcade processes are a class of continuous stochastic processes that interpolate in a strong sense, i.e., omega by omega, between zeros at fixed pre-specified times. Their additive randomisation allows one to match any finite sequence of target random variables, indexed by the given fixed dates, on the whole probability space. The randomised arcade processes (RAPs) can thus be interpreted as a generalisation of anticipative stochastic bridges. The filtrations generated by these processes are utilised to construct a class of martingales that interpolate between the given target random variables. These so-called filtered arcade martingales (FAMs) are almost-sure solutions to the martingale interpolation problem and reveal an underlying stochastic filtering structure. In the special case of conditionally Markov randomised arcade processes, the dynamics of FAMs are informed by Bayesian updating. The same ideas are applied to filtered arcade reverse-martingales, which are constructed in a similar fashion, using reverse-filtrations of RAPs, instead. Several explicit examples for RAPs and FAMs are provided and simulated. This paper concludes with an outlook on potential connections between FAMs and martingale optimal transport, and related applications.
Paper Structure (12 sections, 20 theorems, 149 equations, 11 figures)

This paper contains 12 sections, 20 theorems, 149 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Arcade processes
  3. Randomized arcade processes
  4. Conditionally Markov RAPs
  5. Filtered arcade martingales
  6. The one-arc FAM
  7. Filtered arcade reverse martingales
  8. The n-arc FAM
  9. Outlook on IB-MOT and applications
  10. Conclusions
  11. Arcade processes
  12. Filtered arcade martingales

Key Result

Proposition 2.4

If the driver $(D_t)$ has a mean function $\mu_D$, a variance function $\sigma_D^2$, and a covariance function $K_D$, we have:

Figures (11)

  • Figure 1: Paths simulation of $(I_t^{(5)})$ on $[0,10]$ using the equidistant partition $\{T_i = 2i \mid i=0,1,\ldots, 5\}$, cf., Example \ref{['ex33']}.
  • Figure 2: Paths simulation of $(\tilde{I}_t^{(5)})$ on $[0,10]$ using the equidistant partition $\{T_i = 2i \mid i=0,1,\ldots, 5\}$, cf., Example \ref{['ex33']}.
  • Figure 3: Paths simulation of the signal function of a non-standard $X$-RAP, where $\{T_0,T_2\}_*=\{0,5,10\}$, cf., Example \ref{['exf']}.
  • Figure 4: Paths simulation of a non-standard $X$-RAP, where the noise is rescaled by 0.3 and $\{T_0,T_2\}_*=\{0,5,10\}$, cf., Example \ref{['exf']}.
  • Figure 5: Paths of the $X$-RAP $(I_{t}^{(1)})$, cf., Example \ref{['ex418']}.
  • ...and 6 more figures

Theorems & Definitions (70)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Theorem 2.7
  • proof
  • ...and 60 more