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Some extremal problems for martingale transforms, I

Vasily Vasyunin, Pavel Zatitskii

TL;DR

The paper develops a general algorithm to construct Bellman functions for extremal martingale-transform problems on a horizontal strip by exploiting foliations of diagonally concave functions. It introduces simple and more intricate foliations (right/left, horizontal herringbones, fissures) and derives the governing equations for spines and boundary data, including vector-field dynamics and concavity conditions expressed via $D_\pm$ and related quantities. Through explicit analyses of linear, quadratic, and cubic boundary data, the authors characterize when simple foliations suffice and when fissures or multiple foliations arise, providing concrete formulas for Bellman candidates and conditions for $C^1$-smoothness. The results lay a foundation for sharp distributional estimates of martingale transforms and illuminate the geometric structure of Bellman functions in asymmetrical boundary-value problems, with explicit constructions that will underpin further developments in the series. The approach bridges classical Burkholder-type methods with modern foliation techniques, potentially impacting sharp probabilistic inequalities in martingale theory.

Abstract

With this paper, we begin a series of studies of extremal problems for estimating distributions of martingale transforms of bounded martingales. The Bellman functions corresponding to such problems are pointwise minimal diagonally concave functions on a horizontal strip, satisfying certain given boundary conditions. We describe the basic structures that arise when constructing such functions and present a solution in the case of asymmetric boundary conditions and a sufficiently small width of the strip.

Some extremal problems for martingale transforms, I

TL;DR

The paper develops a general algorithm to construct Bellman functions for extremal martingale-transform problems on a horizontal strip by exploiting foliations of diagonally concave functions. It introduces simple and more intricate foliations (right/left, horizontal herringbones, fissures) and derives the governing equations for spines and boundary data, including vector-field dynamics and concavity conditions expressed via and related quantities. Through explicit analyses of linear, quadratic, and cubic boundary data, the authors characterize when simple foliations suffice and when fissures or multiple foliations arise, providing concrete formulas for Bellman candidates and conditions for -smoothness. The results lay a foundation for sharp distributional estimates of martingale transforms and illuminate the geometric structure of Bellman functions in asymmetrical boundary-value problems, with explicit constructions that will underpin further developments in the series. The approach bridges classical Burkholder-type methods with modern foliation techniques, potentially impacting sharp probabilistic inequalities in martingale theory.

Abstract

With this paper, we begin a series of studies of extremal problems for estimating distributions of martingale transforms of bounded martingales. The Bellman functions corresponding to such problems are pointwise minimal diagonally concave functions on a horizontal strip, satisfying certain given boundary conditions. We describe the basic structures that arise when constructing such functions and present a solution in the case of asymmetric boundary conditions and a sufficiently small width of the strip.
Paper Structure (7 sections, 17 theorems, 155 equations, 16 figures)

This paper contains 7 sections, 17 theorems, 155 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Simplest Foliation
  3. Examples. Linear and Quadratic Functions
  4. Horizontal Herringbones
  5. Investigation of the vector field of equation \ref{['040615']}
  6. Fissures
  7. Examples. Polynomials of third degree

Key Result

Theorem 1.4

The function $\boldsymbol{B}$ is diagonally concave on the strip $\Omega_\varepsilon$ and satisfies the boundary conditions Moreover, it is the pointwise minimal among all functions satisfying these properties.

Figures (16)

  • Figure 1: Vertical herringbones extending from bottom to top and from top to bottom
  • Figure 2: Horizontal herringbones, left and right
  • Figure 3: Fissures: SW, NW, NE, SE
  • Figure 4: Integral curves near the node $(u_0,0)$.
  • Figure 5: Integral curves near the saddle point $(u_+^{\text{\rm L}},\varepsilon)$.
  • ...and 11 more figures

Theorems & Definitions (35)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7: Misha
  • Definition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • ...and 25 more