Stochastic Calculus for the Theta Process

TL;DR

The paper addresses stochastic calculus for the theta process $X$, a Brownian-like, non-semimartingale limit of quadratic Weyl sums. It shows that Itô/Young calculus fail for $X$ and develops a rigorous rough-path framework by constructing a canonical second-order rough path $\mathbb{X}$ above $X$, expressed through higher-rank theta sums along horocycle lifts on the universal Jacobi group. The main result establishes convergence of the lifted processes $\mathbf X_N=(X_N,\mathbb X_N)$ to a geometric rough path above $X$ in the rough-path topology for every $\gamma<\tfrac{1}{2}$, thereby enabling pathwise stochastic calculus for $X$. The analysis intertwines automorphic theta sums, equidistribution on horocycles, and refined decompositions of polygonal indicators (triangles) into corner/edge components, providing a robust rough-path toolkit for number-theoretic stochastic processes with nonstandard regularity and heavy tails.

Abstract

The theta process is a stochastic process of number theoretical origin arising as a scaling limit of quadratic Weyl sums. It can be described in terms of the geodesic flow and an automorphic function on a homogeneous space. This process has several properties in common with Brownian motion such as its Hölder regularity, uncorrelated increments and quadratic variation. However, crucially, we show that the theta process is not a semimartingale, making Itô calculus techniques inapplicable. Instead, we use the celebrated rough paths theory to develop the stochastic calculus for the theta process. We do so by constructing the iterated integrals - the ``rough path" - above the theta process. Rough paths theory takes a signal and its iterated integrals and produces a vast and robust theory of stochastic differential equations. In addition, the rough path we construct can be described in terms of higher rank theta sums, via equidistribution of horocycle lifts.

Figures (8)

• Figure 1: Left panel: the curve $t\mapsto X_N(x;t)$, with $N=50,\!000$, $\alpha=0$, $\beta=\sqrt{2}$, and $x\approx0.42011$ as indicated on the figure. Right panel: a realization of a rescaled simple symmetric random walk, obtained as in \ref{['def-S_N(x,alpha,beta)']}-\ref{['def-X_N(t)']} replacing $\mathrm{e}\!\left(\left(\tfrac{1}{2} n^2+\beta n\right)x+\alpha n\right)$ with $\mathrm{e}\!\left(\xi_n\right)$ for a sequence of i.i.d uniform random variables $(\xi_n)_{n\geq1}$ on $[0,1]$. In both panels, $t$ runs from $0$ (red) to $1$ (blue).
• Figure 2: The functions $x\mapsto f_{\frac{1}{12},\frac{1}{6},\frac{1}{3}}(x)$ and $x\mapsto F_{\frac{1}{12},\frac{1}{6},\frac{1}{3}}(x)$ as in \ref{['fc1c2c3']} and \ref{['Fc1c2c3']}, obtained from the function $f_0(x)=\frac{h(x)}{h(x)+h(1-x)}$, where $h(x)=0$ for $x\leq 0$ and $h(x)=e^{-1/x}$ for $x>0$. These functions will be used Section \ref{['subsection:Triangle']}. In this case $p=2$.
• Figure 3: Top panels: The coloured and shaded area represent the support of the template corner function \ref{['def:TCor']} (left) and of the template line function \ref{['def:TLine']} (right). In both cases, the solid colour represents the region where the function is constant equal to 1. Bottom panels: The graphs of $(x,y)\mapsto T_{\operatorname{Cor}}(x,y)$ and of $(x,y)\mapsto T_{\operatorname{Line}}(x,y)$. The corner template function has jump discontinuities along $\{(x,0):\: 0\leq x< \frac{1}{4}\}\cup\{(0,y):\: 0\leq y< \frac{1}{4}\}$ and is smooth otherwise. The line template function has jump discontinuities along the line segment $\{(x,0):\: \frac{1}{4}< x< \frac{3}{4}\}$ and is smooth otherwise. The boundary of the support of each function from the top panels is also indicated in the bottom panels.
• Figure 4: Top panel: the supports of three corner functions $\mathfrak{C}_{0,1}, \mathfrak{C}_{0,0}, \mathfrak{C}_{1,1}$. Bottom panel: the supports of the three edge functions $\mathfrak{L}_h, \mathfrak{L}_v, \mathfrak{L}_d$. In both panels, the solid colors indicate the regions where the corresponding functions are identically $1$
• Figure 5: The sum $\mathfrak{C}_{0,1}+\mathfrak{L}_h+\mathfrak{L}_v$. Note that this function is identically $1$ on $(0,\frac{1}{6}]\times[\frac{5}{6},1)\cup [\frac{1}{6},\frac{2}{3}]\times[\frac{11}{12},1)\cup(0,\frac{1}{12}]\times[\frac{1}{3},\frac{5}{6}]$ and nonzero only on $(0,\frac{1}{3})\times(\frac{3}{2},1)\cup (\frac{1}{6},\frac{5}{6})\times(\frac{5}{6},1)\cup(0,\frac{1}{6})\times(\frac{1}{6},\frac{5}{6})$.

Theorems & Definitions (147)

• Theorem 1.1: Main Theorem
• Theorem 2.1: Invariance principle for theta sums, Cellarosi-Marklof
• Remark 2.2
• Theorem 2.3: Properties of the theta process, Cellarosi-Marklof
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Definition 3.4
• Theorem 3.5: Young-1936
• Definition 3.6