The stochastic field of aggregate utilities and its saddle conjugate

TL;DR

This paper characterizes the stochastic fields of aggregate utilities and their saddle conjugates in an economy, establishing conjugacy relations crucial for continuous-time price impact modeling.

Contribution

It introduces the stochastic utility fields and their conjugates, providing foundational relations for analyzing price impact in continuous-time economic models.

Findings

Describes sample paths of utility fields $F$ and $G$.

Establishes conjugacy relations between $F$ and $G$.

Provides mathematical framework for price impact models.

Abstract

We describe the sample paths of the stochastic field $F = F_t(v,x,q)$ of aggregate utilities parameterized by Pareto weights $v$ and total cash amounts $x$ and stocks' quantities $q$ in an economy. We also describe the sample paths of the stochastic field $G = G_t(u,y,q)$, which is conjugate to $F$ with respect to the saddle arguments $(v,x)$, and obtain various conjugacy relations between these stochastic fields. The results of this paper play a key role in our study of a continuous-time price impact model.