[Paper Review] Computable Stochastic Processes

Pieter Collins|arXiv (Cornell University)|Sep 15, 2014

Stochastic processes and financial applications27 references10 citations

TL;DR

This paper develops a computable theory of probability, random variables, and stochastic processes using type-two effectivity and computable types, enabling effective computation of solutions to stochastic differential equations (SDEs) with Lipschitz coefficients. By modeling random variables as limits of continuous functions and using contraction mapping in a computable framework, it proves that SDE solutions are effectively computable as random processes in continuous time.

ABSTRACT

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

Motivation & Objective

• To establish a foundational, computable theory of probability and stochastic processes suitable for rigorous numerical analysis.
• To overcome the poor computability of classical σ-algebras by replacing them with topological constructions and completions of continuous functions.
• To develop a framework where random variables and stochastic processes are effectively computable, particularly for SDEs.
• To demonstrate that solutions to stochastic differential equations with Lipschitz drift and diffusion coefficients are computable as random processes in continuous time.

Proposed method

• The theory is built within the framework of type-two effectivity (TTE), using representations of mathematical objects as infinite sequences processed by Turing machines.
• Probability measures are represented via valuations and lower integrals, with integration defined over positive lower-semicontinuous and bounded continuous functions.
• Random variables are defined as limits of almost-everywhere defined continuous partial functions, forming a completion of continuous functions in a Polish space.
• Stochastic processes are modeled as random elements in the space of continuous paths, with norms like $ d_{2,rown} $ measuring pathwise $ L^2 $-distance.
• The Picard iteration for SDEs is shown to be a contraction mapping under $ d_{2,rown} $, ensuring effective convergence for small time intervals.
• Solutions are constructed recursively over small time intervals and joined together to yield global computability.

Experimental results

Research questions

• RQ1 Can a computable theory of probability be developed that avoids the non-computable structure of σ-algebras while preserving classical probabilistic reasoning?
• RQ2 How can random variables over continuous domains be effectively represented and computed using constructive methods?
• RQ3 Can the solution to a stochastic differential equation be effectively computed as a random process in continuous time?
• RQ4 What conditions ensure that the Picard iteration for an SDE converges effectively in a computable framework?
• RQ5 Is the Wiener process itself effectively computable in this framework, enabling the construction of SDE solutions?

Key findings

• The solution to a stochastic differential equation with Lipschitz drift and diffusion coefficients is computable as a random variable taking values in the space of continuous paths $ C([0,rown); ^d) $.
• The Picard operator for SDEs is a contraction mapping under the $ d_{2,rown} $-norm when the time interval $ T $ is small enough, specifically $ T < \min(1/2K, 1/16L^2) $, where $ K $ and $ L $ are Lipschitz constants.
• The non-stochastic integral $ \int_0^t X(s)\,ds $ of a continuous stochastic process is computable, with $ d_{2,rown}(\int Y dt, \int Z dt) \leq T d_{2,rown}(Y,Z) $.
• The stochastic integral $ \int_0^t g(Y(s))\,dW(s) $ is computable, with $ d_{2,rown}(\int g(Y) dW, \int g(Z) dW) \leq 2\sqrt{T} d_{2,rown}(Y,Z) $.
• The Wiener process is constructed effectively, with sample paths being computable, providing a foundation for SDE computation.
• The solution operator $ \mathbb{R} \to R(C([0,T];\mathbb{R})) $ is computable for initial conditions in $ \mathbb{R} $, and extends to random initial conditions via Theorem 69.