A triple comparison between anticipating stochastic integrals in financial modeling

TL;DR

This paper compares different anticipating stochastic integrals in a simplified insider trading model, finding the forward integral more suitable for financial interpretation than the Hitsuda-Skorokhod and Ayed-Kuo integrals.

Contribution

It provides a comparative analysis of three anticipating stochastic integrals in a financial context, highlighting the forward integral's appropriateness and offering new insights into the Ayed-Kuo integral.

Findings

Forward integral yields financially meaningful results.

Hitsuda-Skorokhod and Ayed-Kuo integrals are less appropriate for this problem.

Expectation of Russo-Vallois solution exceeds that of Ayed-Kuo solution.

Abstract

We consider a simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Hitsuda-Skorokhod, the Ayed-Kuo, and the Russo-Vallois forward integrals within this context. Our results give some indication that, while the forward integral yields results with a suitable financial meaning, the Hitsuda-Skorokhod and the Ayed-Kuo integrals do not provide an appropriate formulation of this problem. Further results regarding the use of the Ayed-Kuo integral in this context are also provided, including the proof of the fact that the expectation of a Russo-Vallois solution is strictly greater than that of an Ayed-Kuo solution. Finally, we conjecture the explicit solution of an Ayed-Kuo stochastic differential equation that possesses discontinuous sample paths with finite probability.