A simple comparison between Skorokhod & Russo-Vallois integration for insider trading

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 Skorokhod and the Russo-Vallois forward integrals within this context. We conclude that, while the forward integral yields results with a clear financial meaning, the Skorokhod integral does not provide a suitable formulation for this problem.

A Triple Comparison between Anticipating Stochastic Integrals in Financial Modeling

“…This result is not present in [11] and it is due to the possibility of borrowing money that the insider is allowed for in the present work but not in the previous one. This difference nevertheless highlights the qualitative agreement between the results in the present work and those in [11]. We recall that the analysis of the Ayed-Kuo integral is new and that it agrees with the Hitsuda-Skorohod integral in this setting, what leaves the Russo-Vallois forward integral, among these three, as the natural candidate to model insider trading within the formalism of anticipating stochastic calculus, at least under our present hypotheses.…”

“…The mentioned conjecture is the missing step in the triple comparison between these anticipating stochastic integrals carried out in [11] and herein. Proving or disproving this conjecture is anyway a first step towards the understanding of noise interpretation in anticipating stochastic calculus.…”

“…We readdress now the problem in [11]. Contrary to the previous situation, our trader is now an insider who fully trusts her/his information on the future price of the stock, but s/he is not allowed to borrow any money.…”

Optimal Portfolios for Different Anticipating Integrals under Insider Information

“…Remark 2. The expected wealth of the Russo-Vallois insider under this optimal strategy was computed in [12] and reads:…”

“…Since the preeminent place for this debate on the noise interpretation has been the physics literature, the focus has been put on diffusion processes, perhaps due to the influence of the seminal works by Einstein [9] and Langevin [10]. Herein we move out of even the Markovian setting and, following the steps of [11,12], we concentrate on stochastic differential equations with non-adapted terms. Non-adaptedness arises in financial markets concomitantly to the presence of insider traders.…”