Insider’s problem in the trinomial model: a discrete jump process point of view

In an incomplete market underpinned by the trinomial model, we consider two investors: an ordinary agent whose decisions are driven by public information and an insider who possesses from the beginning a surplus of information encoded through a random variable for which he or she knows the outcome. Through the definition of an auxiliary model based on a marked binomial process, we handle the trinomial model as a volatility one, and use the stochastic analysis and Malliavin calculus toolboxes available in that context. In particular, we connect the information drift, i.e. the drift to eliminate in order to preserve the martingale property within an initial enlargement of filtration in terms of Malliavin’s derivative. We solve explicitly the agent and the insider expected logarithmic utility maximization problems and provide a Ocone-Karatzas type formula for replicable claims. We identify insider’s expected additional utility with the Shannon entropy of the extra information, and examine then the existence of arbitrage opportunities for the insider.