Non-Markov rate kernels: Application to batch auction

Abstract

We introduce a theoretical tool for handling pure-jump processes taking values in complex spaces. We generalize the notion of rate kernels for the non-Markov case, being able to describe any pure-jump process in Borel space with absolutely continuous conditional distribution of jump times. We study the case of two simultaneously running processes where the evolution of the first is locally unaffected on the values of the second; we show that then the conditional distribution of the second can be evaluated as if the first were deterministic. Further we study pure-jump process of bounded atomic measures. We characterize rate kernels ruling processes of completely random atomic measures. Finally, we apply our theory to the model of call auction with the limit order process depending on a common driving factor called fair price; we give analytical formula for the conditional distribution of the order books given the trajectory of the fair price and semi-analytical formulas for both the conditional and unconditional distribution of the settlement price.