We give, in a Markovian setup , some examples of processes which are increasing in the convex order (we call them peacocks). We then establish some relation between the stochastic and convex orders.

Highlights• We recall some properties of R + -valued homogeneous Markov processes which admit a totally positive transition kernel.• We establish the equivalence between the MRL ordering and a log-concavity type condition.• We exhibit new classes of MRL processes, i.e. integrable processes which increase in the MRL order. AbstractWe provide an equivalent log-concavity condition to the mean residual life (MRL) ordering for realvalued processes. This result, combined with classical properties of total positivity of order 2, allows to exhibit new families of integrable processes which increase in the MRL order (MRL processes). Note that MRL processes with constant mean are peacocks to which the Azéma-Yor (Skorokhod embedding) algorithm yields an explicit associated martingale.

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In this work, we generalise the stochastic local time space integration introduced in [8] to the case of Brownian sheet. This allows us to prove a generalised two-parameter Itô formula and derive Davie type inequalities for the Brownian sheet. Such estimates are useful to obtain regularity bounds for some averaging type operators along Brownian sheet curves.

We introduce the notion of weak decreasing stochastic (WDS) ordering for real-valued processes with negative means, which, to our knowledge, has not been studied before. Thanks to Madan-Yor's argument, it follows that the WDS ordering is a necessary and sufficient condition for a process with negative mean to be embeddable in a standard Brownian motion by the Cox and Hobson extension of the Azéma-Yor algorithm. Since the decreasing stochastic order is stronger than the WDS order, then, for every stochastically non-decreasing family of probability measures with densities, the Cox-Hobson stopping times provide an associated Markov process. The quantile process associated to a stochastically non-decreasing process is not necessarily Markovian.

Abstract:We use multivariate total positivity theory to exhibit new families of peacocks. As ). There are many random vectors which are strongly conditionally monotone (SCM). Indeed, we shall prove that multivariate totally positive of order 2 (MTP 2 ) random vectors are SCM. As a consequence, stochastic processes with MTP 2 finite-dimensional marginals are SCM. This family includes processes with independent and log-concave increments, and one-dimensional diffusions which have absolutely continuous transition kernels.Key words: convex order, peacocks, total positivity of order 2 (TP 2 ), multivariate total positivity of order 2 (MTP 2 ), Markov property, strong conditional monotonicity.

In this paper we study path-by-path uniqueness for multidimensional stochastic differential equations driven by the Brownian sheet. We assume that the drift coefficient is unbounded, verifies a spatial linear growth condition and is componentwise nondeacreasing. Our approach consists of showing the result for bounded and componentwise nondecreasing drift using both a local time-space representation and a law of iterated logarithm for Brownian sheets. The desired result follows using a Gronwall type lemma on the plane. As a by product, we obtain the existence of a unique strong solution of multidimensional SDEs driven by the Brownian sheet when the drift is non-decreasing and satisfies a spatial linear growth condition.

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