An application of multivariate total positivity to peacocks

We use multivariate total positivity theory to exhibit new families of peacocks. As the authors of \cite{HPRY}, our guiding example is the result of Carr-Ewald-Xiao \cite{CEX}. We shall introduce the notion of strong conditional monotonicity. This concept is strictly more restrictive than the conditional monotonicity as defined in \cite{HPRY} (see also \cite{Be}, \cite{BPR1} and \cite{ShS1}). 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.Comment: 29 page

Malliavin differentiability of solutions of hyperbolic stochastic partial differential equations with irregular drifts

We prove path-by-path uniqueness of solution to hyperbolic stochastic partial differential equations when the drift coefficient is the difference of two componentwise monotone Borel measurable functions of spatial linear growth. The Yamada-Watanabe principle for SDE driven by Brownian sheet then allows to derive strong uniqueness for such equation and thus extending the results in Bogso-Dieye-Menoukeu Pamen \cite{BDM21b} and Nualart-Tindel \cite{NuTi97}. Assuming further that the drift is globally bounded, we we show that the unique strong solution is Malliavin differentiable. In the case of spatial linear growth drift coefficient, we obtain the Malliavin differentiability of the solution only for sufficiently small time parameters. Such results are new even for bounded and monotone drifts.Comment: 28 pages. arXiv admin note: text overlap with arXiv:2112.0039

Existence of strong solutions of fractional Brownian sheet driven SDEs with bounded drift

We prove the existence of a unique Malliavin differentiable strong solution to a stochastic differential equation on the plane with merely integrable and bounded coefficients driven by the fractional Brownian sheet with Hurst parameter $H=(H_1,H_2)\in(0,\frac{1}{2})^2$. The proof of this result relies on a compactness criterion for square integrable Wiener functionals from Malliavin calculus ([Da Prato, Malliavin and Nualart, 1992]), variational techniques developed in the case of fractional Brownian motion ([Ba\~nos, Nielssen, and Proske, 2020]) and the concept of sectorial local nondeterminism (introduced in [Khoshnevisan and Xiao, 2007]). The latter concept enable us to improve the bound of the Hurst parameter (compare with [Ba\~nos, Nielssen, and Proske, 2020]).Comment: 57 page

Smoothness of solutions of hyperbolic stochastic partial differential equations with L∞L^{\infty}-vector fields

In this paper we are interested in a quasi-linear hyperbolic stochastic differential equation (HSPDE) when the vector field is merely bounded and measurable. Although the deterministic counterpart of such equation may be ill-posed (in the sense that uniqueness or even existence might not be valid), we show for the first time that the corresponding HSPDE has a unique (Malliavin differentiable) strong solution. Our approach for proving this result rests on: 1) tools from Malliavin calculus and 2) variational techniques introduced in [Davie, Int. Math. Res. Not., Vol. 2007] non trivially extended to the case of SDEs in the plane by using an algorithm for the selection of certain rectangles. As a by product, we also obtain the Sobolev differentiability of the solution with respect to its initial value. The results derived here constitute a significant improvement of those in the current literature on SDEs on the plane and can be regarded as an analogous equivalent of the pioneering works by [Zvonkin, Math. URSS Sbornik, 22:129-149] and [Veretennikov, Theory Probab. Appl., 24:354-366] in the case of one-parameter SDEs with singular drift.Comment: 46 page

Peacocks obtained by normalisation and strong peacocks

This paper comprises two parts. In the first one, we exhibit families of peacocks obtained by centering or normalizing of a given process. The constructions we present generally rely on conditionally monotone processes. In the second part, we introduce the notion of strong peacocks, which leads to the study of new classes of processes, and makes it possible to recover many of the peacocks previously described

Path-by-path uniqueness of multidimensional SDE’s on the plane with nondecreasing coefficients

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.Comment: 24 page

Some examples of peacocks in a Markovian set-up

Accepté : Séminaire de Probabilités XLIVWe give, in a Markovian set-up, 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