Anticipated Generalized Backward Doubly Stochastic Differential Equations

In this paper, we explore a new class of stochastic differential equations called anticipated generalized backward doubly stochastic differential equations (AGBDSDEs), which not only involve two symmetric integrals related to two independent Brownian motions and an integral driven by a continuous increasing process but also include generators depending on the anticipated terms of the solution (Y, Z). Firstly, we prove the existence and uniqueness theorem for AGBDSDEs. Further, two comparison theorems are obtained after finding a new comparison theorem for GBDSDEs.

Bankruptcy Prediction with a Doubly Stochastic Poisson Forward Intensity Model and Low-Quality Data

With the record high leverage across all segments of the (global) economy, default prediction has never been more important. The excess cash illusion created in the context of COVID-19 may disappear just as quickly as the pandemic entered our world in 2020. In this paper, instead of using any scoring device to discriminate between healthy companies and potential defaulters, we model default probability using a doubly stochastic Poisson process. Our paper is unique in that it uses a large dataset of non-public companies with low-quality reporting standards and very patchy data. We believe this is the first attempt to apply the Duffie–Duan formulation to emerging markets at such a scale. Our results are comparable, if not more robust, than those obtained for public companies in developed countries. The out-of-sample accuracy ratios range from 85% to 76%, one and three years prior to default, respectively. What we lose in (data) quality, we regain in (data) quantity; the power of our tests benefits from the size of the sample: 15,122 non-financial companies from 2007 to 2017, unique in this research area. Our results are also robust to model specification (with different macro and company-specific covariates used) and statistically significant at the 1% level.

Dynamic Risk Measures for Anticipated Backward Doubly Stochastic Volterra Integral Equations

Inspired by the consideration of some inside and future market information in financial market, a class of anticipated backward doubly stochastic Volterra integral equations (ABDSVIEs) are introduced to induce dynamic risk measures for risk quantification. The theory, including the existence, uniqueness and a comparison theorem for ABDSVIEs, is provided. Finally, dynamic convex risk measures by ABDSVIEs are discussed.

G-tridiagonal majorization on M n,m

For X, Y ∈ M n,m , it is said that X is g-tridiagonal majorized by Y (and it is denoted by X ≺ gt Y) if there exists a tridiagonal g-doubly stochastic matrix A such that X = AY. In this paper, the linear preservers and strong linear preservers of ≺ gt are characterized on M n,m .

The Inequalities of Merris and Foregger for Permanents

Conjectures on permanents are well-known unsettled conjectures in linear algebra. Let A be an n×n matrix and Sn be the symmetric group on n element set. The permanent of A is defined as perA=∑σ∈Sn∏i=1naiσ(i). The Merris conjectured that for all n×n doubly stochastic matrices (denoted by Ωn), nperA≥min1≤i≤n∑j=1nperA(j|i), where A(j|i) denotes the matrix obtained from A by deleting the jth row and ith column. Foregger raised a question whether per(tJn+(1−t)A)≤perA for 0≤t≤nn−1 and for all A∈Ωn, where Jn is a doubly stochastic matrix with each entry 1n. The Merris conjecture is one of the well-known conjectures on permanents. This conjecture is still open for n≥4. In this paper, we prove the Merris inequality for some classes of matrices. We use the sub permanent inequalities to prove our results. Foregger’s inequality is also one of the well-known inequalities on permanents, and it is not yet proved for n≥5. Using the concepts of elementary symmetric function and subpermanents, we prove the Foregger’s inequality for n=5 in [0.25, 0.6248]. Let σk(A) be the sum of all subpermanents of order k. Holens and Dokovic proposed a conjecture (Holen–Dokovic conjecture), which states that if A∈Ωn,A≠Jn and k is an integer, 1≤k≤n, then σk(A)≥(n−k+1)2nkσk−1(A). In this paper, we disprove the conjecture for n=k=4.

Near Optimality of Linear Delayed Doubly Stochastic Control Problem

In this paper, we study a kind of near optimal control problem which is described by linear quadratic doubly stochastic differential equations with time delay. We consider the near optimality for the linear delayed doubly stochastic system with convex control domain. We discuss the case that all the time delay variables are different. We give the maximum principle of near optimal control for this kind of time delay system. The necessary condition for the control to be near optimal control is deduced by Ekeland’s variational principle and some estimates on the state and the adjoint processes corresponding to the system.