General Comparison Theorem Research Articles

Lp solutions of general time interval BSDEs with generators satisfying a p-order weak stochastic-monotonicity condition

This article is interested in solutions of general time interval backward stochastic differential equations (BSDEs) with generator g satisfying a p-order weak stochastic-monotonicity condition and a general growth condition in y. An existence and uniqueness result of the Lp solution is proved for the multidimensional BSDEs where the generator g is stochastic-Lipschitz in z. And, a general comparison theorem and an existence and uniqueness theorem of the Lp solution is established for the one-dimensional BSDEs under a stochastic uniform-continuity condition of the generator g in z. These results strengthen some known works. The stochastic Gronwall-type inequality and the stochastic Bihari-type inequality play an important role in the whole article.

A general comparison theorem for reflected BSDEs

In this paper, we deal with a large class of reflected backward stochastic differential equations (RBSDEs for short) with an arbitrary filtered probability space. We prove the comparison theorem for them in three different methods: a direct argument, characterization method and penalization method.

Refining the general comparison theorem for the Klein–Gordon equation

By recasting the Klein–Gordon equation as an eigen-equation in the coupling parameter [Formula: see text] the basic Klein–Gordon comparison theorem may be written [Formula: see text], where [Formula: see text] and [Formula: see text] are the monotone nondecreasing shapes of two central potentials [Formula: see text] and [Formula: see text] on [Formula: see text]. Meanwhile, [Formula: see text] and [Formula: see text] are the corresponding coupling parameters that are functions of the energy [Formula: see text]. We weaken the sufficient condition for the ground-state spectral ordering by proving (for example in [Formula: see text] dimension) that if [Formula: see text], the couplings remain ordered [Formula: see text], where [Formula: see text] and [Formula: see text] are the ground-states corresponding, respectively to the couplings [Formula: see text] for a given [Formula: see text]. This result is extended to spherically symmetric radial potentials in [Formula: see text] dimensions.

𝕃p solutions of reflected backward stochastic differential equations with jumps

Given p ∈ (1, 2), we study 𝕃p-solutions of a reflected backward stochastic differential equation with jumps (RBSDEJ) whose generator g is Lipschitz continuous in (y, z, u). Based on a general comparison theorem as well as the optimal stopping theory for uniformly integrable processes under jump filtration, we show that such a RBSDEJ with p-integrable parameters admits a unique 𝕃p solution via a fixed-point argument. The Y -component of the unique 𝕃p solution can be viewed as the Snell envelope of the reflecting obstacle 𝔏 under g-evaluations, and the first time Y meets 𝔏 is an optimal stopping time for maximizing the g-evaluation of reward 𝔏.

General comparison theorems for the Klein-Gordon equation in d dimensions

We study bound-state solutions of the Klein-Gordon equation $\varphi^{\prime\prime}(x) =\big[m^2-\big(E-v\,f(x)\big)^2\big] \varphi(x),$ for bounded vector potentials which in one spatial dimension have the form $V(x) = v\,f(x),$ where $f(x)\le 0$ is the shape of a finite symmetric central potential that is monotone non-decreasing on $[0, \infty)$ and vanishes as $x\rightarrow\infty.$ Two principal results are reported. First, it is shown that the eigenvalue problem in the coupling parameter $v$ leads to spectral functions of the form $v= G(E)$ which are concave, and at most uni-modal with a maximum near the lower limit $E = -m$ of the eigenenergy $E \in (-m, \, m)$. This formulation of the spectral problem immediately extends to central potentials in $d > 1$ spatial dimensions. Secondly, for each of the dimension cases, $d=1$ and $d \ge 2$, a comparison theorem is proven, to the effect that if two potential shapes are ordered $f_1(r) \leq f_2(r),$ then so are the corresponding pairs of spectral functions $G_1(E) \leq G_2(E)$ for each of the existing eigenvalues. These results remove the restriction to positive eigenvalues necessitated by earlier comparison theorems for the Klein--Gordon equation.

Representation Theorem for Generators of Quadratic BSDEs

In this paper, we establish a general representation theorem for generator of backward stochastic differential equation (BSDE), whose generator has a quadratic growth in z. As some applications, we obtain a general converse comparison theorem of such quadratic BSDEs and uniqueness theorem, translation invariance for quadratic g-expectation.

A comparison theorem for two divided differences and applications to special functions

In this paper, we present a general comparison theorem for two divided differences of a three times differentiable function. This gives a unified treatment for (logarithmically) complete monotonicity, monotonicity and inequalities involving some special functions including gamma, psi and polygamma functions. As their consequences, we not only refine and generalize some important results, but also present simple and interesting alternative proofs of certain earlier results.

Dynamic programming principle and associated Hamilton-Jacobi-Bellman equation for stochastic recursive control problem with non-Lipschitz aggregator

In this work we study the stochastic recursive control problem, in which the aggregator (or generator) of the backward stochastic differential equation describing the running cost is continuous but not necessarily Lipschitz with respect to the first unknown variable and the control, and monotonic with respect to the first unknown variable. The dynamic programming principle and the connection between the value function and the viscosity solution of the associated Hamilton-Jacobi-Bellman equation are established in this setting by the generalized comparison theorem for backward stochastic differential equations and the stability of viscosity solutions. Finally we take the control problem of continuous-time Epstein−Zin utility with non-Lipschitz aggregator as an example to demonstrate the application of our study.

A general comparison theorem for 1-dimensional anticipated BSDEs

Anticipated backward stochastic differential equation (ABSDE) studied the first time in 2007 is a new type of stochastic differential equations. In this paper, we establish a general comparison theorem for ABSDEs.

A qualitative analysis of sets of trajectories of mechanical systems

The evolution of geometric measures (volume, surface area) of sets of attainability of linear controlled mechanical systems with constant parameters is studied. Lyapunov's direct method, the comparison method, and theory of mixed volumes are used. Based on the general comparison theorem, estimates are obtained for the solutions of differential equations with a generalized Hukuhara derivative that describe the evolution of regions of attainability. For linear controlled systems with one degree of freedom, the maximum boundedness conditions are obtained for the area of the set of attainability. Examples of the application of the obtained results are given.

A Probabilistic Proof of the Fundamental Gap Conjecture Via the Coupling by Reflection

Let \({\Omega }\subset \mathbb{R} ^{n}\) be a strictly convex domain with smooth boundary and diameter D. The fundamental gap conjecture claims that if \(V:\bar {\Omega }\to \mathbb{R} \) is convex, then the spectral gap of the Schrodinger operator −Δ+V with Dirichlet boundary condition is greater than \(\frac {3\pi ^{2}}{D^{2}}\). Using analytic methods, Andrews and Clutterbuck recently proved in (J. Amer. Math. Soc. 24(3), 889–916 2011) a more general spectral gap comparison theorem which implies this conjecture. In the first part of the current work, we shall give a probabilistic proof of their result via the coupling by reflection of the diffusion processes. Moreover, we also present in the second part a simpler probabilistic proof of the original conjecture.

Barrier Reflected Backward Doubly Stochastic Differential Equations WithDiscontinuous Generators

In this paper, we study one-dimensional reflected backward doubly stochastic differential equations(RBDSDEs) with one continuous barrier and discontinuous generator(left or right continuous). By establishing a new comparison theorem for RBDSDEs, we provide a minimal or a maximal solution to RBDSDEs under weaker hypotheses. Furthermore, we obtain a generalized comparison theorem. Some known results axe improved and extended.

Splitting families of sets in ZFC

Miller's 1937 splitting theorem was proved for every finite n>0 for all ρ-uniform families of sets in which ρ is infinite. A simple method for proving Miller-type splitting theorems is presented here and an extension of Miller's theorem is proved in ZFC for every cardinal ν for all ρ-uniform families in which ρ≥ℶω(ν). The main ingredient in the method is an asymptotic infinitary Löwenheim–Skolem theorem for anti-monotone set functions.As corollaries, the use of additional axioms is eliminated from splitting theorems due to Erdős and Hajnal [1], Komjáth [7], Hajnal, Juhász and Shelah [4]; upper bounds are set on conflict-free coloring numbers of families of sets; and a general comparison theorem for ρ-uniform families of sets is proved, which generalizes Komjáth's comparison theorem for ℵ0-uniform families [8].

On the invariance and general cohomology comparison theorems

The Invariance Theorem of Gerstenhaber and Schack states that if \(\mathbb A \) is a diagram of algebras then the subdivision functor induces a natural isomorphism between the Yoneda cohomologies of the category \(\mathbb A \)-\(\mathbf{mod }\) and its subdivided category \(\mathbb A ^{\prime }\)-\(\mathbf{mod }\). In this paper we generalize this result and show that the subdivision functor is a full and faithful functor between two suitable derived categories of \(\mathbb A \)-\(\mathbf{mod }\) and \(\mathbb A ^{\prime }\)-\(\mathbf{mod }\). This result combined with our work in Stancu (Hochschild cohomology and derived categories, PhD thesis, SUNY, Buffalo, 2006; J Homotopy Relat Struct 6(1):39–63, 2011), on the Special Cohomology Comparison Theorem, constitutes a generalization of Gerstenhaber and Schack’s General Cohomology Comparison Theorem (GCCT).

Uniqueness for Integro-PDE in Hilbert Spaces

The paper deals with fully nonlinear integro-PDE of Hamilton-Jacobi-Bellman type with unbounded terms in infinite dimensional Hilbert spaces. A notion of viscosity solution is introduced for such integro-PDE and a general comparison theorem is proved.

Joint torsion of several commuting operators

We introduce the notion of joint torsion for several commuting operators satisfying a Fredholm condition. This new secondary invariant takes values in the group of invertibles of a field. It is constructed by comparing determinants associated with different filtrations of a Koszul complex. Our notion of joint torsion generalize the Carey–Pincus joint torsion of a pair of commuting Fredholm operators. As an example, under more restrictive invertibility assumptions, we show that the joint torsion recovers the multiplicative Lefschetz numbers. Furthermore, in the case of Toeplitz operators over the polydisc we provide a link between the joint torsion and the Cauchy integral formula. We will also consider the algebraic properties of the joint torsion. They include a cocycle property, a triviality property and a multiplicativity property. The proof of these results relies on a quite general comparison theorem for vertical and horizontal torsion isomorphisms associated with certain diagrams of chain complexes.

A general comparison theorem for p-harmonic maps in homotopy class

We prove a general comparison result for homotopic finite p-energy C1p-harmonic maps u,v:M→N between Riemannian manifolds, assuming that M is p-parabolic and N is complete and nonpositively curved. In particular, we construct a homotopy through constant p-energy maps, which turn out to be p-harmonic when N is compact. Moreover, we obtain uniqueness in the case of negatively curved N. This generalizes a well-known result in the harmonic setting due to R. Schoen and S.T. Yau.

General comparison theorem for eigenvalues of a certain class of Hamiltonians

Using the Hellmann-Feynman theorem, a general comparison theorem is established for an eigenvalue equation of the form $(T+V)|\psi> = E|\psi>$, where $T$ is a kinetic part which depends only on momentums and $V$ is a potential which depends only on positions. We assume that $H^{(1)}=T+V^{(1)}$ and $H^{(2)}=T+V^{(2)}$ ($H^{(1)}=T^{(1)}+V$ and $H^{(2)}=T^{(2)}+V$) support both discrete eigenvalues $E^{(1)}_{\{\alpha\}}$ and $E^{(2)}_{\{\alpha\}}$, where ${\{\alpha\}}$ represents a set of quantum numbers. We prove that, if $V^{(1)} \le V^{(2)}$ ($T^{(1)} \le T^{(2)}$) for all position (momentum) variables, then the corresponding eigenvalues are ordered $E^{(1)}_{\{\alpha\}} \le E^{(2)}_{\{\alpha\}}$. Some analytical applications are given.

One-dimensional BSDEs with finite and infinite time horizons

This paper is devoted to solving one-dimensional backward stochastic differential equations (BSDEs), where the time horizon may be finite or infinite and the assumptions on the generator g are not necessary to be uniform on t . We first show the existence of the minimal solution for this kind of BSDEs with linear growth generators. Then, we establish a general comparison theorem for solutions of this kind of BSDEs with weakly monotonic and uniformly continuous generators. Finally, we give an existence and uniqueness result for solutions of this kind of BSDEs with uniformly continuous generators.

Oscillation of equations with an infinite distributed delay

For the equation with a finite or infinite distributed delay ẋ(t)+∫−∞tx(s)dsR(t,s)=0 the existence of nonoscillatory solutions is studied. A general comparison theorem is obtained which allows to compare oscillation properties of equations with concentrated delays to integrodifferential equations. Sharp nonoscillation conditions are deduced for some autonomous integrodifferential equations. Using comparison theorems, an example is constructed where oscillation properties of an integrodifferential equation are compared to equations with several concentrated delay which can be treated as its finite difference approximations.