A Lévy process on $R^d$ with distribution $μ$ at time 1 is denoted by $X^{(μ)}=\{X_t^{(μ)}\}$. If the improper stochastic integral $\int_0^{\infty-} f(s)dX_s^{(μ)}$ of $f$ with respect to $X^{(μ)}$ is definable, its distribution is denoted by $Φ_f(μ)$. The class of all infinitely divisible distributions $μ$ on $R^d$ such that $Φ_f(μ)$ is definable is denoted by $D(Φ_f)$. The class $D(Φ_f)$, its two extensions $D_c(Φ_f)$ and $D_e(Φ_f)$ (compensated and essential), and its restriction $D^0(Φ_f)$ (absolutely definable) are studied. It is shown that $D_e(Φ_f)$ is monotonic with respect to $f$, which means that $|f_2|\leq |f_1|$ implies $D_e(Φ_{f_1})\subset D_e(Φ_{f_2})$. Further, $D^0(Φ_f)$ is monotonic with respect to $f$ but neither $D(Φ_f)$ nor $D_c(Φ_f)$ is monotonic with respect to $f$. Furthermore, there exist $μ$, $f_1$, and $f_2$ such that $0\leq f_2\leq f_1$, $μ\in D(Φ_{f_1})$, and $μ\not\in D(Φ_{f_2})$. An explicit example for this is related to some properties of a class of martingale Lévy processes.
We study stochastic monotonicity and propagation of order for Markov processes with respect to stochastic integral orders characterized by cones of functions satisfying $Φf \geq 0$ for some linear operator $Φ$. We introduce a new functional analytic technique based on the generator $A$ of a Markov process and its resolvent. We show that the existence of an operator $B$ with positive resolvent such that $ΦA - B Φ$ is a positive operator for a large enough class of functions implies stochastic monotonicity. This establishes a technique for proving stochastic monotonicity and propagation of order…
Stochastic monotonicity is a well known partial order relation between probability measures defined on the same partially ordered set. Strassen Theorem establishes equivalence between stochastic monotonicity and the existence of a coupling compatible with respect to the partial order. We consider the case of a countable set and introduce the class of \emph{finitely decomposable flows} on a directed acyclic graph associated to the partial order. We show that a probability measure stochastically dominates another probability measure if and only if there exists a finitely decomposable flow having…
This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator $g$ satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable $y$, and a stochastic-Lipschitz condition in the state variable $z$. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in \citet{XiaoFan2017Stochastics} and \citet{LiuLiFan2019CAM}. In particular, we put forward and prove…
In this note we establish several inequalities and monotonicity properties for the free energy of directed polymers under certain stochastic orders: the usual stochastic order, the Laplace transform order and the convex order. For the latter our results cover also many classical disordered systems.
This paper establishes a Freidlin-Wentzell large deviation principle for stochastic differential equations(SDEs) under locally weak monotonicity conditions and Lyapunov conditions. We illustrate the main result of the paper by showing that it can be applied to SDEs with non-Lipschitzian coefficients, which can not be covered in the existing literature. These include the interesting biological models like stochastic Duffing-van der Pol oscillator model, stochastic SIR model, etc.
We consider the problem of viscosity solution of integro-partial differential equation(IPDE in short) with one obstacle via the solution of reflected backward stochastic differential equations(RBSDE in short) with jumps. We show existence and uniqueness of a continuous viscosity solution of equation with non local terms, in case the generator is not monotonous and Levy's measure is infinite.
We formalize and analyze the notions of stochastic monotonicity and realizable mono-tonicity for Markov Chains in continuous-time, taking values in a finite partially ordered set. Similarly to what happens in discrete-time, the two notions are not equivalent. However, we show that there are partially ordered sets for which stochastic monotonicity and realizable monotonicity coincide in continuous-time but not in discrete-time.
Multi-class queueing networks (McQNs) extend the classical concept of Jackson network by allowing jobs of different classes to visit the same server. While such a generalization seems rather natural, from a structural perspective there is a significant gap between the two concepts. Nice analytical features of Jackson networks, such as stability conditions, product-form equilibrium distributions, and stochastic monotonicity do not immediately carry over to the multi-class framework. The aim of this paper is to shed some light on this structural gap, focusing on monotonicity properties. To this …
Given two random variables $X$ and $Y$, stochastic monotonicity describes a monotone influence of $X$ on $Y$. We prove two different characterizations of stochastically monotone $2$-copulas using the isomorphism between $2$-copulas and Markov operators. The first approach establishes a one-to-one correspondence between stochastically monotone copulas and monotonicity-preserving Markov operators. The second approach characterizes stochastically monotone copulas by their monotonicity property with respect to the Markov product. Applying the latter result, we identify all idempotent stochasticall…
By using the Picard iteration scheme, this article establishes the existence and uniqueness theory for solutions to stochastic functional differential equations driven by G-Browniain motion. Assuming the monotonicity conditions, the boundedness and existence-uniqueness results of solutions have been derived. The error estimation between Picard approximate solution $y^k(t)$ and exact solution $y(t)$ has been determined. The $L^2_G$ and exponential estimates have been obtained. The theory has been further generalized to weak monotonicity conditions. The existence, uniqueness and exponential esti…
Higher order moment estimates for solutions to nonlinear SPDEs governed by locally-monotone operators are obtained under appropriate coercivity condition. These are then used to extend known existence and uniqueness results for nonlinear SPDEs under local monotonicity conditions to allow derivatives in the operator acting on the solution under the stochastic integral.
Zhongtai Securities Institute for Financial Studies, Shandong University（山东大学中泰金融研究院）；Department of Mathematics, National University of Defense Technology（国防科技大学数学系）
Viscosity solution of systems of integral-partial differential equations with interconnected obstacles without Monotonicity Conditions and infinite L{é}vy measure
Le Mans Université（勒芒大学）；University of Tunis El Manar（突尼斯艾曼大学）
We consider the class of (possibly killed) spectrally positive Lévy process that have been time-changed by the inverse of an integral functional. Within this class we characterize the family of those processes which satisfy the following property: as functions of point of issue, the Laplace transforms of their first-passage times downwards are completely monotone. A wide (dense, in a sense) subfamily of this family admits closed form expressions for said Laplace transforms.
We study stochastic particle systems that conserve the particle density and exhibit a condensation transition due to particle interactions. We restrict our analysis to spatially homogeneous systems on finite lattices with stationary product measures, which includes previously studied zero-range or misanthrope processes. All known examples of such condensing processes are non-monotone, i.e. the dynamics do not preserve a partial ordering of the state space and the canonical measures (with a fixed number of particles) are not monotonically ordered. For our main result we prove that condensing ho…
In the present paper we prove that the probabilities of the Pólya urn distribution (with negative replacement) satisfy a monotonicity property similar to that of the binomial distribution (Pólya urn distribution with no replacement). As a consequence, we show that the random variables with Pólya urn distribution (with negative replacement) are stochastically ordered with respect to the parameter giving the initial distribution of the urn. An equivalent formulation of this result shows that the new Bernstein operator recently introduced in [3] is a monotone operator. The proofs are probabilisti…
In this paper, we study the evolution of a Finitary Random Interlacement (FRI) with respect to the expected length of each fiber. In contrast to the previously proved phase transition between sufficiently large and small fiber length, we show that for $d=3,4$, FRI is NOT stochastically monotone as fiber length increasing. At the same time, numerical evidences still strongly support the existence of a unique and sharp phase transition on the existence of a unique infinite cluster, while the critical value for phase transition is estimated to be an inversely proportional function with respect to…
In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDE where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.
We prove that distribution dependent (also called McKean--Vlasov) stochastic delay equations of the form \begin{equation*} \mathrm{d}X(t)= b(t,X_t,\mathcal{L}_{X_t})\mathrm{d}t+ σ(t,X_t,\mathcal{L}_{X_t})\mathrm{d}W(t) \end{equation*} have unique (strong) solutions in finite as well as infinite dimensional state spaces if the coefficients fulfill certain monotonicity assumptions.
The article is devoted to the integration order replacement technique for iterated Ito stochastic integrals and iterated stochastic integrals with respect to martingales. We consider the class of iterated Ito stochastic integrals, for which with probability 1 the formulas of integration order replacement corresponding to the rules of classical integral calculus are reasonable. The theorems on integration order replacement for the class of iterated Ito stochastic integrals is proven. Many examples of this theorems usage have been considered. These results are generalized for the class of iterat…
It was recently proven that the correlation function of the stationary version of a reflected Lévy process is nonnegative, nonincreasing and convex. In another branch of the literature it was established that the mean value of the reflected process starting from zero is nonnegative, nondecreasing and concave. In the present paper it is shown, by putting them in a common framework, that these results extend to substantially more general settings. Indeed, instead of reflected Lévy processes, we consider a class of more general stochastically monotone Markov processes. In this setup we show monot…
