In this paper, we establish existence and uniqueness results for generalized backward doubly stochastic differential equations (GRBDSDEs) with a right upper semi-continuous and left-limited (r.u.s.c.l.l.) barrier and stochastic Lipschitz coefficients. The analysis is based on Mertens' decomposition, tools from optimal stopping theory, and a Picard iteration scheme. In addition, we prove a comparison theorem and use it to establish the existence of a minimal solution when the coefficients satisfy a stochastic linear growth condition.
Stochastic primitive equations are fundamental for modelling geophysical fluid flows in the ocean and atmosphere. In large-scale systems, where horizontal motion dominates, we consider only horizontal viscosity to capture the anisotropic nature of the flow. The nonlinear convection term possesses an intrinsic anisotropic structure, which, combined with the anisotropic viscosity, presents substantial analytical challenges. In this paper, we establish a moderate deviation principle through a careful decomposition of the nonlinear terms, together with anisotropic estimates and the weak convergence method. This work provides a refined asymptotic description that bridges the central limit and large deviation regimes.
In this paper we study partially observable discrete-time zero-sum games with Borel state spaces and unbounded reward functions. The optimality criterion is the first passage risk probability criterion. We first introduce a new auxiliary zero-sum game problem and establish the corresponding Shapley equation. Then via an approximation approach we obtain the existence of a solution to the Shapley equation under the first passage risk probability criterion. Moreover, applying the Shapley equation we prove the existence of a saddle-point equilibrium.
This paper investigates bounds on the expected p-variation of partial sums process for series of vector random variables under moment conditions. As a special case, we obtain a non-orthogonal generalization of results known for orthogonal systems. A variational strengthening of Kolmogorov continuity theorem for stochastic processes is also deduced.
Recently, a new concept for some stochastic process called fractional G-Brownian motion (fGBm) was developed, which generalizes the concepts of standard Brownian motion (Bm), fractional Brownian motion (fBm) and G-Brownian motion (GBm) under the framework of sublinear expectation. The fGBm can exhibit long-range dependence and feature the volatility uncertainty of financial markets simultaneously. Thus it can be a better alternative stochastic process in the financial applications. In this paper, some financial markets driven by the fGBm were considered and the corresponding arbitrage opportunities were discussed. Specifically, the financial market $ \mathcal {M} $ M that consists a risk free asset and a risky asset will admit some arbitrage opportunity if the stochastic integrals with respect to the fGBm were established in the path-wise sense, and the arbitrage opportunity can be excluded if the stochastic integrals were established in the Wick calculus sense. What is more, the arbitrage opportunity for the financial market $ (\mathcal {M}, C) $ (M,C), consisting of the original market $ \mathcal {M} $ M and some contingent claim C were also investigated in the general case, and some interval of free arbitrage prices for the claim were derived by applying the generalized fractional G-Girsanov theorem and fractional G-Clark-Ocone theorem. This study generalized the well-known existing results of arbitrage theory in financial markets driven by the standard Bm, fBm and GBm.
This paper is devoted to the existence and uniqueness of invariant measures for a class of stochastic functional hydrodynamical type systems perturbed by degenerate white noise. This abstract model covers many incompressible fluid flows such as 2D Navier-Stokes equations, 2D magneto-hydrodynamic equations, 2D viscous lake equation, 2D B & eacute;nard magnetic problems, 3D Leray alpha-model. By a weak type of irreducibility, we apply the asymptotic strong Feller property introduced by Hairer and Mattingly [Ergodicity of the 2D Navier-Stokes equations with degenerate stochastic forcing, Ann. Math. 164 (2006), pp. 993-1032.] to prove our main results.
The primary objective of this article is to investigate the convergence of higher-order moments of the diffusion processes within the state space $ \mathbb {R} $ R. This analysis will subsequently demonstrate weak convergence forms of the diffusion processes. The methodology employed in this paper is relatively straightforward, involving the Poisson equation associated with the infinitesimal generator as presented in Depauw and Derrien (Variance limite d'une marche al & eacute;atoire r & eacute;versible en milieu al & eacute;atoire sur $ \mathbb {Z} $ Z, C. R. Acad. Sci. Paris, Ser. I 347 (2009), pp. 401-406).
In this paper, we prove a strong law of large numbers for m-dependent random variables with a general moment condition that is the weakest one under sub-linear expectation. The result can be viewed as a natural generalization of Hu (2016).
In this paper, we derive a second-order asymptotic formula for the tail probability of a randomly weighted sum, where the primary random variables follow second-order subexponential distributions and exhibit a multivariate Farlie-Gumbel-Morgenstern dependence structure. This result significantly improves existing first-order asymptotic results. Simulation studies are conducted to assess the accuracy of our asymptotic formula, demonstrating its improvement over first-order counterparts. As an application, we investigate a nonstandard continuous-time risk model with a constant force of interest and establish the second-order asymptotics of the discounted aggregate claims.
In this paper we consider the stochastic 3D globally modified Navier-Stokes equations with finite delays in bounded domains. We give a sufficient condition for the existence, uniqueness and exponential stability of the stationary solution in the sense of mean square and almost sure. Moreover, when the sufficient condition for exponential stability is not satisfied, we use a linear internal feedback controller with support large enough to stabilize an unstable stationary solution.
In this article, we consider Carath & eacute;odory scheme for scalar Caputo stochastic fractional differential equations (CSFDE) of order nu is an element of (7/8,1) in L-p spaces with p is an element of (2/2 nu-1,1/1-nu) of the form D-C(nu)0+y(t)=g(t,y(t))+& planckh;(t,y(t))dW(t)/dt, t is an element of [0,T], (1) where T > 0 is arbitrary, (W-t)(t is an element of[0,T]) denotes a standard Brownian motion on a completely filtered probability space (Omega,& Fouriertrf;, := {& Fouriertrf;(t)}(t is an element of[0,T]),& Popf;) and g, & planckh;:[0,T]x & Ropf;->& Ropf; are measurable functions. Based on the techniques of fractional calculus and Malliavin calculus, we establish an upper bound for |[sigma(y(t))]-[sigma(y(t))]|, where y(t) represents the accurate solution of (Equation1) and y(t) denotes the numerical solution of (Equation1), as defined by (Equation5) below, with sigma is an element of & Bernoullis;. Here, & Bernoullis; is defined as follows: & Bernoullis;:= {sigma: & Ropf; -> & Ropf; is measurable function: integral(+infinity)(-infinity)|sigma(x)|dx < infinity and & Vert;sigma & Vert;infinity := sup x is an element of & Ropf; |sigma(x)| <= 1}.
The quantum exclusion semigroup constructed from quantum Bernoulli noise decomposes into distinct actions on diagonal and off-diagonal operator spaces. Its restriction to the diagonal subspace corresponds to a classical Markov process. In this paper, we establish explicit contraction rates for this semigroup under the Wasserstein-1 distance. For arbitrary initial states, the Wasserstein distance between evolved states decays exponentially with a rate combining the Wasserstein curvature of the classical Markov process and the decay rate of quantum off-diagonal operators. The contraction features a prefactor constant containing a quantum correction term that quantifies how coherence affects the convergence dynamics.
In this paper, we derive the stability and instability criteria for hybrid stochastic systems driven by the Ornstein-Uhlenbeck process. The results indicate that almost surely exponentially stable and instability are dependent on the intensities of both the Markovian jump noise and the Ornstein-Uhlenbeck process noise. Furthermore, sufficient conditions for asymptotic stability in distribution are derived using Lyapunov functions and M-matrices. To validate our results, we present several representative examples along with corresponding simulation outcomes.
Stochastic orders provide a powerful method for comparing random variables based on their distributions, extensively studied by many scholars. However, in reality, it is often difficult to obtain the true distribution of a random variable due to data limitations. To address this challenge, we introduce a novel concept of robust stochastic order, where the distribution of the random variable is estimated within a family of distributions. Specifically, we study the relationships of several important robust stochastic orders and smooth generators of robust integral stochastic orders, which generalize the corresponding results of classical stochastic orders. The main focus of this study is on various robust stochastic orderings of elliptical distributions under parametric ambiguity, including robust usual stochastic order, robust convex order, robust supermodular order, robust directionally convex order, robust componentwise convex order, and so on. As an application grounded in expected utility theory, we present some examples, providing more reasonable decision-making guidances for risk-averse investors.
We introduce and study a fractional variant of the linear birth-death process, namely, the generalized fractional linear birth-death process (GFLBDP). It is defined by taking the regularized Hilfer-Prabhakar derivative in the system of differential equations that governs the state probabilities of linear birth-death process. For a particular choice of parameters, the GFLBDP reduces to the fractional linear birth-death process that involves the Caputo derivative. Its time-changed representation is obtained and utilized to derive the explicit expressions of its state probabilities. The explicit expressions for its mean and variance are derived. In a particular case, it is observed that the limiting distribution of the time changing process coincides to that of an inverse stable subordinator. A relation between the extinction probability of GFLBDP and the density of inter arrival times of a generalized fractional Poisson process is obtained. Later, we study some integrals of the GFLBDP and discuss the asymptotic distributional characteristics for a particular integral process. Also, an application of the path integral at random time to a genetic population with an upper bound is discussed.
This work focuses on a class of semi-linear functional stochastic partial differential equations with Markovian switching, in which the switching component may have finite or countably infinite states. The well-posedness of the underlying process is obtained by Skorokhod's representation of the switching component. Then, the exponential mixing of such processes in a finite state space is derived by using the so-called remote start method proposed firstly by Da Prato and Zabczyk [Ergodicity for Infinite-Dimensional Systems, London Mathematical Society Lecture Note Series Vol. 229, Cambridge University Press, Cambridge, 1996]. Finally, the corresponding result in a countable infinite state space is further obtained via the finite partition method.
We consider a limit theorem for a triangular array of point processes generated by non-identically distributed random variables, and apply the result for the analysis of the limiting behavior of the Argmaximum of independent random variables, as well as for some step processes.
We establish analytic characterization theorems for symbols of linear operators acting on Mittag-Leffler distribution spaces, relying on the corresponding characterization theorem via the $ S_{\mu _\beta } $ S mu beta-transform. Several explicit examples are discussed, including operators that admit an interpretation as integral kernel operators. As an important application, we analyse translation operators in the Mittag-Leffler setting.
The main purpose of this work is the derivation of a path-dependent partial differential equation for the calculations of equity-linked insurance policies, where the payment stream may depend on the whole past history of the financial asset. To this end, we employ variational techniques from the theory of functional It & ocirc; calculus.
