Generalized Modulated Random Measures and Their Potentials

Abstract

This article deals with a class of random measures formed of doubly stochastic marked random measures that assumes parameters in accordance with the evolution of some stochastic process, called a “modulator.” Throughout the paper, restrictions imposed on random measures (to be modulated) and the modulator are kept to a minimum. One of the objective of these studies are intensities and reward rates of modulated random measures that can play a significant role in stochastic control and optimization. Analytically tractable formulas for such functionals are obtained and examples and applications are discussed and treated in details.     

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

Abstract

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.     

Averaging principle for equation driven by a stochastic measure

ABSTRACT

Equation with the symmetric integral with respect to stochastic measure is considered. For the integrator, we assume only σ-additivity in probability and continuity of the paths. It is proved that the averaging principle holds for this case, the rate of convergence to the solution of the averaged equation is estimated.     

The Mean-Variance Hedging of a Defaultable Option with Partial Information

Abstract

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q ⁰, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.     

General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.     

Vague Convergence of Semimartingale Random Measures

Abstract

In this paper, we introduce and research the vague convergence of semimartingale random measures in distribution. The conditions are provided for the vague convergence of semimartingale random measures and the convergence of stochastic integrals with respect to semimartingale random measures in distribution.     

Characterization of Multidimensional Stable Random Measures by Means of Vector Measures

Abstract

In connection with a symmetric α stable random measure Φ on a measurable space (F, ?) with values in R ^d , a complete metric space of symmetric finite measures on S _d?1 is constructed, and is employed to characterize the law of Φ by a unique positive measure on ? and a unique function on F × R ^d . The stochastic integral ∫ _F f d Φ is also defined for certain d × d matrix valued functions f, which for α = 2 reduces to the Wiener–Masani integral.     

Dynamic risk measure for BSVIE with jumps and semimartingale issues

Risk measure is a fundamental concept in finance and in the insurance industry. It is used to adjust life insurance rates. In this article, we will study dynamic risk measures by means of backward stochastic Volterra integral equations (BSVIEs) with jumps. We prove a comparison theorem for such a type of equations. Since the solution of a BSVIEs is not a semimartingale in general, we will discuss some particular semimartingale issues.     

Integral representation of random set-valued measures

Abstract

We consider random set-valued measures with values in a separable Banach space. We prove two integral representation theorems using measurable multifunctions and set-valued integrals. The first theorem is valid for all separable Banach spaces, while the second holds for reflexive separable Banach spaces.     

Stochastic Integrals and Stochastic Differential Equations over the Field of p-Adic Numbers

We construct, for various classes of p-adic-valued functions, stochastic integrals with respect to the Poisson random measure. This leads to the construction of Markov processes over the field of p-adic numbers by means of stochastic differential equations.     

Continuity properties of decomposable probability measures on euclidean spaces

It is shown that every full e^A decomposable probability measure on R^k, where A is a linear operator all of whose eigenvalues have negative real part, is either absolutely continuous with respect to Lebesgue measure or continuous singular with respect to Lebesgue measure. This result is used to characterize the continuity properties of random variables that are limits of solutions of certain stochastic difference equations.     

On the Theory of (Dual) Projection for Fuzzy Stochastic Processes

Abstract

In a theory similar to one of real-valued stochastic processes, in this paper, we investigate the projection and dual projection for fuzzy stochastic processes. First, the related concepts of fuzzy stochastic processes are introduced, such as adaption, measurability, optionality, predictability, etc. Subsequently, we study fuzzy stochastic integral and fuzzy measure generated by increasing fuzzy stochastic processes. Moreover, (dual) projection w.r.t. (increasing) fuzzy stochastic processes are discussed. We prove the existence and uniqueness of (dual) optional (predictable) projection for (increasing) fuzzy stochastic processes.     

Investment with Sequence Losses in an Uncertain Environment and Mean-Variance Hedging

Abstract

In a market with a discontinuous filtration, whose price is influenced by a random factor, we study an optimization problem of an investor who is facing a sequence of losses driven by a Cox process. We give a form of variance-optimal martingale measure by changing the filtration. By using the solutions of the stochastic Riccati equation and another associated backward stochastic equation, we obtain a solution of the optimization problem of the investor.     

On a Class of Stochastic Semilinear PDEs

Abstract

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L ²(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.     

On the Lattice Approximation for Certain Generalized Vector Markov Fields in Four Spacetime Dimensions

We extend previous results by Albeverio, Iwata and Schmidt on the construction of a convergent lattice approximation for invariant scalar 3-vector generalized random fields F of an infinitely divisible type and apply them to the construction of convergent lattice approximation for the generalized random vector field A determined by the stochastic quaternionic Cauchy–Riemann equation A = F.     

Existence of Random Solutions of a General Class of Stochastic Functional Integral Equations

Abstract

In this paper, we establish the existence of solutions of a more general class of stochastic functional integral equations. The main tools here are the measure of noncompactness and the fixed point theorem of Darbo type. The results of this paper generalize the results of Rao–Tsokos [Rao, A.N.V.; Tsokos, C.P. A class of stochastic functional integral equations. Coll. Math. 1976, 35, 141–146.] and Szynal–Wedrychowicz [Szynal, D.; Wedrychowicz, S. On existence and an asymptotic behaviour of random solutions of a class of stochastic functional integral equations. Coll. Math. 1987, 51, 349–364.].     

Existence and uniqueness of path wise solutions for stochastic integral equations driven by Lévy noise on separable Banach spaces

The stochastic integrals of M- type 2 Banach valued random functions w.r.t. compensated Poisson random measures introduced in (Rüdiger, B., 2004, In: Stoch. Stoch. Rep., 76, 213–242.) are discussed for general random functions. These are used to solve stochastic integral equations driven by non Gaussian Lévy noise on such spaces. Existence and uniqueness of the path wise solutions are proven under local Lipshitz conditions for the drift and noise coefficients on M-type 2 as well as general separable Banach spaces. The continuous dependence of the solution on the initial data as well as on the drift and noise coefficients are shown. The Markov properties for the solutions are analyzed.     

Double stabilities of pullback random attractors for stochastic delayed p-Laplacian equations

We provide a method to study the double stabilities of a pullback random attractor (PRA) generated from a stochastic partial differential equation (PDE) with delays, such a PRA is actually a family of compact random sets A_ϱ(t,·), where t is the current time and ϱ is the memory time. We study its longtime stability, which means the attractor semiconverges to a compact set as the current time tends to minus infinity, and also its zero-memory stability, which means the delayed attractor semiconverges to the nondelayed attractor as the memory time tends to zero. The stochastic nonautonomous p-Laplacian equation with variable delays on an unbounded domain will be applied to illustrate the method and some suitable assumptions about the nonlinearity and time-dependent delayed forces can ensure existence, backward compactness, and double stabilities of a PRA.     

Random point fields with Markovian refinements and the geometry of fractally disordered media

We give a general construction of the probability measure for describing stochastic fractals that model fractally disordered media. For these stochastic fractals, we introduce the notion of a metrically homogeneous fractal Hansdorff-Karathéodory measure of a nonrandom type. We select a classF[q] of random point fields with Markovian refinements for which we explicitly construct the probability distribution. We prove that under rather weak conditions, the fractal dimension D for random fields of this class is a self-averaging quantity and a fractal measure of a nonrandom type (the Hausdorff D-measure) can be defined on these fractals with probability 1. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 490–505, September, 2000.     

Itô formula for stochastic integrals w.r.t. compensated Poisson random measures on separable Banach spaces

We prove the Ito formula (1.3) for Banach valued functions acting on stochastic processes with jumps, the martingale part given by stochastic integrals of time dependent Banach valued random functions w.r.t. compensated Poisson random measures. Such stochastic integrals have been discussed by Mandrekar and Rüdiger, Stochastics and Stochastic Reports 78(4), 189–212 (2006) and Rüdiger (2004), Stochastics and Stochastic Reports, 76, pp. 213–242.