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The spike variation technique plays a crucial role in deriving Pontryagin's type maximum principle of optimal controls for ordinary differential equations (ODEs), partial differential equations (PDEs), stochastic differential equations (SDEs), and (deterministic forward) Volterra integral equations (FVIEs), when the control domains are not assumed to be convex. It is natural to expect that such a technique could be extended to the case of (forward) stochastic Volterra integral equations (FSVIEs). However, by mimicking the case of SDEs, one encounters an essential difficulty of handling an involved quadratic term. To overcome this difficulty, we introduce an auxiliary process for which one can use It's formula, and develop new technologies inspired by stochastic linear-quadratic optimal control problems. Then the suitable representation of the above-mentioned quadratic form is obtained, and the second-order adjoint equations are derived. Consequently, the maximum principle of Pontryagin type is established. Some relevant extensions are investigated as well. [ABSTRACT FROM AUTHOR]

We consider the optimal control problem of stochastic evolution equations in a Hilbert space under a recursive utility, which is described as the solution of a backward stochastic differential equation (BSDE). A very general maximum principle is given for the optimal control, allowing the control domain to not be convex and the generator of the BSDE to vary with the second unknown variable z. The associated second-order adjoint process is characterized as a unique solution of a conditionally expected operator-valued backward stochastic integral equation. [ABSTRACT FROM AUTHOR]

We present an approach to proving parabolic Aleksandrov estimates with mixed norms for stochastic integrals with singular "moderated" drift. [ABSTRACT FROM AUTHOR]

This paper deals with the expectation of monomials with respect to the stochastic area integral A1,2(t,t+h)=∫t+ht∫stdW1(r)dW2(s)-∫t+ht∫stdW2(r)dW1(s) and the increments of two Wiener processes, ΔWi(t,t+h)=Wi(t+h)-Wi(t), i=1,2. In a monomial, if the exponent of one of the Wiener increments or the stochastic area integral is an odd number, then the expectation of the monomial is zero. However, if the exponent of any of them is an even number, then the expectation is nonzero and its exact value is not known in general. In the present paper, we derive formulae to give the value in general. As an application of the formulae, we will utilize the formulae for a careful stability analysis on a Magnus-type Milstein method. As another application, we will give some mixed moments of the increments of Wiener processes and stochastic double integrals. [ABSTRACT FROM AUTHOR]

A linear-quadratic optimal control problem for a forward stochastic Volterra integral equation (FSVIE) is considered. Under the usual convexity conditions, open-loop optimal control exists, which can be characterized by the optimality system, a coupled system of an FSVIE and a type-II backward SVIE (BSVIE). To obtain a causal state feedback representation for the open-loop optimal control, a path-dependent Riccati equation for an operator-valued function is introduced, via which the optimality system can be decoupled. In the process of decoupling, a type-III BSVIE is introduced whose adapted solution can be used to represent the adapted M-solution of the corresponding type-II BSVIE. Under certain conditions, it is proved that the path-dependent Riccati equation admits a unique solution, which means that the decoupling field for the optimality system is found. Therefore, a causal state feedback representation of the open-loop optimal control is constructed. An additional interesting finding is that when the control only appears in the diffusion term, not in the drift term of the state system, the causal state feedback reduces to a Markovian state feedback. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the asymptotic behavior of stochastic pantograph delay evolution equations driven by a tempered fractional Brownian motion (tfBm) with Hurst parameter H > 1/2. First of all, the global existence, uniqueness, and mean-square stability with general decay rate of mild solutions are established. In particular, we would like to point out that our analysis is not necessary to construct Lyapunov functions, but we deal directly with stability via the Banach fixed point theorem, the fractional power of operators, and the semigroup theory. It is worth emphasizing that a novel estimate of stochastic integrals with respect to tfBm is presented, which greatly contributes to the stability analyses. Then after extending the factorization formula to the tfBm case, we construct the nontrivial equilibrium solution, defined for t R, by means of an approximation technique and a convergence analysis. Moreover, we analyze the Hölder regularity in time and general stability (including both polynomial and logarithmic stability) of the nontrivial equilibrium solution in the sense of mean-square. As an example of application, the reaction diffusion neural network system with pantograph delay is considered, and the nontrivial equilibrium solution and general stability of the system are proved under the Lipschitz assumption. [ABSTRACT FROM AUTHOR]

In this article, a class of backward stochastic Volterra integro-differential equations (BSVIDEs) is introduced and studied. It is worthy mentioning that the proposed BSVIDEs cannot be covered by the existing backward stochastic Volterra integral equations (BSVIEs), and they also have the nice flow property such that Itô's formula becomes quite applicable. It is found that BSVIDEs can provide a neat sufficient condition for the solvability of BSVIEs with generator depending on the diagonal value of the solutions. As applications, the optimal control problems in terms of maximum principles and linear quadratic control problems of optimal control for forward stochastic Volterra integro-differential equations are investigated. In contrast with the BSVIEs in current literature, some interesting phenomena and advantages of BSVIDEs are revealed. [ABSTRACT FROM AUTHOR]

The full discretization of the semilinear stochastic wave equation is considered. The discontinuous Galerkin finite element method is used in space and analyzed in a semigroup framework, and an explicit stochastic position Verlet scheme is used for the temporal approximation. We study the stability under a CFL condition and prove optimal strong convergence rates of the fully discrete scheme. Numerical experiments illustrate our theoretical results. Further, we analyze and bound the expected energy and numerically show excellent agreement with the energy of the exact solution. [ABSTRACT FROM AUTHOR]

We prove that the set of laws of stochastic integrals H ·W, where W is a multidimensional Wiener process and H2 takes values in a compact convex subset D of the set of symmetric positive semidefinite matrices, is weakly dense in the set of laws of martingales M with d⟨M⟩/dt taking values in D. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analysis and verification of stability for a class of stochastic hybrid systems (SHSs), which contain not only continuous flows described by stochastic differential equations but also Markovian switching and discrete jumps together. To start with, taking advantage of multiple Lyapunov functions (MLFs), we propose a series of sufficient conditions for (uniform) stability in probability, (uniform) asymptotic stability in probability, stochastically asymptotic stability in the large and (uniform) exponential stability in probability, respectively, in which indefinite scalar functions are utilized to relax the conservativeness of the classical MLFs. Then, based on the generalized exponential martingale inequality for L\'evy type stochastic integral, we extend our less conservative MLFs and indefinite scalar functions based method to obtain a sufficient condition for globally almost sure exponential stability. Further, we investigate the mechanical verification of all the above stabilities for rational SHSs by proposing a semidefinite programming based computational approach. Especially, our mechanical verification approach can also be applied to deal with certain systems beyond rational SHSs. The applicabilities of both our theoretical results and mechanical approach are demonstrated by academic examples. [ABSTRACT FROM AUTHOR]

A classical problem in stochastic ergodic control consists of studying the limit behavior of the optimal value of a discounted integral in infinite horizon (the so called Abel mean of an integral cost) as the discount factor tends to zero or the value defined with a Cesaro mean of an integral cost when the horizon T tends to +1. We investigate the possible limits in the norm of uniform convergence topology of values defined through Abel means or Cearo means when ! 0+ and T ! +1, respectively. Here we give two types of new representation formulas for the accumulation points of the values when the averaging parameter converges. We show that there is only one possible accumulation point which is the same for Abel means or Cesaro means. The first type of representation formula is based on probability measures on the product of the state space and the control state space, which are limits of occupational measures. The second type of representation formula is based on measures which are the projection of invariant measure on the space of relaxed controls. We also give a result comparing the both sets of measures involved in both classes of representation formulas. An important consequence of the representation formulas is the existence of the limit value when one has the equicontinuity property of Abel or Cesaro mean values. This is the case, for example, for nonexpansive stochastic control systems. In the end some insightful examples are given which help to better understand the results. [ABSTRACT FROM AUTHOR]

We propose and study a new multivariate stochastic scattered data quasi-interpolation scheme that is reminiscent of the classical Monte Carlo method for estimating integrals. We first employ a convolution operator to approximate (deterministically) Sobolev space functions and use a result of Cheney, Light, and Xu [On kernels and approximation orders, in Approximation Theory, Lecture Notes in Pure Appl. Math. 138, Dekker, 1992, pp. 227--242] and Cheney and Lei [Quasiinterpolation on irregular points, in Approximation and Computation, Internat. Ser. Numer. Math. 119, Birkh\"auser Boston, 1994, pp. 121--135] to obtain an approximation error estimate in terms of moment conditions. We then approximate (stochastically) the convolution integral using a Monte Carlo method and derive the maximal mean squared error (M-MSE) estimate and mean Lp-error estimate on bounded domains which are in line with those obtained by the classical Monte Carlo method for estimating multivariate integrals. The introduction of convolution operators is solely for the purpose of facilitating error analysis. The implementation of this scheme does not require any numerical handling of the convolution integral involved. Our final approximant is in the form of scattered data quasi-interpolation. It enjoys a simple construction and optimal convergence rate, yet it provides an efficient tool in various computing environments. Asymptotic normality and confidence interval test results show that the scheme is computationally stable. Numerical simulation results show that the scheme is robust in the presence of noise. [ABSTRACT FROM AUTHOR]

We present remarkably simple proofs of Burkholder–Davis–Gundy inequalities for stochastic integrals and maximal inequalities for stochastic convolutions in Banach spaces driven by Lévy-type processes. Exponential estimates for stochastic convolutions are obtained and two versions of Itô’s formula in Banach spaces are also derived. Based on the obtained maximal inequality, the existence and uniqueness of mild solutions of stochastic quasi-geostrophic equation with Lévy noise is established. [ABSTRACT FROM AUTHOR]

In this paper we propose a method for finding an integrand in the Clark stochastic integral representation of quadratic integrable Brownian functional F with Malliavin differentiable conditional mathematical expectation E[F|JtB], t < T, where (JtB) is a natural Brownian filtration. The proposed method allows us to determine the explicit form of the integrand in the case when F has no Malliavin derivative. We give several illustrations as examples. [ABSTRACT FROM AUTHOR]

We derive exponential upper bounds for the distribution tails of multiple stochastic integrals of bounded nonrandom functions. In this paper we consider integrating processes from certain wide classes. Some important examples of integrating processes are given. Multiple stochastic integrals with respect to increments of such processes arise in statistical applications. [ABSTRACT FROM AUTHOR]

In this work, we are concerned with the high-order numerical methods for coupled forward-backward stochastic differential equations (FBSDEs). Based on the FBSDEs theory, we derive two reference ordinary differential equations (ODEs) from the backward SDE, which contain the conditional expectations and their derivatives. Then, our high-order multistep schemes are obtained by carefully approximating the conditional expectations and the derivatives, in the reference ODEs. Motivated by the local property of the generator of diffusion processes, the Euler method is used to solve the forward SDE; however, it is noticed that the numerical solution of the backward SDE is still of high-order accuracy. Such results are obviously promising: on one hand, the use of the Euler method (for the forward SDE) can dramatically simplify the entire computational scheme, and on the other hand, one might be only interested in the solution of the backward SDE in many real applications such as option pricing. Several numerical experiments are presented to demonstrate the effectiveness of the numerical method. [ABSTRACT FROM AUTHOR]

A minimum contrast criterion for consistently estimating the spatial fractal dimension in the wavelet domain is presented in [M. D. Ruiz-Medina and R. M. Crujeiras, Comm. Statist. Theory Methods., 40 (2011), pp. 3599–3613] for a class of semiparametric fractal Gaussian random fields, which includes fractional Brownian motion and Riesz–Bessel motion as particular cases. In this work, asymptotic normality of this minimum contrast estimator is obtained using central limit results for multiple stochastic integrals (see [N. Nualart and G. Peccati, Ann. Probab., 33 (2005), pp. 91–115]). [ABSTRACT FROM AUTHOR]

A generalized stochastic integral of predictable generalized random processes (predictable random processes) with respect to a real Wiener process is defined. The question of the existence of the stochastic integral in a Banach space is reduced to the problem of decomposability of the generalized random element. The stochastic differential equation for generalized random processes is introduced and existence and uniqueness of solutions are developed. As a consequence the corresponding results on stochastic differential equations in a Banach space are given. [ABSTRACT FROM AUTHOR]

This paper contains sufficient conditions of measurability with respect to a parameter of the limit of a sequence of random fields with trajectories in D. Measurability with respect to a parameter of the quadratic variation of the two-parameter strong martingale was proved. We obtain sufficient conditions of measurability with respect to a parameter of the stochastic integral with respect to the two-parameter strong martingale. [ABSTRACT FROM AUTHOR]

By using multiple Wiener-Itô stochastic integrals, we study the cubic variation of a class of self-similar stochastic processes with stationary increments (the Rosenblatt process with self-similarity order H∊(½, 1)). This study is motivated by statistical purposes. We prove that this renormalized cubic variation satisfies a noncentral limit theorem, and its limit is (in the L²(Ω) sense) still the Rosenblatt process. [ABSTRACT FROM AUTHOR]

A numerical predictor-corrector method is presented for the integration of the Langevin equations with holonomic constraints, specifically Kramers' ball-rod model for long-chain polymers. In the predictor, constraint forces are expressed as Lagrange multipliers and evaluated as implicit functions of particle coordinates. The resulting expressions, with and without the Duhamel form, are developed as a strong second-order Itô-Taylor method by the expansion technique of Wagner and Platen. The corrector is a projection that enforces constraints to machine precision. The numerical evaluation of the stochastic integrals, as well as the "coarsening" relations used to recursively construct integral sets when the time step is doubled, are described in detail. We present numerical simulations that demonstrate the strong and weak orders of convergence. With parameters approximating λ-phage DNA, the Duhamel formulation is stable with time steps exceeding the relaxation time. As a validation study we evolve initially linear polymers and observe that they evolve to configurations with mean square end-to-end distance predicted by analytical theory. We also simulate a velocity correlation function in agreement with classical theory. [ABSTRACT FROM AUTHOR]

Symmetric integrals of the Stieltjes type for an arbitrary continuous function and which are determinate versions of the Stratonovich stochastic integrals are defined. It is shown that the solution of a pathwise analogue of stochastic differential equations with a symmetric integral and Itô stochastic differential equations is reduced to a solution of a system of ordinary differential equations. The relation between improper symmetric integrals, which is extended to symmetric integrals and Hellinger integrals, is established. [ABSTRACT FROM AUTHOR]

Construction of splitting-step methods and properties of related nonnegativity and boundary preserving semianalytic numerical algorithms for solving stochastic differential equations (SDEs) of Itô type are discussed. As the crucial assumption, we oppose conditions such that one can decompose the original system of SDEs into subsystems for which one knows either the exact solution or its conditional transition probability. We present convergence proofs for a newly designed splittingstep algorithm and simulation studies for numerous well-known numerical examples ranging from stochastic dynamics occurring in asset pricing in mathematical finance (Cox-Ingersoll-Ross (CIR) and constant elasticity of variance (CEV) models) to measure-valued diffusion and super-Brownian motion (stochastic PDEs (SPDEs)) as met in biology and physics. [ABSTRACT FROM AUTHOR]

In the first part of this paper [A. N. Shiryaev and M. Yor, Theory Probab. Appl., 48 (2004), pp. 304–313], a method of obtaining stochastic integral representations of functionals S(w) of Brownian motion B = (Bt)t≧0 was stated. Functionals maxt≦T Bt and maxt≦T-a Bt, where T-a = inf{t: Bt = -a}, a > 0, were considered as an illustration. In the present paper we state another derivation of representations for these functionals and two proofs of representation for functional maxt≦gT St, where (non-Markov time) gT = sup{0 ≦ t ≦ T: Bt = 0} are given. [ABSTRACT FROM AUTHOR]

In the paper the limiting behavior of finite sums with averaging containing Lévy processes is considered. For this purpose a new class of stochastic integrals for Lévy processes including, in particular, the Itô integral, Stratonovich integral, and others, is defined. By using these integrals complete classification of the limiting behavior of the considered sums is given. [ABSTRACT FROM AUTHOR]

We study various partially ordered spaces of probability measures and we determine which of them are lattices. This has important consequences for optimization problems with stochastic dominance constraints. In particular we show that the space of probability measures on ℝ is a lattice under most of the known partial orders, whereas the space of probability measures on ℝd typically is not. Nevertheless, some subsets of this space, defined by imposing strong conditions on the dependence structure of the measures, are lattices. [ABSTRACT FROM AUTHOR]

In this paper the construction of a stochastic integral of a nonrandom function is suggested without the classical orthogonality condition of the noise. This construction includes some known constructions of univariate and multiple stochastic integrals. Conditions providing the existence of this integral are specified for noises generated by random processes with nonorthogonal increments from certain classes which are rich enough. [ABSTRACT FROM AUTHOR]

This paper introduces the concept of σ-localization, which is a generalization of localization in the general theory of stochastic processes. The σ-localized class derived from the set of martingales is the class of σ-martingales, which plays an important role in mathematical finance. These processes and the corresponding σ-martingale measures are considered in detail. By extending the stochastic integral with respect to compensated random measures, a canonical representation of σ-martingales as for local martingales is derived. [ABSTRACT FROM AUTHOR]

The definition of a generalized additive homogeneous functional of a Wiener process is introduced. It is shown that a generalized functional is uniquely specified by its characteristics. In this case the functions from the Schwartz space S* of slowly growing generalized functions play the role of generating functions. [ABSTRACT FROM AUTHOR]

By the Dambis-Dubins-Schwarz theorem, any stochastic integral M ≔&inte;[SUP.],[SSUB0]H[SUBs]Dw[sSUBs] of Brownian motion can be written as a time-changed Brownian motion, i.e., M = (Ŵ[SUB&Tcirc;[SUBt]])[SUBt∊[SUBR+]] for some Brownian motion (Ŵ[SUBθ])[SUBθ∊R[SUB+]] and some time change (&Tcirc;[SUBt])[SUBt∊R[SUB+]. In [J. Jacod and A. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin-Heidelberg, 1987] and [O. Kallenberg, Stochastic Process. Appl., 40 (1992), pp. 199-223] it is shown that in this statement Brownian motion can be replaced with (symmetric) α-stable Léevy motion. Using the cumulant process of a semimartingale, we give new short proofs. Moreover, we show that the statement cannot be extended to any other Léevy processes. [ABSTRACT FROM AUTHOR]

The theme of providing predictable criteria for absolute continuity and for mutual singularity of two density processes on a filtered probability space is extensively studied, e.g., in the monograph by J. Jacod and A. N. Shiryaev [Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987]. While the issue of absolute continuity is settled there in full generality, for the issue of mutual singularity one technical difficulty remained open [J. Jacod and A. N. Shiryaev, Limit Theorems for Stochastic Processes, Springer-Verlag, Berlin, 1987, p. 210]: "We do not know whether it is possible to derive a predictable criterion (necessary and sufficient condition) for P'[SUBT] ⊥ P[SUBT],...". It turns out that there are two answers to this question raised in the monograph of J. Jacod and A. N. Shiryaev: On the negative side, we give an easy example showing that in general the answer is no, even when we use a rather wide interpretation of the concept of "predictable criterion." The difficulty comes from the fact that the density process of a probability measure P with respect to another measure P' may suddenly jump to zero. On the positive side, we can characterize the set where P' becomes singular with respect to P -- provided this happens in a continuous way rather than suddenly -- as the set where the Hellinger process diverges, which certainly is a "predictable criterion." This theorem extends results in the monograph of J. Jacod and A. N. Shiryaev. [ABSTRACT FROM AUTHOR]

A condition is given in terms of the absolute moments of stochastic integrals for two stochastic processes, continuous in probability with independent stationary symmetric increments, to be identical. characterization, stochastic integral, stochastic process, absolute moment [ABSTRACT FROM AUTHOR]

Information on the Fifth International Conference titled Limit Theorems in Probability Theory and Their Applications, which was held at the Sobolev Institute of Mathematics in Novosibirsk, Russia from August 15-18, 2011 is presented. Topics include central limit theorem, random walks, and stochastic integrals. The conference featured several speakers including mathematicians A. A. Borovkov and T. Mikosh, and author I. A. Ibragimov.