Stochastic integral representations of quantum martingales on multiple Fock space

In this paper a quantum stochastic integral representation theorem is obtained for unbounded regular martingales with respect to multidimensional quantum noise. This simultaneously extends results of Parthasarathy and Sinha to unbounded martingales and those of the author to multidimensions. Dedicated to Professor Kalyan B Sinha on the occasion of his 60th birthday     

The optional stopping theorem for quantum martingales

In classical probability theory, a random time T is a stopping time in a filtration (Ft)_t?0 if and only if the optional sampling holds at T for all bounded martingales. Furthermore, if a process (Xt)_t?0 is progressively measurable with respect to (Ft)_t?0, then XT is FT-measurable. Unfortunately, this is not the case in noncommutative probability with the definition of stopped process used until now. It is shown in this article that we can define the stopping of noncommutative processes in Fock space in such a way that all the bounded martingales can be stopped at any stopping time T, are adapted to the filtration of the past before T and satisfy the optional stopping theorem.     

A Note on the LaSalle-Type Theorems for Stochastic Differential Delay Equations

The main aim of this note is to improve some results obtained in the author's earlier paper (1999, J. Math. Anal. Appl.236, 350-369). From the improved result follow some useful criteria on the stochastic asymptotic stability and boundedness.     

On Non-Continuous Dirichlet Processes

We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C ¹ transformation.     

Little's theorem: A stochastic integral approach

In a previous paper we have given a unified approach to the PASTA and the conditional PASTA property that is based upon the observation that the difference between the two limits can be represented as a stochastic integral with respect to a square integrable martingale. The equality of the two limits is then a consequence of a strong law of large numbers for martingales. In this paper we derive a non-standard version of Little's theorem via the same method. The moral of the story is that each of these theorems is but a particular case of a more general theory.     

Exponential stability for nonlinear stochastic differential equations with respect to semimartingales

Consider a nonlinear stochastic differential equations with respect to semimartingales (1) dY(t) = F{Y(t),t)dti(t) + G(t)dM(t)+f(Y(t),t)dti(t)+g(Y(t),t)dM(t), which might be regarded as a stochastic perturbed system of (2) dX(t) = F(X(t),t)dfi(t) + G(t)dM(t). Suppose Eq. (2) is exponentially stable almost surely. Under what conditions is Eq. (1) still exponentially stable almost surely? In this paper we will give some sufficient conditions. As an application we also discuss the almost sure exponential stability for semilinear stochastic systems and small stochastic perturbed systems     

The Itô integral for Brownian motion in vector lattices: Part 1

In this paper the Itô integral for Brownian motion is constructed in a vector lattice and some of its properties are derived. The assumption is that there exists a conditional expectation operator on the vector lattice and the construction does not depend on a probability measure space. The classical case of the Itô integral is a special case of the constructed integral in the vector lattice.     

Interpolation solution in generalized stochastic exponential population growth model

In this paper, first we consider model of exponential population growth, then we assume that the growth rate at time t is not completely definite and it depends on some random environment effects. For this case the stochastic exponential population growth model is introduced. Also we assume that the growth rate at time t depends on many different random environment effect, for this case the generalized stochastic exponential population growth model is introduced. The expectations and variances of solutions are obtained. For a case study, we consider the population growth of Iran and obtain the output of models for this data and predict the population individuals in each year.     

Stochastic moment problem and hedging of generalized Black-Scholes options

In mathematical finance one is interested in the quadratic error which occurs while replacing a continuously adjusted portfolio by a discretely adjusted one. We first study higher order approximations of stochastic integrals. Then we apply the results to quantify quadratic error which occurs in estimating the discretely adjusted hedging risk in pricing European options in a generalized Black-Scholes market.     

Stochastic integral with respect to set-valued square integrable martingales

In this paper, we shall firstly illustrate why we should consider integral of a stochastic process with respect to a set-valued square integrable martingale. Secondly, we shall prove the representation theorem of set-valued square integrable martingale. Thirdly, we shall give the definition of stochastic integral of a stochastic process with respect to a set-valued square integrable martingale and the representation theorem of this kind of integrals. Finally, we shall prove that the stochastic integral is a set-valued sub-martingale.     

Weak convergence of the Stratonovich integral with respect to a class of Gaussian processes

For a Gaussian process X$X$ and smooth function f$f$, we consider a Stratonovich integral of f(X)$f\left(X\right)$, defined as the weak limit, if it exists, of a sequence of Riemann sums. We give covariance conditions on X$X$ such that the sequence converges in law. This gives a change-of-variable formula in law with a correction term which is an Itô integral of f^?$f^{?}$ with respect to a Gaussian martingale independent of X$X$. The proof uses Malliavin calculus and a central limit theorem from Nourdin and Nualart (2010) [8]. This formula was known for fBm with H=1/6$H=1/6$ Nourdin et al. (2010) [9]. We extend this to a larger class of Gaussian processes.     

Stochastic Analysis of the Fractional Brownian Motion

Since the fractional Brownian motion is not a semi-martingale, the usual Ito calculus cannot be used to define a full stochastic calculus. However, in this work, we obtain the Itô formula, the Itô–Clark representation formula and the Girsanov theorem for the functionals of a fractional Brownian motion using the stochastic calculus of variations.     

Sesquilinear quantum stochastic analysis in Banach space

A theory of quantum stochastic processes in Banach space is initiated. The processes considered here consist of Banach space valued sesquilinear maps. We establish an existence and uniqueness theorem for quantum stochastic differential equations in Banach modules, show that solutions in unital Banach algebras yield stochastic cocycles, give sufficient conditions for a stochastic cocycle to satisfy such an equation, and prove a stochastic Lie–Trotter product formula. The theory is used to extend, unify and refine standard quantum stochastic analysis through different choices of Banach space, of which there are three paradigm classes: spaces of bounded Hilbert space operators, operator mapping spaces and duals of operator space coalgebras. Our results provide the basis for a general theory of quantum stochastic processes in operator spaces, of which Lévy processes on compact quantum groups is a special case.     

Stochastic PDEs Driven by Nonlinear Noise and Backward Doubly SDEs

We study a new kind of backward doubly stochastic differential equations, where the nonlinear noise term is given by Itô–Kunita's stochastic integral. This allows us to give a probabilistic interpretation of classical and Sobolev's solutions of semilinear parabolic stochastic partial differential equations driven by a nonlinear space-time noise.     

Stochastic integration on partially ordered sets

The stochastic integral is introduced with respect to a stochastic process X = (X_s)_sεV, where V is any general partially ordered set satisfying some mild regularity conditions. As important examples the stochastic integral is constructed with respect to a class of Gaussian processes having similarities to the Brownian motion on the real line, and also with respect to L²-martingales under an assumption of conditional independence on the underlying σ-fields.     

The Itô integral for Brownian motion in vector lattices: Part 2

The Itô integral for Brownian motion in a vector lattice, as constructed in Part 1 of this paper, is extended to accommodate a larger class of integrands. This extension provides an analogue of the indefinite Itô integral in the classical setting which yields a local martingale. The assumption is that there exists a conditional expectation operator on the vector lattice and the construction does not depend on a probability measure space. The classical case of the extended Itô integral is a special case of the constructed integral in the vector lattice.     

Fractional smoothness for the generalized local time of the indefinite Skorohod integral

Let be the indefinite Skorohod integral on Wiener space (Ω,H,P), and let Lt(x) be its the generalized local time introduced by Tudor in [C.A. Tudor, Martingale-type stochastic calculus for anticipating integral processes, Bernoulli 10 (2004) 313-325]. We prove that the generalized local time, as a nonlinear functional of ω, is in the fractional Sobolev spaces Dα,p ( and p>2) under some conditions imposed on the anticipating integrand u via the technique of Malliavin calculus and the K-method in the real interpolation theory. The result is optimal for the fractional Brownian motion with the Hurst parameter .     

Wiener distributions and white noise analysis

The paper describes the structure of a new space of generalized Wiener functionals, , called the Wiener algebra, or space of Wiener distributions, and demonstrates its use in the white noise analysis. The concepts of derivatives and integrals for multi-time parameter generalized stochastic process:^N are introduced, and a derivative version of Itô's lemma is proved. The algebraic structure of and its lattice of subspaces is elaborated, and within this framework a generalized version of the Malliavin calculus is presented.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.