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 integral representation theorem for quantum semimartingales

The quantum stochastic integral of Itô type formulated by Hudson and Parthasarathy is extended to a wider class of adapted quantum stochastic processes on Boson Fock space. An Itô formula is established and a quantum stochastic integral representation theorem is proved for a class of unbounded semimartingales which includes polynomials and (Wick) exponentials of the basic martingales in quantum stochastic calculus.     

On Dirichlet Processes Associated with Second Order Divergence Form Operators

Let be a d - dimensional Markov family corresponding to a uniformly elliptic second order divergence form operator. We show that for any quasi continuous in the Sobolev space the process (X) admits under P ^x a decomposition into a martingale additive functional (AF) M and a continuous AF A of zero quadratic variation for almost every starting point x if q=2, for quasi every x if q>2 and for every if is continuous, d=1 and or d>1 and q>d. Our decomposition enables us to show that in the case of symmetric operator the energy of A equals zero if q=2 and that the decomposition of (X) into the martingale AF M and the AF of zero energy A is strict if for some q>d. Moreover, our decomposition provides a probabilistic representation of A .     

Markov dilation of Diffusion Type Processes and Its Application to the Financial Mathematics

The Markov dilation of diffusion type processes is defined. Infinitesimal operators and stochastic differential equations for the obtained Markov processes are described. Some applications to the integral representation for functionals of diffusion type processes and to the construction of a replicating portfolio for a non-terminal contingent claim are considered.     

Integration with respect to local time

Let be the local time process of a linear Brownian motion B. We integrate the Borel functions on with respect to . This allows us to write Itôrs formula for new classes of functions, and to define a local time process of B on any borelian curve. Some results are extended from deterministic to random functions.     

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.     

Change of variable formulas for non-anticipative functionals on path space

We derive a change of variable formula for non-anticipative functionals defined on the space of Rd-valued right-continuous paths with left limits. The functionals are only required to possess certain directional derivatives, which may be computed pathwise. Our results lead to functional extensions of the Itô formula for a large class of stochastic processes, including semimartingales and Dirichlet processes. In particular, we show the stability of the class of semimartingales under certain functional transformations.     

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 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.     

A note on a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation

In this paper, we show there is a stationary distribution of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation and it has ergodic property.     

Stochastic calculus for Markov processes associated with non-symmetric Dirichlet forms

Nakao’s stochastic integrals for continuous additive functionals of zero energy are extended from the symmetric Dirichlet forms setting to the non-symmetric Dirichlet forms setting.It? ’s formula in terms of the extended stochastic integrals is obtained.     

An Extension of Itô's Formula for Anticipating Processes

In this paper we introduce a class of square integrable processes, denoted by L^F, defined in the canonical probability space of the Brownian motion, which contains both the adapted processes and the processes in the Sobolev space L^2,2. The processes in the class L^F satisfy that for any time t, they are twice weakly differentiable in the sense of the stochastic calculus of variations in points (r, s) such that r s t. On the other hand, processes belonging to the class L^F are Skorohod integrable, and the indefinite Skorohod integral has properties similar to those of the Ito integral. In particular we prove a change-of-variable formula that extends the classical Itô formula. Those results are generalization of similar properties proved by Nualart and Pardoux⁽⁷⁾ for processes in L^2,2.     

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.     

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.     

非对称Dirichlet型的扰动及其结合的Markov过程

该文研究非对称Ｄｉｒｉｃｈｌｅｔ型的扰动及其结合的Ｍａｒｋｏｖ过程．讨论一般状态空间上的拟正则Ｄｉｒｉｃｈｌｅｔ型（ε,犇（ε））关于光滑符号测度μ的扰动,这里μ＝μ＋ －μ－ ,μ＋ 为光滑测度,μ－ 属于Ｋａｔｏ类,证明了扰动型（εμ,犇（εμ））犺 结合犿 胎紧犿 特别标准的Ｍａｒｋｏｖ过程．     

Dual variational formulas for the first Dirichlet eigenvalue on half-line

The aim of the paper is to establish two dual variational formulas for the first Dirichlet eigenvalue of the second order elliptic operators on half-line. Some explicit bounds of the eigenvalue depending only on the coefficients of the operators are presented. Moreover, the corresponding problems in the discrete case and the higher-order eigenvalues in the continuous case are also studied.     

The Exterior Dirichlet Problem for the Biharmonic Equation: Solutions with Bounded Dirichlet Integral

We study the unique solvability of the Dirichlet problem for the biharmonic equation in the exterior of a compact set under the assumption that a generalized solution of this problem has a bounded Dirichlet integral with weight |x|^a. Depending on the value of the parameter ^a,a we prove uniqueness theorems or present exact formulas for the dimension of the solution space of the Dirichlet problem.     

Quadratic Covariation and Itô's Formula for Smooth Nondegenerate Martingales

In this paper we prove the existence of the quadratic covariation [f(X),X], where f is a locally square integrable function and X _t = ^t ₀ u _s dW _s is a smooth nondegenerate Brownian martingale. This result is based on some moment estimates for Riemann sums which are established by means of the techniques of the Malliavin calculus.     

On the Dirichlet problem for asymmetric zero-range process on increasing domains

In order to obtain hitting time estimates for the asymmetric zero-range process (AZRP) on ^d, in dimensions d3, we characterize the principal eigenvalue of the generator of the AZRP with Dirichlet boundary on special domains. We obtain a Donsker-Varadhan variational representation and show that the corresponding eigenfunction is unique in a natural class of functions.Mathematics Subject Classification (2000):60K35, 82C22, 60J25