Showing papers in "Stochastic Analysis and Applications in 2002"
TL;DR: In this paper, the authors consider linear controlled stochastic systems subjected both to white noise disturbance and Markovian jumping, and derive necessary and sufficient conditions for the zero solution of a linear stochnastic system with multiplicative white noise perturbations.
Abstract: In this paper we consider linear controlled stochastic systems subjected both to white noise disturbance and Markovian jumping. Our aim is to provide a mathematical background in order to give unified approach for a large class of problems associated to linear controlled systems subjected both to multiplicative white noise perturbations and Markovian jumping. First we prove an Ito type formula. Our result extends the result of Ref. [24], to the case when the stochastic process x(t) has not all moments bounded. Necessary and sufficient conditions assuring the exponential stability in mean square for the zero solution of a linear stochastic system with multiplicative white noise and Markovian jumping are provided. Some estimates for solutions of affine stochastic systems are derived, and necessary and sufficient conditions assuring the stochastic stabilizability and stochastic detectability are given. A stochastic version of Bounded Real Lemma is proved and several aspects of the problem of robust stabiliza...
98 citations
TL;DR: In this paper, the existence and uniqueness of a backward stochastic nonlinear Volterra integral equation under global Lipschitz condition was studied, where W t is a standard k-dimensional Wiener process defined on a probability space and X is a measurable d-dimensional random vector.
Abstract: This paper studies the existence and uniqueness of the following kind of backward stochastic nonlinear Volterra integral equation under global Lipschitz condition, where {W t ;t∈[0,T]} is a standard k-dimensional Wiener process defined on a probability space {Ω,F,F t ,P}, and X is {F T } measurable d-dimensional random vector. The problem is to look for an adapted pair of processes {X(t),Z(t,s);t∈[0,T],s∈[t,T]} with values in R d and R d×k respectively, which solves the above equation. This paper also generalize our results to the following equation: under rather restrictive assumptions on g. *This research was supported by the National Natural Science Foundation of China, program no. 79790130.
89 citations
TL;DR: A strong law of large numbers for arrays of rowwise negatively dependent random variables is obtained in this article, and the moment conditions of the main result are similar to previous results, and the stochastic bounded condition also provides a relaxation of the usual distributional assumptions.
Abstract: A strong law of large numbers for arrays of rowwise negatively dependent random variables is obtained which relaxes the usual assumption of rowwise independence. The moment conditions of the main result are similar to previous results, and the stochastic bounded condition also provides a relaxation of the usual distributional assumptions.
64 citations
TL;DR: In this paper, the chaos expansion of local time l T (H)(x,·) of fractional Brownian motion with Hurst coefficient H ∈(0, 1) at a point x∈R d was studied.
Abstract: We find the chaos expansion of local time l T (H)(x,·) of fractional Brownian motion with Hurst coefficient H∈(0,1) at a point x∈R d . As an application we show that when H 0 d<1 then l T (H)(x,·)∈L 2(μ). Here μ denotes the probability law of B (H) and H 0=max{H 1,…,H d }. In particular, we show that when d=1 then l T (H)(x,·)∈L 2(μ) for all H∈(0,1).
37 citations
TL;DR: This paper considers M/G/1 queuing systems governed by a stochastic clearing mechanism, called "disaster", which removes all workload in the system whenever it occurs to the system, and presents the system size distribution and the sojourn time distribution.
Abstract: In this paper, we consider M/G/1 queuing systems governed by a stochastic clearing mechanism, called “disaster,” which removes all workload in the system whenever it occurs to the system. The clearing mechanism of disasters can be applied to computer systems in the presence of a virus as a clearing operation of all stored messages present in the system. We present the system size distribution and the sojourn time distribution.
36 citations
TL;DR: In this article, the existence and uniqueness of solutions for stochastic evolution equations containing some hereditary characteristics are proved from a variational point of view and in a general functional setting which permit us to deal with several kinds of delay terms in a unified formulation.
Abstract: Some results on the existence and uniqueness of solutions for stochastic evolution equations containing some hereditary characteristics are proved. In fact, our theory is developed from a variational point of view and in a general functional setting which permit us to deal with several kinds of delay terms in a unified formulation.
29 citations
TL;DR: In this article, the authors studied the order of uniform strong convergence of two Local Linear (LL) approximations to the solution of stochastic differential equations (SDEs) with additive noise.
Abstract: This paper studies the order of uniform strong convergence of two Local Linear (LL) approximations to the solution of stochastic differential equations (SDEs) with additive noise. The results obtained cover multi-dimensional and non-autonomous SDEs, and also ordinary differential equations with random initial conditions. It is demonstrated that the global order of convergence of one of the LL approximations considered is actually larger than that reported in an earlier paper, so solving an apparent discrepancy between theory and recent simulation studies.
24 citations
TL;DR: In this article, the authors focus on the countably infinite state space case and show that stochastic stability and mean square stability are not equivalent in this setting, unlike in the finite state space setting.
Abstract: We deal with linear systems with Markovian Jump Parameters (LSMJP). Most of the literature on this matter adopts a finite state space for the Markov chain. In this paper we focus on the countably infinite state space case showing that, unlike the finite state space case, two important concepts in optimal control theory, namely, stochastic stability (SS) and mean square stability (MSS) are no longer equivalent in this setting. *Research supported in part by the Brazilian National Research Council—CNPq and PRONEX.
24 citations
TL;DR: In this article, the authors established an existence and uniqueness result for semilinear stochastic differential equations in Hilbert space dX=(AX+f(X))dt+g(X)dW under weaker conditions than the Lipschitz one by investigating the convergence of successive approximations.
Abstract: In this paper we first shall establish an existence and uniqueness result for the semilinear stochastic differential equations in Hilbert space dX=(AX+f(X))dt+g(X)dW under weaker conditions than the Lipschitz one by investigating the convergence of the successive approximations. Secondly we show, under the assumption of so-called pathwise uniqueness (PU), the convergence of the Euler and Lie-Trotter schemes in L p , p>2 and the continuous dependence of the solutions on the initial data and on the coefficients for such equation. Finally we study the existence of the solutions when the coefficients f and g are only defined on a subset of the state Hilbert space.
19 citations
TL;DR: In this article, the authors introduce a general framework for stochastic volatility models, with the risky asset dynamics given by: where (ω,η)∈(Ω×H,F Ω⊗F H,P Ω ⊗P H ).
Abstract: We introduce a general framework for stochastic volatility models, with the risky asset dynamics given by: where (ω,η)∈(Ω×H,F Ω⊗F H ,P Ω⊗P H ). In particular, we allow for random discontinuities in the volatility σ and the drift μ. First we characterize the set of equivalent martingale measures, then compute the mean–variance optimal measure P˜, using some results of Schweizer on the existence of an adjustment process β. We show examples where the risk premium λ=(μ−r)/σ follows a discontinuous process, and make explicit calculations for P˜. †The first draft of this paper was completed while the second author was affiliated to Scuola Normale Superiore. The views expressed in this paper are those of the authors and do not involve the responsibility of the Bank of Italy.
18 citations
TL;DR: In this paper, the authors studied the asymptotic behavior of the initial boundary value problem for a stochastic partial differential equation in a sequence of perforated domains.
Abstract: We study the asymptotic behaviour of the initial boundary value problem for a stochastic partial differential equation in a sequence of perforated domains. We prove that the sequence of solutions of the problem converges in appropriate topologies to the solution of a limit stochastic initial boundary value problem of the same type as the original problem, but containing an additional term expressed in terms of some characteristics of the perforated domain.
TL;DR: In this article, the authors proved that the solution of the stochastic pressure equation of Wick-type belongs to a space of generalized random fields having square integrable homogeneous chaos kernels.
Abstract: We prove that the solution of the stochastic pressure equation of Wick-type belongs to a space of generalized random fields having square integrable homogeneous chaos kernels. We find the chaos expansion, and calculate its stochastic regularity in distributional sense. Furthermore, we show that the solution is stable under perturbations of the permeability field and the source term.
TL;DR: In this paper, the authors consider stochastic semilinear functional differential equations in a Hilbert space and prove the existence and uniqueness of a mild solution under two sets of hypotheses.
Abstract: In this paper, we study stochastic semilinear functional differential equations in a Hilbert space. First, we prove the existence and uniqueness of a mild solution under two sets of hypotheses. We then consider the exponential stability of the second moment of the solution process of such equations as well as the exponential stability and asymptotic stability in probability of its sample paths. We further consider global stability in the mean. Such results are obtained using both local Lipschitz and non-Lipschitz nonlinearities. Our method is an interplay of the method of successive approximations and a comparison principle. Two applications are included to motivate this study.
TL;DR: In this paper, the authors study a dynamic model of asset pricing which is driven by two characteristic market features: the law of investor demand (e.g. "buy low, sell high" and the market institution) which codifies the trading rules under which the market operates.
Abstract: We study a dynamic model of asset pricing which is driven by two characteristic market features: the law of investor demand (e.g. ’buy low, sell high’) and the law of the market institution (which codifies the trading rules under which the market operates). We demonstrate in a simple investor-specialist trading market that these features are sufficient to guarantee an equilibrium where investors’ trading strategies and the specialist’s rule of price adjustments are best responses to each other. The drift term appearing in the resulting equation of the asset price process may be interpreted using Newtonian mechanics as the acceleration of a ’market force’. If either of the market participants is risk-neutral, the result leads to risk-neutral asset pricing (e.g. the Black and Scholes option pricing formula).
TL;DR: In this article, the uniqueness and existence results of a large class of backward stochastic differential equations with Poisson jumps in a Banach space setting are presented, which are called BSPDE's with random coefficients (P for partial).
Abstract: The objective of this note is to show uniqueness and existence results of a large class of backward stochastic differential equations with Poisson jumps in a Banach space setting. Among this classes of equations the so-called BSPDE's with random coefficients (P for partial) which we illustrate with several examples.
TL;DR: In this article, a stochastic control problem to maximize expected utility from terminal and/or consumption is studied, where the portfolio is allowed to anticipate the future with constraints and a higher interest rate for borrowing.
Abstract: We study a stochastic control problem to maximize expected utility from terminal and/or consumption. The novel feature of our work is that the portfolio is allowed to anticipate the future with constraints and a higher interest rate for borrowing. The investor possesses information about the terminal values of the components of the Brownian motion, possibly distorted by ‘noise’. We use the technique from the so-called enlargement of filtrations, to model our problem. General existence results are established for optimal portfolio and consumption strategies. Equivalent conditions for optimality are obtained, and explicit solutions leading to feedback formulae are derived for special utility functions and for deterministic coefficients.
TL;DR: In this article, the authors studied the time dependent behavior of three stochastic models of manpower systems and provided the asymptotic variability of the vector of means, variances and covariances in the manpower system responding to promotion blockages.
Abstract: In this paper, time dependent behavior of three stochastic models of manpower systems is studied. In model 1, the pressure for promotion in a particular grade is contributed by the employees in that grade alone. In model 2, the pressure for promotion to a particular grade is proportional to the number of employees in the lower grades who have eligibility to get promoted to that grade. In model 3, the pressure for promotion is considered to be proportional to the number of employees in a particular grade as in model 1, but incorporating weightage for the number of employees in the different categories in the same grade. The weight is proportional to the length of the minimum service of an employee in that category. For all these models, we provide the asymptotic variability of the vector of means, variances and covariances in the manpower system responding to promotion blockages. Numerical examples illustrate the results.
TL;DR: In this article, the authors consider the case when the cost of interfering with an impulse control of size ζ∈R is given by with c≥0,λ>0 constants, and show that V c is very sensitive (non-robust) to an increase in c near c=0 in the sense that
Abstract: We study how the value function (minimal cost function) V c of certain impulse control problems depends on the intervention cost c. We consider the case when the cost of interfering with an impulse control of size ζ∈R is given by with c≥0,λ>0 constants, and we show (under some assumptions) that V c is very sensitive (non-robust) to an increase in c near c=0 in the sense that The paper provides a general scheme to prove results of this type. In particular, we show that the scheme applies to several of the standard processes used in financial markets, i.e., Brownian motion, Geometric Brownian motion with jumps, and the Ornstein–Uhlenbeck process.
TL;DR: In this article, set-valued semimartingales are introduced, as an extension of single-valued ones, and a stochastic inclusion with mixed type of integrals is considered.
Abstract: Set-valued semimartingales are introduced, as an extension of single-valued ones. For such multivalued processes, say X, we define a set-valued stochastic integral and study its selection properties. Finally, a stochastic inclusion, with mixed type of integrals: is considered.
TL;DR: In this article, the authors dealt with the steady state behavior of an M/G/1 queuing system with two different vacation times under multiple vacation policy, where length of the first vacation is different from the second and subsequent vacations.
Abstract: This paper deals with the steady state behavior of an M/G/1 queuing system with two different vacation times under multiple vacation policy, where length of the first vacation is different from the second and subsequent vacations. In this paper, attempts have been made to obtain the additional queue size distribution, distribution of additional delay and waiting time distribution of this model. Also, we obtain some important measures of this model.
TL;DR: In this article, the authors study the stochastic optimization problem of renewable resources to maximize the expected discounted utility of exploitation, and develop the viscosity solution method to the associated Hamilton-Jacobi-Bellman equation.
Abstract: We study the stochastic optimization problem of renewable resources to maximize the expected discounted utility of exploitation. We develop the viscosity solution method to the associated Hamilton–Jacobi–Bellman equation and further show the C 2-regularity of the viscosity solution under the strict concavity of the utility function. The optimal policy is shown to exist and given in a feedback form or a stochastic version of Hotelling's rule.
TL;DR: Characteristics of Jacod-Grigelionis type for quasi-left continuous semi-martingales with values in a separable Hilbert space are introduced in this article, where definitions of characteristics and proofs of their existence are presented.
Abstract: Characteristics of Jacod–Grigelionis type for quasi-left continuous semi-martingales with values in a separable Hilbert space are introduced. Definitions of characteristics and proofs of their existence are the main aims of this paper.
TL;DR: In this article, a generalization of truncated deterministic Taylor expansions, truncated expansions about a point for sufficiently smooth functions of a solution of a stochastic differential equation, was proposed.
Abstract: We propose, as a generalization of truncated deterministic Taylor expansions, truncated expansions about a point for sufficiently smooth functions of a solution of a stochastic differential equation. The first order expansion will be the natural one according to Ito's formula. The second order one has been obtained in the multi-dimensional case and the third order one in the scalar case. In a similar way higher order truncated expansions would be obtained. Finally, with the same procedure, we obtain with Stratonovich calculus truncated Stratonovich-Taylor expansions. These expansions, as it's to be expected, coincide with deterministic Taylor expansions.
TL;DR: In this article, the authors focus on stability in expectation (e.g., stability) of numerical methods of second-order accuracy in the weak sense, and the possibility of enlarging regions of stability for these methods is discussed.
Abstract: In recent years, many numerical methods for solving stochastic differential equations have been developed. Some of these methods converge in the weak sense and some others converge in the mean square sense. One of the important features of numerical methods is their stability behavior. In this paper, we focus our attention on stability in expectation (e. stability) of numerical methods of second-order accuracy in the weak sense. The region of e. stability for these methods will be discussed. The possibility of enlarging regions of e. stability will be described. Some numerical examples will be discussed to support the theoretical study.
TL;DR: In this paper, the generalized stochastic differential equations of the Itoˆ type whose coefficients are additionally perturbed and dependent on a small parameter are compared with the solutions of the corresponding unperturbed equations.
Abstract: The paper is devoted to the generalized stochastic differential equations of the Itoˆ type whose coefficients are additionally perturbed and dependent on a small parameter. Their solutions are compared with the solutions of the corresponding unperturbed equations. We give conditions under which the solutions of these equations are close in the (2m)-th moment sense on finite intervals or on intervals whose length tends to infinity as the small parameter tends to zero. We also give the degree of the closeness of these solutions.
TL;DR: In this paper, a variety of random fixed point results are presented for continuous, countably condensing multivalued maps, relying on a new result for hemicompact maps and on a deterministic fixed point result of Agarwal and O'Regan.
Abstract: A variety of random fixed point results are presented for continuous, countably condensing multivalued maps. Our arguments rely on a new result for hemicompact maps and on a new deterministic fixed point result of Agarwal and O'Regan.
TL;DR: In this article, the authors obtained a complete convergence of weighted sums for arrays of rowwise independent Banach space valued random elements with no assumptions on the geometry of the underlying Banach spaces.
Abstract: We obtain a result on complete convergence of weighted sums for arrays of rowwise independent Banach space valued random elements No assumptions are given on the geometry of the underlying Banach space The result generalizes the main results of Ahmed et al (1), Chen et al (2), and Volodin et al (14)
TL;DR: In this article, the authors apply the change of numeraire technique to the analysis of the value and the hedging strategies of American options, and give numerical results for the pricing and hedging of such a kind of option.
Abstract: The change of numeraire technique is a standard tool in mathematical finance. We apply it to the analysis of the value and the hedging strategies of American options. The change of numeraire is particularly powerful if the option is written on more assets and has a positively homogeneous payoff. In this case, the option writer doesn't need the riskless bond to hedge his position. We treat some examples as the Margrabe option on two stocks paying continuous dividends and the best of two assets option. Thanks to variational inequalities we are able to give numerical results for the pricing and the hedging of such a kind of American options.
TL;DR: In this paper, a discrete-time version of Ito's formula was established for martingale-driven stochastic difference equation, and this version converges to a continuous-time generalized Ito formula derived in Kunita.
Abstract: We establish here a discrete-time version of Ito's formula that is needed in our study of a martingale-driven stochastic difference equation, and show that this version converges, in the limit case, to a continuous-time generalized Ito formula derived in Kunita.[4]
TL;DR: In this article, a representation of the output or response process of a quadratic filter driven by a stationary Gaussian process is established to facilitate the calculation of statistical descriptors of the response process.
Abstract: The main focus of the present paper is to establish a representation of the output or response process of a quadratic filter driven by a stationary Gaussian process. The specific goal of such a representation is to facilitate the calculation of statistical descriptors of the response process. In particular, the statistical moments and characteristic functions are key elements to be calculated for a practical analysis of the response statistics of quadratic filters.