Showing papers in "Stochastics and Stochastics Reports in 1993"
TL;DR: In this paper, a stochastic optimal switching and impulse control problem in a finite horizon is studied, and the continuity of the value function, which is by no means trivial, is proved.
Abstract: A stochastic optimal switching and impulse control problem in a finite horizon is studied. The continuity of the value function, which is by no means trivial, is proved. The Bellman dynamic programming principle is shown to be valid for such a problem. Moroever, the value function is characterized as the unique viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation.
135 citations
TL;DR: In this article, the authors consider a simple mathematical model for the following economic problem: a company wants to expand its production capacity in an uncertain market The investments needed are irreversible in the sense that if the market later drops, the company can not get the invested capital back by reducing the capacity.
Abstract: We consider a simple mathematical model for the following economic problem: A company wants to expand its production capacity in an uncertain market The investments needed are irreversible in the sense that if the market later drops, the company can not get the invested capital back by reducing the capacity Which strategy should the company follow in order to maximize the long term expected profit? In our model, the state of the market is described by a one dimensional, geometric Brownian motion We show that under some monotonocity conditions the optimal strategy can be described as a deflection off a “forbidden region”, and that the boundary of this region can be computed quite explicitly
104 citations
TL;DR: In this article, the existence of solutions for nonlinear evolution equations with nonhomogenous boundary conditions of white noise type is studied. And it is shown that if the nonlinearity satisfies appropriate dissipativity conditions, then it has a solution as well.
Abstract: The paper is devoted to nonlinear evolution equations with nonhomogenous boundary conditions of white noise type Necessary and sufficient conditions for the existence of solutions in the linear case are given It is also shown that if the nonlinearity satisfies appropriate dissipativity conditions the nonlinear equation has a solution as well The results are applied to equations with polynomial nonlinearities
97 citations
TL;DR: In this article, basic properties of transition probability functions of Markov processes corresponding to solutions of semilinear stochastic evolution equations with a general Gaussian noise were studied.
Abstract: Basic properties of transition probability functions of Markov processes corresponding to solutions of semilinear stochastic evolution equations with a general Gaussian noise are studied Condition
35 citations
TL;DR: In this article, large deviation limit theorems are established for the occupation times of both the measure-valued and discrete, particle models of critical branching Brownian motion in low dimensions (3 or 4).
Abstract: Large deviation limit theorems are established for the occupation times of both the measure-valued and discrete, particle models of critical branching Brownian motion in low dimensions (3 or 4). The decay of tail probabilities is shown to be slower than exponential and the rate functions are characterized.
35 citations
TL;DR: In this article, the authors prove a tightness result for a sequence of interacting branching-diffusion processes, where the limit points are continuous measure-valued processes and satisfy a martingale property.
Abstract: We prove a tightness result for a sequence of interacting branching-diffusion processes. The limit points are continuous measure-valued processes and satisfy a martingale property. We state that th...
30 citations
TL;DR: In this paper, the average cost per unit time problem for controlled and uncontrolled open queueing networks in heavy traffic is treated. And the authors provide a particularly strong approach to this problem via a "functional occupation measure" method via a fairly general cost functional.
Abstract: We treat the average cost per unit time problem for controlled and uncontrolled open queueing networks in heavy traffic The usual heavy traffic theorems prove that a suitably scaled and normalized sequence of queue length processes converges weakly in the Skorohod topology to a certain reflected diffusion, as the traffic intensity goes to unity The Skorohod topology essentially discounts the distant future, and the usual weak convergence methods are not well suited for dealing with the average cost problem over an infinite time interval We provide a particularly strong approach to this problem via a “functional occupation measure” method A fairly general cost functional is used For the uncontrolled problem, it is shown that the average pathwise (ie, no mathematical expectation is used) cost per unit time converges in probability to an ergodic cost for the limit reflected diffusion, no matter how the time goes to infinity or the traffic intensity goes to unity The methods which are introduced are n
30 citations
TL;DR: Asymptotic efficiency of a stochastic approximation algorithm is proved under weak conditions: no growth rate restriction is imposed on the regression function; wide class of observation noises is treated.
Abstract: Asymptotic efficiency of a stochastic approximation algorithm is proved under weak conditions: no growth rate restriction is imposed on the regression function; wide class of observation noises is ...
28 citations
TL;DR: The filtering problem is considered for a nonlinear model with state and observation equations driven by noise processes having a Markovian representation; this includes the case of systems driven by Gaussian noise processes with stationary increments having rational spectral densities.
Abstract: The filtering problem is considered for a nonlinear model with state and observation equations driven by noise processes having a Markovian representation; this includes the case of systems driven by Gaussian noise processes with stationary increments having rational spectral densities. Under some natural regularity conditions the model is shown to be equivalent to one in which the observation has a noisy as well as a noiseless component. For the model in this latter equivalent form and where the coefficients are supposed to depend causally on the observations, we study an approximation scheme, based on periodic sampling. An error bound for the approximation is derived in the case when there are only noisy observations. In this latter case we consider also the situation when the system depends on an unknown random parameter and derive a robustness result in terms of the probability distribution of this parameter.
25 citations
TL;DR: In this article, it is shown that strong Markov processes on sets E\{infE,supE} being compact are constructed from Wiener processes by means of time change, killing and space transformation.
Abstract: Strong Markov processes on sets E\{infE,supE} being compact are constructed from Wiener processes by means of time change, killing and space transformation. Such processes are called one-dimensional quasidiffusions. Their infinitesimal generators are Krein-Feller generalized second order differential operators of the form . In particular, selfajfine diffusions and Brownian motion on the Cantor set (these are quasidiffusions which are preserved under certain transformations of space and time) are characterized by determining their parameters n,s,k.. It is shown that such processes are traces of diffusions with state interval (0, 1)
23 citations
TL;DR: In this article, the amplitude of a real process X is defined as the difference between its maximum process and its minimum process, and the law of the first time when X reaches a fixed level is characterized.
Abstract: The amplitude of a real process X, is the increasing process A defined as the difference between its maximum process and its minimum process. We study the Brownian case, and we characterize, in particular, the law of the first time when A reaches a fixed level a ≷0. We apply this basic study to the case when X is a continuous diffusion and also when X is the angular part of a planar Brownian motion
TL;DR: In this paper, the authors investigated the asymptotic behavior of a diffusion process on a Riemannian manifold having an infinitesimal generator and showed that the strength β of the drift becomes large under the situation where β is large.
Abstract: Let be a diffusion process on a Riemannian manifold Nhaving an infinitesimal generator . We investigate the asymptotic behavior of as the strength β of the drift becomes large under the situation t...
TL;DR: In this article, the authors studied one dimensional diffusion with controlled drift and gave definitions of almost surely and almost surely optimal policy and a policy optimal in probability for the stationary linear regulator with quadratic cost.
Abstract: This paper studies one dimensional diffusion with controlled drift. We give definitions of an almost surely optimal policy and a policy optimal in probability. These types of optimality are much stronger than the classical optimality for the expected limiting average per unit time cost (optimality in mean on [0,∞)). To analyse when an optimal in mean on [0,∞) policy is also optimal almost surely and in probability, we offer a simple direct method. This method is later applied for the stationary linear regulator with quadratic cost. The question of overtaking optimality is also discussed
TL;DR: In this paper, a class of finite-fuel singular stochastic control problems for diffusions is studied, and the existence of the optimal control is derived as a direct consequence of the comparison theorems without any extra conditions.
Abstract: We study two kinds of Discontinuous Reflecting Problem (DRP for short), defined by Chaleyat-Maurel et al. [3] and Dupuis and Ishii [5] (reduced to the one-dimensional case) and the related Stochastic Differential Equations with Discontinuous Paths and Reflecting Boundary Conditions (SDEDR for short). We compare the properties of the solutions to the two DRP's as well as the two SDEDR's. Some comparison theorems, either for the solutions of different kinds of SDEDR's or for the same kind of SDEDR but with different data, are derived. As an application, we consider a class of finite-fuel singular stochastic control problems for diffusions. With the new approach, the complete class of admissible controls can now be obtained as a direct consequence of our comparison theorems without any extra conditions. The existence of the optimal control (in a wider sense) is derived
TL;DR: In this article, a one-dimensional stochastic differential equation without drift but with time-dependent diffusion coefficient is considered and a sufficient condition for the existence of a possibly drift-free variant is given.
Abstract: Consider a one-dimensional stochastic differential equation without drift but with time-dependent diffusion coefficient. In Section 3 we give a sufficient condition for the existence of a possibly ...
TL;DR: In this paper, the problem of minimax estimation of parameters in linear regression models with uncertain second order statistics is studied and the solution to the problem is shown to be the least squares estimator corresponding to the least favorable matrix of the second moments.
Abstract: This paper is devoted to the problem of minimax estimation of parameters in linear regression models with uncertain second order statistics. The solution to the problem is shown to be the least squares estimator corresponding to the least favourable matrix of the second moments. This allows us to construct a new algorithm for minimax estimation closely connected with the least squares method. As an example, we consider the problem of polynomial regression introduced by A. N. Kolmogorov
TL;DR: In this paper, the authors improved some known properties of the Snell envelopes based on which they proved the main result of this article, namely the inequality where X 1 and X 2 are two arbitrary semimartingales belonging to the space H p P> 1Y 1 and Y 2 are their corresponding Snell envelope and C p are absolute constants.
Abstract: On the finite time interval [0,T] we consider the class of right continuous (with left limits) processes and for each element its Snell envelope, that is the smallest supermartingale bounding X from above. We improve some known properties of the Snell envelopes based on which we prove the main result of this article, namely the inequality where X 1 and X 2 are two arbitrary semimartingales belonging to the space H p P> 1Y 1 and Y 2 are their corresponding Snell envelopes and C p are absolute constants. The key step in proving this result is a bound of the distance in variation between the predictable components of the Snell envelopes Y l and Y 2
TL;DR: In this article, an asymptotic analysis of controlled diffusions couled by a parameter process is presented, in which the oscillation rate of the process is assumed to be very large and the stochastic parameter process can be replaced by its averaged value.
Abstract: This paper presents an asymptotic analysis of controlled diffusions couled by a parameter process. The oscillation rate of the parameter process is assumed to be very large. This gives rise to a limiting problem in which the stochastic parameter process can be replaced by its averaged value. A control for the original problem can be constructed from the optimal control of the limiting problem in a way which guarantees its asymptotic optimality. It is shown that the value function of the original problem converges to the value function of the limiting problem. The convergence rate of the value function and the error estimate of the constructed asymptotically optimal control are obtained. Finally, the results are applied to an adaptive control problem.
TL;DR: In this paper, the Cox-Russ jump process pricing model is analyzed using methods of nonstandard (i.e. infinitesimal) analysis, following the methodology developed by the authors in earlier papers.
Abstract: The Cox-Russ jump process pricing model is analysed using methods of nonstandard (i.e. infinitesimal) analysis, following the methodology developed by the authors in earlier papers. Explicit formulae are found for replicating strategies for European options. The existence of liftings for claims and strategies, and the concepts of D2-convergence and adapted discretisation schemes, introduced earlier by the authors to discuss the convergence of random walk pricing models to continuous-time models driven by Brownian motion, are analysed for Poisson jump price models
TL;DR: In this paper, the existence of weak solutions of the stochastic differential equation was proved for a standard Wiener process and a measurable function, assuming that f 2 is neither locally bounded above nor bounded below by a strictly positive constant.
Abstract: Let W be a standard Wiener process and let be a measurable function. We prove existence of weak solutions of the stochastic differential equation . As compared with earlier works we assume that f 2 is neither locally bounded above nor bounded below by a strictly positive constant
TL;DR: In this paper, the authors prove a strong invariance principle for smooth functions of certain chaotic dynamical systems, and show that solutions of dynamical system which are coupled to such chaotic systems may be approximated by solutions of stochastic differential equations.
Abstract: In this paper we prove results regarding certain precise relationships between random motion and chaotic motion. In particular we prove a strong invariance principle for smooth functions of certain chaotic dynamical systems, and show that solutions of dynamical systems which are coupled to such chaotic systems may be approximated by solutions of stochastic differential equations
TL;DR: In this paper, the authors show that if X 1 and X 1 are independent semimartingales and both filtrations have the martingale representation property, then the filtration also has the same representation property.
Abstract: Suppose that X 1and X 1 are independent semimartingales and both filtrations and have the martingale representation property, then the filtration also has the martingale representation property, wh...
TL;DR: In this article, the expectation of the solution of a stochastic differential equation conditioned on complete knowledge of the path of one of its components is established, and it is shown that any appropriately regular solution of this polynomial p.d. must be given by the conditional expectation.
Abstract: This paper establishes an anticipating stochastic differential equation of parabolic type for the expectation of the solution of a stochastic differential equation conditioned on complete knowledge of the path of one of its components. Conversely, it is shown that any appropriately regular solution of this stochastic p.d.e. must be given by the conditional expectation. These results generalize the connection, known as the Feynman-Kac formula, between parabolic equations and expectations of functions of a diffusion. As an application, we derive an equation for the unnormalized smoothing law of a filtering problem with observation feedback.
TL;DR: The strong law of large numbers for multivariate martingales with deterministic quadratic variation was proved in this article, along the same lines as in Lai, Wei and Robbins (1979), though the setting here is more general.
Abstract: The strong law of large numbers is proved for multivariate martingales with deterministic quadratic variation, along the same lines as in Lai, Wei and Robbins (1979), though the setting here is more general
TL;DR: In this paper, the authors studied discrete and continuous time parameter queueing processes by employing and developing semi-regenerative techniques and the theory of fluctuations, and established a necessary and sufficient ergodicity criterion for both processes.
Abstract: The paper deals with a class of single-server queueing systems with compound input process, a queue length dependent service delay and a random server capacity. Under the assumption that the server does not begin to process a group of units until the number of available units matches the server capacity (r≥1), such a model generalizes many known queueing systems. Since units arrive in batches, the queue length at the instant of the first passage time is more likely to exceed rather than to reach level r. Consequently, the analysis of models like this requires knowledge of the first excess level process and other related processes which were studied in the recent work of Abolnikov and Dshalalow [1] on the first passage problem. The authors study discrete and continuous time parameter queueing processes by employing and developing semi-regenerative techniques and the theory of fluctuations. They establish a necessary and sufficient ergodicity criterion for both processes and obtain explicit expressions for ...
TL;DR: In this paper, an anticipative stochastic calculus for manifold valued processes is defined; if uses both, second-order geometry defined by P.-A. Meyer [8] and Nualart-Pardoux's duality definition of Skorohod Integral [11].
Abstract: An anticipative stochastic calculus for manifold valued processes is defined; if uses both, second-order geometry defined by P.-A. Meyer [8] and Nualart-Pardoux's duality definition of Skorohod Integral [11] This stochastic calculus allows us to integrate non-adapted processes taking their values in the space of second-order I-forms that are above a manifold valued semi-martingale
TL;DR: For the R d-valued semimartingaleN be the standard d-dimensional normal r.v. as discussed by the authors, the distance of a real bounded Borel measurable function f on R d was considered and exact upper bounds for Δ f (X t ) were given in terms of some functional on the triple of of X.
Abstract: Let be the R d-valued semimartingaleN be the standard d-dimensional normal r.v. For any real bounded Borel measurable function f on R d we consider the distance . Exact upper bounds for Δ f (X t ) are given in terms of some functional on the triple of of X. The bounds for Δ f (X t ) are derived provided G being convex Borel set in R d
TL;DR: In this paper, a measure-valued process was constructed as a weak limit of the Ohta-Kimura stepwise mutation model, under scalings of time and space, and the support of the random measure has Hausdorff dimension not greater than two.
Abstract: In Fleming and Viot [4] a measure-valued process was constructed as a weak limit of the Ohta-Kimura stepwise mutation model, under scalings of time and space. Dawson and Hochberg [3] obtained further results, including the fact that for fixed times, almost surely, the support of the random measure has Hausdorff dimension not greater than two. This paper uses a non standard construction of the Fleming-Viot process to establish this result for all times simultaneously
TL;DR: In this article, the existence and uniqueness of a solution for a one-dimensional quasilinear stochastic differential equation of the Skorohod type, with a non-linear boundary condition of the form.
Abstract: In this paper we prove the existence and uniqueness of a solution for a one-dimensional quasilinear stochastic differential equation of the Skorohod type, with a non-linear boundary condition of the form . The method of proof uses the Girsanov transformation and some basic exponential estimates. When the coefficients of the equation are deterministic we show that the solution is weakly differentiable on the Wiener space and possesses a continuous modification.
TL;DR: In this paper, the class of regular estimators having a martingale representation is defined and the best among them is sought in the sense of asymptotic second order.
Abstract: In the partially specified statistical models the class of regular estimators having a martingale representation is defined and the best among them is sought in the sense of asymptotic second order...