Search

For a risk-sensitive, partially observed stochastic control problem, the modified Zakai equation includes an extra term related to the exponential running cost. The finite-dimensional solutions of this modified Zakai equation are obtained. These are analogs of the Kalman and Benes filters. The small noise limits of the finite-dimensional risk-sensitive problems are then obtained. These lead to differential games with deterministic disturbances.

A partially observed stochastic control problem with exponential running cost is considered. The dynamics are linear and the running cost is quadratic, although the control may enter nonlinearly. Explicit solutions are found to a modified Zakai equation and a backward adjoint equation. This enables the problem to be expressed in terms of observable finite- dimensional dynamics and a separation principle to be applied.

A discrete-time partially observed stochastic control problem with exponential running cost is considered. The dynamics are linear and the running cost quadratic in the state variable, but the control may enter nonlinearly. Explicit solutions for a forward Zakai equation and a backward adjoint equation are derived in terms of finite-dimensional dynamics. This enables the partially observed problem to be expressed in finite-dimensional terms and a separation principle applied.

It is shown how, even when the market is incomplete, certain contingent claims are attainable: that is, they can be represented as stochastic integrals with respect to the process which describes the evolution of the asset prices.

Because of their low cost, photovoltaic (PV) cells made from upgraded metallurgical grade silicon (UMG-Si) are a promising alternative to conventional solar grade silicon-based PV cells. This study investigates the outdoor performance of a 1.26 kW grid-connected UMG-Si PV system over five years, reporting the energy yields and performance ratio and estimating the long-term performance degradation rate. To make this investigation more meaningful, the performance of a mono-Si PV system installed at the same place and studied during the same period of time is presented for reference. Furthermore, this study systematizes and rationalizes the necessity of a data selection and filtering process to improve the accuracy of degradation rate estimation. The impact of plane-of-array irradiation threshold for data filtering on performance ratio and degradation rate is also studied. The UMG-Si PV system’s monthly performance ratio after data filtering ranged from 84% to 93% over the observation period. The annual degradation rate was 0.44% derived from time series of monthly performance ratio using the classical decomposition method. A comparison of performance ratio and degradation rate to conventional crystalline silicon-based PV systems suggests that performance of the UMG-Si PV system is comparable to that of conventional systems.

We propose a general framework to assess the value of the financial claims issued by the firm, European equity options and warrantsin terms of the stock price. In our framework, the firm's asset is assumed to follow a standard stationary lognormal process with constant volatility. However, it is not the case for equity volatility. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. In a previous paper we studied the stochastic process for equity volatility, and proposed analytic approximations for different capital structures. In this companion paper we derive analytic approximations for the value of European equity options and warrants for a firm financed by equity, debt and warrants. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities either as a function of the stock price, or as a function of the firm's total assets. Since stock prices a...

The purpose of this article is to survey some results related to the theory of stochastic Navier-Stokes equations (SNSE). The interest of SNSE arises from modelling turbulence. We begin to show how SNSE can be introduced intuitively from the random motion of particles. We then review briefly the deterministic theory and present the main core of existence theory for NSE. We also discuss uniqueness issues. We end up by showing how the splitting-up method provides a useful constructive approach to existence, and by presenting some extensions, like weakening assumptions or considering the special case of small initial data.

Computing is undergoing a substantial shift from client/server to the cloud. The enthusiasm for cloud infrastructures is not only present in the business world, but also extends to government agencies. Managers of both segments thus need to have a clear view of how this new era will evolve in the coming years, in order to appropriately react to a changing economic and technological environment. In this study, we explore the dynamic equilibrium of cloud computing adoption through the application of Mean Field Games. In our formulation, each agent (i.e., each firm or government agency) arbitrates between “continuing to implement the traditional on-site computing paradigm” and “moving to adopt the cloud computing paradigm”. To decide on his level of moving to the cloud computing paradigm, each agent will optimize a total cost that consists of two components: the effort cost of moving to the cloud computing paradigm and the adoption cost of implementing the cloud computing paradigm. In the formulation, the adoption cost is linked to the general trend of decisions on the computing paradigm adoption. Thus, an agent's optimal level of transition to the cloud computing paradigm is not only dependent on his own effort and adoption costs but also affected by the general trend of adoption decisions. The problem is solved by a system of partial differential equations (PDEs), that is, mean field games PDEs, which consists of a backward PDE, the Hamilton Jacobi Bellman equation for a controlled problem, and a forward Fokker-Planck equation transported by the optimal control from the backward HJB equation. Thus, the solution to the forward Fokker-Planck equation enables us to study the dynamic evolution of the density of the cloud computing adoption. It therefore allows us to investigate the impact of the general trend of technology adoption decisions on a firm's optimal decision of technology transition.

International audience; In an earlier work made by the first author with J. Turi (Degenerate Dirichlet Problems Related to the Invariant Measure of Elasto-Plastic Oscillators, AMO, 2008), the solution of a stochastic variational inequality modeling an elasto-perfectly-plastic oscillator has been studied. The existence and uniqueness of an invariant measure have been proven. Nonlocal problems have been introduced in this context. In this work, we present a new characterization of the invariant measure. The key finding is the connection between nonlocal PDEs and local PDEs which can be interpreted with short cycles of the Markov process solution of the stochastic variational inequality.

We model an impulse control problem when the controller's action affects the state as well as the dynamics of the state process for a random amount of time. We apply our model to solve a central bank intervention problem in the foreign exchange market when the market observes and reacts to the bank's interventions.

We consider a discrete-time inventory problem with infinite horizon, with a fixed lead time of several periods. When there is no restriction on the times of order, in particular when it is possible to make an order while some previous order is not yet delivered, then it is possible to reduce the problem to a standard one with no lead time (more precisely a lead time of one period) by considering the inventory position instead of the real inventory. This is a well-known result, which requires however the possibility of backlog. We consider here the same problem, this time imposing a constraint on the order times. It is not possible to make an order while some previous deliveries are pending. With this constraint, the problem cannot be reduced to a standard one for the inventory position.We solve the problem and show cases when the optimal policy is of (s, S) type.

It is known that cybercriminal black markets that trade in illicit digital goods and services belong to markets for lemons due to the information asymmetry of quality of goods and services between sellers and buyers. Based on the seminal work of Akerlof [1], Franklin et al. [3] suggests that “Lemonizing the Market” be an effective way to crack down the well-developed cybercriminal underground market. In our work, we provide a game theoretical framework to analyze whether cybercriminal black markets can be effectivitely lemonized. First, we investigate if signaling quality through an extra provision, such as the offer of trial periods or a money-back guarantee, observed in this marketplace (see the Panda security report [6]) provides cybercriminals selling real illicit data (i.e., the peach group) with a solution to address the lemon market problem. We also study the relation between the market lemonization and the cost constraint on seller’s implementation of signaling of quality. We find that, because of the effectiveness of resolving quality uncertainty through perfect signaling of quality, law enforcement cannot clamp down the operation of this underground economy through “Lemonizing the Market” by joining the group of “pure lemons”, that is, joining the group of sellers with no crime products offered to sell (i.e., ripoff sellers). If no information of quality is disclosed, the market demand shrinks increasingly as lemons in the market increases. However, to secure the market demand, cybercriminals with real illicit data for sale always attempt to implement quality signaling to single out their quality products, accepting a higher amount of cost constraints on applying quality signaling as the portion of lemons in the market escalates.

We consider inventory control problems in discrete time. The horizon is infinite, and we consider discounted payoffs as well nondiscounted payoffs (ergodic control). We may have backlog or not. We may have set-up costs or not. In the traditional framework, the demand is modeled as a sequence of i.i.d. random variables. The ordering strategy is given by a base stock policy or an s, S policy, whether or not there is a set-up cost. We consider here the situation when the demand is modeled by a Markov chain. We show how the base stock policy and the s, S policy can be extended.

This paper develops a general model for equity volatility when the firm is financed by equity, debt and any other financial instruments like warrants and convertible bonds. The stochastic nature of equity volatility is endogenous and comes from the impact of a change in the value of the firm’s assets on the financial leverage. We first present the basic model to value corporate securities, which is an extension of the Black-Scholes model. Then, we are able to propose an analytic approximation for equity volatility, which is shown to be extremely precise. Finally, we study the behaviour of equity volatility when the firm is financed by equity and debt.

Cloud computing has been a paradigm shift in the information technology domain. It offers potential benefits to users in terms of instant availability, scalability and resource sharing, while potentially posing security issues. Especially, recent events like Amazons system failure increased the concerns related to cloud computing1. Given these security and reliability concerns, we explore the optimal decision rule for moving certain IT function to public clouds. We formulate the problem as an entrepreneurial decision for an optimal stopping time at which the entrepreneur shall migrate to the cloud computing paradigm. Two different models are presented. Recognizing that an important and specific issue related to different computing paradigm is the potential “security” risk posed by each technology, we consider security risks in both models. The first model approaches the optimal adoption problem from assessing the cloud computing adoption under project value uncertainty. The entrepreneur has the timing flexibility and solves his optimal adoption decision under uncertainty. The optimal adoption rule obtained is a threshold strategy. A firm should adopt the cloud computing only if the value from the adoption exceeds the threshold level. The second model builds on a comprehensive assessment of two different computing paradigms. The entrepreneur can either keep the traditional on-site computing paradigm or migrate to the cloud computing paradigm. His problem is to make the paradigm shift optimally. We model such a problem as optimally swapping two “risky” assets, which refer to benefits of the traditional on-site computing paradigm and those of the cloud computing paradigm. The term “risky” captures the fact that actual benefits can only be resolved through time, and thus estimates of benefits are embedded with uncertainty. We obtain the optimal swapping rule as a threshold strategy, defined in terms of the two benefit ratio. A firm should only shift the part of its business to the cloud computing service if the ratio, the benefit from the cloud computing paradigm over that from the traditional on-site computing paradigm, exceeds the threshold. In both models, both the extent of riskiness (i.e. uncertainty) and the significance of security risks (both in terms of potential occurrence probability and the severity of damage) affect the threshold level, thus the entrepreneurial adoption decision.

We develop a general methodology for a partially observed stochastic control problem. The dynamics is governed by a discrete-time Markov process. We describe an application to an inventory system with possibility of shrinkage, and introduce unnormalized conditional probabilities to transform the nonlinear state evolution into a linear one. We then prove the existence and uniqueness of the solution for the Bellman equation between the floor and the ceiling functions in the case of unbounded costs.

Inventory inaccuracy is common at retailers. At many retailers, a cash register records incoming orders and outgoing sales, but not the demand or the shrinkage. The shrinkage refers to spoilage or pilferage of inventory. The demand differs from the sales in the periodic-review lost-sales inventory model presented in this paper. When the inventory exceeds the demand, the remaining inventory is subject to a shrinkage unobserved by the inventory manager. Otherwise, the unmet demand is lost and unobserved. Our objective is to minimize the expected discounted cost over an infinite horizon. We use dynamic programming along with the concept of unnormalized probability, and prove the existence of an optimal feedback policy and that the value function is the solution of the dynamic programming equation. We obtain a theoretical lower bound for the cost via the formulation of a fictitious inventory problem. For discrete-valued inventories, we provide the analogous formulation, develop an iterative algorithm, and compare its solution to a myopic solution and a lower bound. We identify many cases of parameter values for which the solution of the iterative algorithm performs significantly better than the myopic solution. Moreover, the achieved cost is close to the lower bound.

This paper proposes a new methodology to solve partially observed inventory problems. Generally, these problems have infinitedimensional states that are conditional distribution of the inventory level. Our methodology involves linearizing the state transitions via unnormalized probabilities. It then uses an appropriate functional basis to represent the state. Considering the speed and stability of computations, we choose truncated Chebyshev polynomials as the basis. We use Fast Fourier Transforms along with an appropriate discretization of inventory levels to speed up the computations. These main ideas are to obtain an iterative algorithm to solve a partially observed inventory model with rain checks. In this model, the inventory manager (IM) does not know the inventory level when it is positive. Otherwise, the IM fully observes it. This model provides a context to illustrate our methodology, which applies to other such models. Although this model has been studied mathematically in the literature, the use of our algorithm provides a numerical approximation of the optimal order quantities.These are compared to the orders released under a base mean-stock policy, where the IM replaces the unobserved inventory level with its mean and applies the well-known base stock policy. We show numerically that the optimal order quantity is very close to the base mean-stock order quantity, when the variance of the inventory distribution is small. When the mean of the inventory distribution is large, the optimal order quantity is more than the base mean-stock quantity, and it is the other way around when the mean is small or negative. These insights are explained via uncertainty and information effects and their interplay. We expect this interplay to show up in other partially observed inventory models.

In this paper we deal with the convergence of some iterative schemes suggested by Lie-Trotter product formulas for stochastic differential equations of parabolic type. The stochastic equation is split into two problems which are simpler for numerical computations, as already shown, for example, for the Zakai equation. An estimate of the approximation error is given in a particular case.

We consider a mutual insurance system whose reserve is determined by a Brownian motion. The controller tries to minimize the total cost by increase or reduce the reserve instantly. Both cases of zero and positive fixed cost are investigated. By applying the theory of stochastic control with QVI approach, we make the connection between singular control and impulse control. The procedures to solve these systems are also presented in this paper.

We establish existence and uniqueness of solutions of a class of partial differential equations with nonlocal Dirchlet conditions in weighted function spaces. The problem is motivated by the study of the probability distribution of the response of an elasto-plastic oscillator when subjected to white noise excitation (see [1,2] on the derivation of the boundary value problem). Note that the developments in [1,2] are based on an extension of Khasminskii’s method (see, e.g. [5]) and in this paper we use a direct approach to achieve our objectives.

We give in this chapter a presentation of the capital asset pricing model in discrete time. The presentation is usually done in continuous time. However, the discrete time model is not just a discrete time version of the continuous time model. Some significant differences occur. They are related to the fact that the usual assumption of complete markets is not satisfied in discrete time unless the randomness is modelled by a finite number of events.

This paper describes our approach to assured information sharing. The research is being carried out under a MURI 9Multiuniversiyt Research Initiative) project funded by the Air Force Office of Scientific Research (AFOSR). The main objective of our project is: define, design and develop an Assured Information Sharing Lifecycle (AISL) that realizes the DoD's information sharing value chain. In this paper we describe the problem faced by the Department of Defense and our solution to developing an AISL System.

Under many circumstances, demand observations are often censored due to the lack of tracking lost sales caused by stockouts. To understand the impact of the lost sales information on the ordering decisions, a periodic-review inventory model is formulated in which only the sales information is obtained immediately upon the realization of the demand. This is equivalent to observing the demand when the sales are less than the available stock and to inferring that the demand is higher than the stock when there is a stockout. Subsequently, the lost sales information is obtained after a delay. In the resulting model, an optimal policy, if exists, reveals a very complex structure. By decomposing the derivative of the value function, we demonstrate two different roles of inventory in our model: satisfying the demand and extracting the demand information. We show that the optimal inventory levels under the delayed observation of the lost sales are always higher than those for which the demands are fully observed. Moreover, as illustrated in numerical examples, the optimal policy possesses a counterintuitive behavior with respect to the problem parameters. To understand the key drivers of the optimal decisions, we further compare the costs under different demand observations. Two important observations are made. First, a lower cost is obtained when the realized demand is observed than when the demand is only observed to be higher than the inventory level, and, furthermore, the cost difference represents the value of demand information. Second, while a higher inventory level induces a more accurate demand forecast, the value of exact demand observation is not monotone in the procurement cost. Consequently, the optimal ordering quantity is not always decreasing in the procurement cost.

In many inventory control contexts, inventory levels are only partially (i.e., not fully) observed. This may be due to non-observation of demand, spoilage, misplacement, or theft of inventory. We study a periodic review inventory system where the unmet demand is backordered. When inventory level is nonnegative, the inventory manager does not know the exact inventory level. Otherwise, inventory shortages occur and the inventory manager issues rain checks to customers. The shortages are fully observable via the rain checks. The inventory manager determines the order quantity based on the partial information on the inventory level. The objective is to minimize the expected total discounted cost over an infinite horizon. The dynamic programming formulation of this problem has an infinite dimensional state space. We use the methodology of the unnormalized probability to establish the existence of an optimal feedback policy when the periodic cost has linear growth. Moreover, uniqueness and continuity of the solution to dynamic programming equations are proved when the discount factor is sufficiently small.

We develop a general filtering framework for the problem of estimating the state of a system whose dynamics is governed by a discrete-time Markov process. We describe applications to inventory control systems with partial observations. We introduce conditional distributions and unnormalized conditional probabilities to transform nonlinear transition equations into linear ones. Moreover, this transformation greatly facilitates our study of the stochastic optimal control problems governed by nonlinear transition equations.

In this preliminary article we consider the problem of estimating an unknown noise-gain for a Markov chain observed through a scaled Brownian motion. It is assumed that the unknown noise-gain is time invariant. Two objectives are addressed in this work, 1) compute an estimation scheme that is fast, and 2) compute an estimation scheme without recourse to stochastic integration. To address the first objective we avoid the Expectation Maximization (EM) algorithm, instead we develop an estimation scheme for a finite number of candidate model hypotheses. To address the second objective we develop a version of the Gauge-Transformation technique introduced by J. M. C. Clark.

This paper introduces recent developments in the analysis of inventory systems with partial observations. The states of these systems are typically conditional distributions, which evolve in infinite dimensional spaces over time. Our analysis involves introducing unnormalized probabilities to transform nonlinear state transition equations to linear ones. With the linear equations, the existence of the optimal feedback policies are proved for two models where demand and inventory are partially observed. In a third model where the current inventory is not observed but a past inventory level is fully observed, a sufficient statistic is provided to serve as a state. The last model serves as an example where a partially observed model has a finite dimensional state. In that model, we also establish the optimality of the basestock policies, hence generalizing the corresponding classical models with full information.

Managing customer satisfaction in a cost effective way has always been a major challenge faced by inventory managers. This paper studies long-term service performance of a two-stage newsvendor selling a perishable product with short-term demand patterns. We characterize the optimal inventory policy to minimize the expected inventory cost such that a long-term stock availability target is satisfied. Both in-stock probability and fill rate targets are examined and compared. In particular, we address the following questions: How should an inventory manager evaluate his long-term fill rate performance without observing the lost sales? How are in-stock probabilities and fill rates connected with respect to different demand patterns? How does the forecast update impact the evaluation of the long-term service performance? How do the short-term cost trade-offs under different long-term service targets depend on the monotone structures of the forecast signal?

We introduce a general formalism for linear evolution equations with skew adjoint operators. We make explicit the controllability operator as an expansion with respect to eigenfunctions. Using the fact that the eigenvalues are purely imaginary, we give sufficient controllability conditions. This approach is convenient for studying the asymptotic behaviour of the optimal control.

We formulate the problem of control of diffusions with partial observation as a problem of control with full observation, for a state representing the unnormalized conditional probability. We extend the Dynamic Programming approach to this problem. A Bellman principle is shown, which amounts to defining a nonlinear semigroup. We also consider the existence of an optimal relaxed control, in a generalized sense

In this article, we complete the theory of estimation of Linear Random Functionals, introduced by the Author, in order to extend the Kalman filter to infinite dimensional linear systems. The objective is to show that all properties for the finite dimensional case remain valid in the framework of Linear Random Functionals (this is thanks to linearity of course).

Let O be an open bounded smooth domain of ℝ n , and let Γ = αO be its boundary. We denote by n the normal vector at the boundary Γ, oriented towards the outside of O. Let us consider the canonical process $$\begin{gathered} \Omega = C^0 ([0,\infty );\bar O), \hfill \\ y(t,w) \equiv w(t), if w \in \Omega , \hfill \\ \mathfrak{F}^t = \sigma (y(s),0 \leqslant s \leqslant t). \hfill \\ \end{gathered} $$

In this paper, we consider the class of systems of nonlinear partial differential equations, which has been lengthily studied by Prof J. FREHSE and the A. , with application to stochastic differential games with N players. In particular, we refer to the book, A.BENSOUSSAN, J. FREHSE [1]. The regularity theory is instrumental to prove the existence of equilibriums in noncooperative games. The objective in this paper is to show that regularity theory is also extremely useful for obtaining the limit of problems with small parameters, like in homogenization. The methods used for scalar equations cannot extend, and the regularity results become instrumental