This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called BrennanSchwartz algorithm for the valuation of American put options. (1950). 90A09, 60G40, 60J60, 65K10, 65M 10. AMS subject classifications

I n the energy markets, in particular the electricity and natural gas markets, many contracts incorporate flexibility-of-delivery options known as "swing" or "take-or-pay" options. Subject to daily as well as periodic constraints, these contracts permit the option holder to repeatedly exercise the right to receive greater or smaller amounts of energy. We extract market information from forward prices and volatilities and build a pricing framework for swing options based on a one-factor mean-reverting stochastic process for energy prices that explicitly incorporates seasonal effects. We present a numerical scheme for the valuation of swing options calibrated for the case of natural gas.

In this paper we formally introduce a generic real-time multi-vehicle truckload pick-up and delivery problem. The problem includes the consideration of various costs associated with trucks' empty travel distances, jobs' delayed completion times, and job rejections. Although very simple, the problem captures most features of the operational problem of a real-world trucking fleet that dynamically moves truckloads between different sites according to customer requests that arrive continuously over time.We propose a mixed integer programming formulation for the off-line version of the problem. We then consider and compare five rolling horizon strategies for the real-time version. Two of the policies are based on a repeated reoptimization of various instances of the off-line problem, while the others use simpler local (heuristic) rules. One of the re-optimization strategies is new while the other strategies have recently been tested for similar real-time fleet management problems.The comparison of the policies is done under a general simulation framework. The analysis is systematic and consider varying traffic intensities, varying degrees of advance information, and varying degrees of flexibility for job rejection decisions. The new re-optimization policy is shown to systematically outperform the others under all these conditions.

Abstract-In many Intelligent Transportation System (ITS)applications that crowd-source data from probe vehicles, a crucial step is to accurately map the GPS trajectories to the road network in real time. This process, known as mapmatching, often needs to account for noise and sparseness of the data because (1) highly precise GPS traces are rarely available, and (2) dense trajectories are costly for live transmission and storage.We propose an online map-matching algorithm based on the Hidden Markov Model (HMM) that is robust to noise and sparseness. We focused on two improvements over existing HMM-based algorithms: (1) the use of an optimal localizing strategy, the variable sliding window (VSW) method, that guarantees the online solution quality under uncertain future inputs, and (2) the novel combination of spatial, temporal and topological information using machine learning. We evaluated the accuracy of our algorithm using field test data collected on bus routes covering urban and rural areas. Furthermore, we also investigated the relationships between accuracy and output delays in processing live input streams.In our tests on field test data, VSW outperformed the traditional localizing method in terms of both accuracy and output delay. Our results suggest that it is viable for lowlatency applications such as traffic sensing.

We consider variants of the online stochastic bipartite matching problem motivated by Internet advertising display applications, as introduced in Feldman et al. [6]. In this setting, advertisers express specific interests into requests for impressions of different types. Advertisers are fixed and known in advance while requests for impressions come online. The task is to assign each request to an interested advertiser (or to discard it) immediately upon its arrival.In the adversarial online model, the ranking algorithm of Karp et al.[11] provides a best possible randomized algorithm with competitive ratio 1−1/e ≈ 0.632. In the stochastic i.i.d. model, when requests are drawn repeatedly and independently from a known probability distribution over the different impression types, Feldman et al. [6] prove that one can do better than 1 − 1/e. Under the restriction that the expected number of request of each impression type is an integer, they provide a 0.670-competitive algorithm, later improved by Bahmani and Kapralov [3] to 0.699, and by Manshadi et al.[13] to 0.705. Without this integrality restriction, Manshadi et al. [13] are able to provide a 0.702-competitive algorithm.In this paper we consider a general class of online algorithms for the i.i.d. model which improve on all these bounds and which use computationally efficient offline procedures (based on the solution of simple linear programs of maximum flow types). Under the integrality restriction on the expected number of impression types, we get a 1 − 2e −2 (≈ 0.729)-competitive algorithm. Without this restriction, we get a 0.706-competitive algorithm.Our techniques can also be applied to other related problems such as the online stochastic vertex-weighted bipartite matching problem as defined in Aggarwal et al. [1]. For this problem, we obtain a 0.725-competitive algorithm under the stochastic i.i.d. model with integral arrival rate.Finally we show the validity of all our results under a Poisson arrival model, removing the need to assume that the total number of arrivals is fixed and known in advance, as is required for the analysis of the stochastic i.i.d. models described above.