The Skorokhod map is a convenient tool for constructing solutions to stochastic differential equations with reflecting boundary conditions. In this work, an explicit formula for the Skorokhod map Γ0,a on [0, a] for any a > 0 is derived. Specifically, it is shown that on the space D[0, ∞) of right-continuous functions with left limits taking values in R, Γ0,a = Λa • Γ0, where Λa :In addition, properties of Λa are developed and comparison properties of Γ0,a are established.
A general approach for calibrating Monte Carlo models to the market prices of benchmark securities is presented. Starting from a given model for market dynamics (price diffusion, rate diffusion, etc.), the algorithm corrects price-misspecifications and finite-sample effects in the simulation by assigning "probability weights" to the simulated paths. The choice of weights is done by minimizing the Kullback–Leibler relative entropy distance of the posterior measure to the empirical measure. The resulting ensemble prices the given set of benchmark instruments exactly or in the sense of least-squares. We discuss pricing and hedging in the context of these weighted Monte Carlo models. A significant reduction of variance is demonstrated theoretically as well as numerically. Concrete applications to the calibration of stochastic volatility models and term-structure models with up to 40 benchmark instruments are presented. The construction of implied volatility surfaces and forward-rate curves and the pricing and hedging of exotic options are investigated through several examples.
This paper presents a heavy-traffic analysis of the behavior of a
single-server queue under an Earliest-Deadline-First (EDF) scheduling policy in
which customers have deadlines and are served only until their deadlines
elapse. The performance of the system is measured by the fraction of reneged
work (the residual work lost due to elapsed deadlines) which is shown to be
minimized by the EDF policy. The evolution of the lead time distribution of
customers in queue is described by a measure-valued process. The heavy traffic
limit of this (properly scaled) process is shown to be a deterministic function
of the limit of the scaled workload process which, in turn, is identified to be
a doubly reflected Brownian motion. This paper complements previous work by
Doytchinov, Lehoczky and Shreve on the EDF discipline in which customers are
served to completion even after their deadlines elapse. The fraction of reneged
work in a heavily loaded system and the fraction of late work in the
corresponding system without reneging are compared using explicit formulas
based on the heavy traffic approximations. The formulas are validated by
simulation results.Comment: Published in at http://dx.doi.org/10.1214/10-AAP681 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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