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Convergence Analysis of Smoothing Methods for Optimal Control of Stationary Variational Inequalities
(2012)
In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise
control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to
the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained. These rates coincide with
rates obtained by numerical experiments, which are included in the paper.
We derive a-priori estimates on the length of the primal-dual path that results from a Moreau-Yosida approximation of the feasible set for state constrained optimal control problems. These bounds depend on the regularity of the state and the dimension of the problem. Comparison with numerical results indicates that these bounds are sharp and are attained for the case of a single active point.
We analyze a remarkable class of centrally symmetric polytopes, the Hansen
polytopes of split graphs. We confirm Kalai's 3^d-conjecture for such polytopes
(they all have at least 3^d nonempty faces) and show that the Hanner polytopes
among them (which have exactly 3^d nonempty faces) correspond to threshold
graphs. Our study produces a new family of Hansen polytopes that have only
3^d+16 nonempty faces.
In this article we propose a novel approach to reduce the computational complexity
of the dual method for pricing American options. We consider a sequence of
martingales that converges to a given target martingale and decompose the original
dual representation into a sum of representations that correspond to dierent levels
of approximation to the target martingale. By next replacing in each representation
true conditional expectations with their Monte Carlo estimates, we arrive at what
one may call a multilevel dual Monte Carlo algorithm. The analysis of this algorithm
reveals that the computational complexity of getting the corresponding target upper
bound, due to the target martingale, can be signicantly reduced. In particular, it
turns out that using our new approach, we may construct a multilevel version of the
well-known nested Monte Carlo algorithm of Andersen and Broadie (2004) that is,
regarding complexity, virtually equivalent to a non-nested algorithm. The performance
of this multilevel algorithm is illustrated by a numerical example.
In this paper, we study the dual representation for generalized multiple stopping problems,
hence the pricing problem of general multiple exercise options. We derive a dual representation which allows for cashflows which are subject to volume constraints modeled by
integer valued adapted processes and refraction periods modeled by stopping times. As
such, this extends the works by Schoenmakers (2010), Bender (2011a), Bender (2011b),
Aleksandrov and Hambly (2010), and Meinshausen and Hambly (2004) on multiple exercise
options, which either take into consideration a refraction period or volume constraints, but
not both simultaneously. We also allow more flexible cashflow structures than the additive
structure in the above references. For example some exponential utility problems are covered
by our setting. We supplement the theoretical results with an explicit Monte Carlo algorithm
for constructing confidence intervals for the price of multiple exercise options and exemplify
it by a numerical study on the pricing of a swing option in an electricity market.
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
(2012)
In this paper we introduce and study the concept of optimal and surely
optimal dual martingales in the context of dual valuation of Bermudan
options, and outline the development of new algorithms in this context.
We provide a characterization theorem, a theorem which gives conditions
for a martingale to be surely optimal, and a stability theorem concerning martingales which are near to be surely optimal in a sense. Guided
by these results we develop a framework of backward algorithms for constructing such a martingale. In turn this martingale may then be utilized
for computing an upper bound of the Bermudan product. The methodology is pure dual in the sense that it doesn’t require certain (input)
approximations to the Snell envelope.
In an Ito-Levy environment we outline a particular regression based
backward algorithm which allows for computing dual upper bounds without nested Monte Carlo simulation. Moreover, as a by-product this algorithm also provides approximations to the continuation values of the
product, which in turn determine a stopping policy. Hence, we may obtain lower bounds at the same time.
In a first numerical study we demonstrate a backward dual regression algorithm in a Wiener environment that is easy to implement and
is regarding accuracy comparable with the method of Belomestny et. al.
(2009).
Primal-dual linear Monte Carlo algorithm for multiple stopping - An application to flexible caps
(2012)
In this paper we consider the valuation of Bermudan callable derivatives with
multiple exercise rights. We present in this context a new primal-dual linear
Monte Carlo algorithm that allows for ecient simulation of lower and upper price
bounds without using nested simulations (hence the terminology). The algorithm
is essentially an extension of a primal{dual Monte Carlo algorithm for standard
Bermudan options proposed in Schoenmakers et al. (2011), to the case of multiple
exercise rights. In particular, the algorithm constructs upwardly a system of dual
martingales to be plugged into the dual representation of Schoenmakers (2010).
At each level the respective martingale is constructed via a backward regression
procedure starting at the last exercise date. The thus constructed martingales are
nally used to compute an upper price bound. At the same time, the algorithm
also provides approximate continuation functions which may be used to construct
a price lower bound. The algorithm is applied to the pricing of
exible caps
in a Hull and White (1990) model setup. The simple model choice allows for
comparison of the computed price bounds with the exact price which is obtained
by means of a trinomial tree implementation. As a result, we obtain tight price
bounds for the considered application. Moreover, the algorithm is generically
designed for multi-dimensional problems and is tractable to implement.
We study minimal supersolutions of backward stochastic differential equations. We show the existence and uniqueness of the minimal supersolution, if the generator is jointly lower semicontinuous, bounded from below by an affine function of the control variable, and satisfies a specific normalization property. Semimartingale convergence is used to establish the main result.
We study a nonlinear operator defined via minimal supersolutions of backward stochastic differential equations with generators that are monotone in y, convex in z, jointly lower semicontinuous, and bounded below by an affine function of the control variable. We show existence, uniqueness, monotone convergence, Fatou’s Lemma and lower semicontinuity of this functional. We provide a comparison principle for the underlying minimal supersolutions of BSDEs, which we illustrate by maximizing expected exponential utility.