260 research outputs found
Projected particle methods for solving McKean-Vlasov stochastic differential equations
We propose a novel projection-based particle method for solving the
McKean-Vlasov stochastic differential equations. Our approach is based on a
projection-type estimation of the marginal density of the solution in each time
step. The projection-based particle method leads in many situation to a
significant reduction of numerical complexity compared to the widely used
kernel density estimation algorithms. We derive strong convergence rates and
rates of density estimation. The convergence analysis in the case of linearly
growing coefficients turns out to be rather challenging and requires some new
type of averaging technique. This case is exemplified by explicit solutions to
a class of McKean-Vlasov equations with affine drift. The performance of the
proposed algorithm is illustrated by several numerical examples
Statistical Skorohod embedding problem and its generalizations
Given a L\'evy process , we consider the so-called statistical Skorohod
embedding problem of recovering the distribution of an independent random time
based on i.i.d. sample from Our approach is based on the genuine
use of the Mellin and Laplace transforms. We propose a consistent estimator for
the density of derive its convergence rates and prove their optimality. It
turns out that the convergence rates heavily depend on the decay of the Mellin
transform of We also consider the application of our results to the
problem of statistical inference for variance-mean mixture models and for
time-changed L\'evy processes
Representations for optimal stopping under dynamic monetary utility functionals
In this paper we consider the optimal stopping problem for general dynamic monetary utility functionals. Sufficient conditions for the Bellman principle and the existence of optimal stopping times are provided. Particular attention is payed to representations which allow for a numerical treatment in real situations. To this aim, generalizations of standard evaluation methods like policy iteration, dual and consumption based approaches are developed in the context of general dynamic monetary utility functionals. As a result, it turns out that the possibility of a particular generalization depends on specific properties of the utility functional under consideration.monetary utility functionals, optimal stopping, duality, policy iteration
A jump-diffusion Libor model and its robust calibration
In this paper we propose a jump-diffusion Libor model with jumps in a high-dimensional space (Rm) and test a stable non-parametric calibration algorithm which takes into account a given local covariance structure. The algorithm returns smooth and simply structured LĂ©vy densities, and penalizes the deviation from the Libor market model. In practice, the procedure is FFT based, thus fast, easy to implement, and yields good results, particularly in view of the severe ill-posedness of the underlying inverse problem.Libor market model, calibration, correlation, jump-diffusion
Dynamic programming for optimal stopping via pseudo-regression
We introduce new variants of classical regression-based algorithms for
optimal stopping problems based on computation of regression coefficients by
Monte Carlo approximation of the corresponding inner products instead of
the least-squares error functional. Coupled with new proposals for simulation
of the underlying samples, we call the approach "pseudo regression". A detailed
convergence analysis is provided and it is shown that the approach
asymptotically leads to less computational cost for a pre-specified error
tolerance, hence to lower complexity. The method is justified by numerical
examples
Generalized Post-Widder inversion formula with application to statistics
In this work we derive an inversion formula for the Laplace transform of a
density observed on a curve in the complex domain, which generalizes the well
known Post-Widder formula. We establish convergence of our inversion method and
derive the corresponding convergence rates for the case of a Laplace transform
of a smooth density. As an application we consider the problem of statistical
inference for variance-mean mixture models. We construct a nonparametric
estimator for the mixing density based on the generalized Post-Widder formula,
derive bounds for its root mean square error and give a brief numerical
example
Optimal dual martingales, their analysis and application to new algorithms for Bermudan products
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 It\^o-L\'evy
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 the backward dual regression algorithm in a Wiener environment
at well known benchmark examples. It turns out that the method is at least
comparable to the one in Belomestny et. al. (2009) regarding accuracy, but
regarding computational robustness there are even several advantages.Comment: This paper is an extended version of Schoenmakers and Huang, "Optimal
dual martingales and their stability; fast evaluation of Bermudan products
via dual backward regression", WIAS Preprint 157
Robust option replication for a Black-Scholes model extended with nondeterministic trends
Statistical analysis on various stocks reveals long range dependence behavior of the stock prices that is not consistent with the classical Black and Scholes model. This memory or nondeterministic trend behavior is often seen as a reflection of market sentiments and causes that the historical volatility estimator becomes unreliable in practice. We propose an extension of the Black and Scholes model by adding a term to the original Wiener term involving a smoother process which accounts for these effects. The problem of arbitrage will be discussed. Using a generalized stochastic integration theory [8], we show that it is possible to construct a self financing replicating portfolio for a European option without any further knowledge of the extension and that, as a consequence, the classical concept of volatility needs to be re-interpreted.
AMS subject classifications: 60H05, 60H10, 90A09
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