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 $L$, we consider the so-called statistical Skorohod embedding problem of recovering the distribution of an independent random time $T$ based on i.i.d. sample from $L_{T}.$ Our approach is based on the genuine use of the Mellin and Laplace transforms. We propose a consistent estimator for the density of $T,$ 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 $T.$ 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

Uniform approximation of the Cox-Ingersoll-Ross process

The Doss-Sussmann (DS) approach is used for uniform simulation of the Cox-Ingersoll-Ross (CIR) process. The DS formalism allows to express trajectories of the CIR process through solutions of some ordinary differential equation (ODE) depending on realizations of a Wiener process involved. By simulating the first-passage times of the increments of the Wiener process to the boundary of an interval and solving the ODE, we uniformly approximate the trajectories of the CIR process. In this respect special attention is payed to simulation of trajectories near zero. From a conceptual point of view the proposed method gives a better quality of approximation (from a path-wise point of view) than standard, or even exact simulation of the SDE at some discrete time grid.Comment: 24 page

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 Lévy 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 $L^2$ 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

Efficient and accurate log-L\'evy approximations to L\'evy driven LIBOR models

The LIBOR market model is very popular for pricing interest rate derivatives, but is known to have several pitfalls. In addition, if the model is driven by a jump process, then the complexity of the drift term is growing exponentially fast (as a function of the tenor length). In this work, we consider a L\'evy-driven LIBOR model and aim at developing accurate and efficient log-L\'evy approximations for the dynamics of the rates. The approximations are based on truncation of the drift term and Picard approximation of suitable processes. Numerical experiments for FRAs, caps, swaptions and sticky ratchet caps show that the approximations perform very well. In addition, we also consider the log-L\'evy approximation of annuities, which offers good approximations for high volatility regimes.Comment: 32 pages, 21 figures. Added an example of a path-dependent option (sticky ratchet caplet). Forthcoming in the Journal of Computational Financ

Monte Carlo Greeks for financial products via approximative transition densities

In this paper we introduce efficient Monte Carlo estimators for the valuation of high-dimensional derivatives and their sensitivities (''Greeks''). These estimators are based on an analytical, usually approximative representation of the underlying density. We study approximative densities obtained by the WKB method. The results are applied in the context of a Libor market model.Comment: 24 page

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