Papers by Denis Belomestny
SIAM Journal on Control and Optimization, 2020
In this paper we study optimal stopping problems for nonlinear Markov processes driven by a McKea... more In this paper we study optimal stopping problems for nonlinear Markov processes driven by a McKean-Vlasov SDE and aim at solving them numerically by Monte Carlo. To this end we propose a novel regression algorithm based on the corresponding particle system and prove its convergence. The proof of convergence is based on perturbation analysis of a related linear regression problem. The performance of the proposed algorithms is illustrated by a numerical example.
SIAM Journal on Numerical Analysis, 2018
We propose a novel projection-based particle method for solving McKean-Vlasov stochastic differen... more We propose a novel projection-based particle method for solving 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, particularly 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.
Journal of Mathematical Analysis and Applications, 2017
In this work we derive an inversion formula for the Laplace transform of a density observed on a ... more 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.
SIAM/ASA Journal on Uncertainty Quantification, 2015
This paper presents a novel approach to reduce the complexity of simulation based policy iteratio... more This paper presents a novel approach to reduce the complexity of simulation based policy iteration methods for solving optimal stopping problems. Typically, Monte Carlo construction of an improved policy gives rise to a nested simulation algorithm. In this respect our new approach uses the multilevel idea in the context of the nested simulations, where each level corresponds to a specific number of inner simulations. A thorough analysis of the convergence rates in the multilevel policy improvement algorithm is presented. A detailed complexity analysis shows that a significant reduction in computational effort can be achieved in comparison to the standard Monte Carlo based policy iteration. The performance of the multilevel method is illustrated in the case of pricing a multidimensional American derivative.
Finance and Stochastics, 2015
In this article we propose a novel approach to reduce the computational complexity of the dual me... more 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 different 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 significantly 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.
Proceedings Title: Proceedings of the 2012 Winter Simulation Conference (WSC), 2012
This paper is an overview of recent results by Belomestny and Schoenmakers 2011 and Belomestny, L... more This paper is an overview of recent results by Belomestny and Schoenmakers 2011 and Belomestny, Ladkau, and Schoenmakers 2012, on dual and primal Monte Carlo evaluation of American style derivatives using multilevel principles. It presents a novel and generic approach to reduce the complexity of nested simulations problems arising in Monte Carlo pricing of American options. The approach genuinely uses the multilevel idea where each level corresponds to a given number of inner simulations. A thorough complexity analysis of the respective nested dual algorithm and nested policy improvement algorithm shows that a significant complexity reduction can be achieved by using the multilevel versions of the algorithms.
SSRN Electronic Journal, 2008
In this paper we develop several regression algorithms for solving general stochastic optimal con... more In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithms is particulary useful for problems with high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea of the algorithms is to simulate a set of trajectories under some reference measure P * and to use a dynamic program formulation combined with fast methods for approximating conditional expectations and functional optimizations on these trajectories. Theoretical properties of the presented algorithms are investigated and convergence to the optimal solution is proved under mild assumptions. Finally, we present numerical results showing the efficiency of regression algorithms in a case of a highdimensional Bermudan basket options, in a model with a large investor and transaction costs.
Finance and Stochastics, 2013
In this article we propose a novel approach to reduce the computational complexity of the dual me... more 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 different 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 significantly 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.
Decisions in Economics and Finance, 2009
In this article we propose several pathwise and finite difference based methods for calculating s... more In this article we propose several pathwise and finite difference based methods for calculating sensitivities of Bermudan options using regression methods and Monte Carlo simulation. These methods rely on conditional probabilistic representations which allow, in combination with a regression approach, for efficient simultaneous computation of sensitivities at many initial positions. Assuming that the price of a Bermudan option can be evaluated sufficiently accurate, we develop a method for constructing deltas based on least squares. We finally propose a testing procedure for assessing the performance of the developed methods.
arXiv (Cornell University), Oct 22, 2018
In this paper we revisit the well-known constrained projection approximation subspace tracking al... more In this paper we revisit the well-known constrained projection approximation subspace tracking algorithm (CPAST) and derive, for the first time, non-asymptotic error bounds. Furthermore, we introduce a novel sparse modification of CPAST which is able to exploit sparsity in the underlying covariance structure. We present a nonasymptotic analysis of the proposed algorithm and study its empirical performance on simulated and real data.
ArXiv, 2021
We develop an Explore-Exploit Markov chain Monte Carlo algorithm (ExMCMC) that combines multiple ... more We develop an Explore-Exploit Markov chain Monte Carlo algorithm (ExMCMC) that combines multiple global proposals and local moves. The proposed method is massively parallelizeable and extremely computationally efficient. We prove V -uniform geometric ergodicity of ExMCMC under realistic conditions, and compute explicit bounds on the mixing rate showing the improvement brought by the multiple global moves. We show that ExMCMC allows fine-tuning of exploitation (local moves) and exploration (global moves) via a novel approach to proposing dependent global moves. Finally, we develop an adaptive scheme, FlExMCMC, that learns the distribution of global moves using normalizing flows. We illustrate the efficiency of ExMCMC and its adaptive versions on many classical sampling benchmarks. We also show that these algorithms improve the quality of sampling GANs as energy-based models.
The Annals of Statistics, 2021
In this paper we study the problem of density deconvolution under general assumptions on the meas... more In this paper we study the problem of density deconvolution under general assumptions on the measurement error distribution. Typically deconvolution estimators are constructed using Fourier transform techniques, and it is assumed that the characteristic function of the measurement errors does not have zeros on the real line. This assumption is rather strong and is not fulfilled in many cases of interest. In this paper we develop a methodology for constructing optimal density deconvolution estimators in the general setting that covers vanishing and non-vanishing characteristic functions of the measurement errors. We derive upper bounds on the risk of the proposed estimators and provide sufficient conditions under which zeros of the corresponding characteristic function have no effect on estimation accuracy. Moreover, we show that the derived conditions are also necessary in some specific problem instances.
In this note we consider the problem of using regression on interacting particles to compute cond... more In this note we consider the problem of using regression on interacting particles to compute conditional expectations for McKean-Vlasov SDEs. We prove general result on convergence of linear regression algorithms and establish the corresponding rates of convergence. Application to optimal stopping and variance reduction are considered.
arXiv: Computation, 2019
In this paper we propose an efficient variance reduction approach for additive functionals of Mar... more In this paper we propose an efficient variance reduction approach for additive functionals of Markov chains relying on a novel discrete time martingale representation. Our approach is fully non-asymptotic and does not require the knowledge of the stationary distribution (and even any type of ergodicity) or specific structure of the underlying density. By rigorously analyzing the convergence properties of the proposed algorithm, we show that its cost-to-variance product is indeed smaller than one of the naive algorithm. The numerical performance of the new method is illustrated for the Langevin-type Markov Chain Monte Carlo (MCMC) methods.
arXiv: Statistics Theory, 2018
Given discrete time observations over a growing time interval, we consider a nonparametric Bayesi... more Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the L\'evy density of a L\'evy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical results in challenging simulation examples. On the theoretical side, we establish that our nonparametric Bayesian procedure is consistent: in the low frequency data setting, with equispaced in time observations and intervals between successive observations remaining fixed, the posterior asymptotically, as the sample size $n\rightarrow\infty$, concentrates around the L\'evy density ...
In this paper we propose a Libor model with a high-dimensional specially structured system of dri... more In this paper we propose a Libor model with a high-dimensional specially structured system of driving CIR volatility processes. A stable calibration procedure which takes into account a given local correlation structure is presented. The calibration algorithm is FFT based, so fast and easy to implement.
arXiv: Statistics Theory, 2014
Given a L\'evy process $L$, we consider the so-called statistical Skorohod embedding problem ... more 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.
SIAM Journal on Financial Mathematics, 2021
In this paper we study randomized optimal stopping problems and consider corresponding forward an... more In this paper we study randomized optimal stopping problems and consider corresponding forward and backward Monte Carlo based optimisation algorithms. In particular we prove the convergence of the proposed algorithms and derive the corresponding convergence rates.
Journal of Mathematical Analysis and Applications, 2020
We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer... more We study linear rough partial differential equations in the setting of [Friz and Hairer, Springer, 2014, Chapter 12]. More precisely, we consider a linear parabolic partial differential equation driven by a deterministic rough path W of Hölder regularity α with 1/3 < α ≤ 1/2. Based on a stochastic representation of the solution of the rough partial differential equation, we propose a regression Monte Carlo algorithm for spatio-temporal approximation of the solution. We provide a full convergence analysis of the proposed approximation method which essentially relies on the new bounds for the higher order derivatives of the solution in space. Finally, a comprehensive simulation study showing the applicability of the proposed algorithm is presented.
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Papers by Denis Belomestny