Univ.-Prof. Dr. Denis Belomestny

Tel.:  +49 201 183-7416

Nicole Obszanski
Tel.:  +49 201 183-7425
Fax.: +49 201 183-2426

Universit?t Duisburg-Essen
Fachbereich Mathematik
Thea-Leymann-Str. 9
D-45127 Essen


Ausgewählte Veröffentlichungen

Referierte Veröffentlichungen

Belomestny, D. and Nagapetyan, T. (2015). Multilevel path simulation for weak approximation schemes with application to Levy-driven SDEs, Bernoulli Journal
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In this paper we discuss the possibility of using multilevel Monte Carlo (MLMC) approach for weak approximation schemes. It turns out that by means of a simple coupling between consecutive time discretisation levels, one can achieve the same complexity gain as under the presence of a strong convergence. We exemplify this general idea in the case of weak Euler schemes for Levy-driven stochastic differential equations. The numerical performance of the new ``weak'' MLMC method is illustrated by several numerical examples.

Belomestny, D., Ladkau, M. and Schoenmakers, J. (2015). Multilevel simulation based policy iteration for optimal stopping convergence and complexity, SIAM Journal of Uncertainty Quantification, 460-483
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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.

Denis Belomestny, Fabian Dickmann, and Tigran Nagapetyan (2015). Pricing Bermudan options via multilevel approximation methods, SIAM Journal of Financial Mathematics, 448-466
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In this article we propose a novel approach to reduce the computational complexity of various approximation methods for pricing discrete time American or Bermudan options. Given a sequence of continuation values estimates corresponding to different levels of spatial approximation, we propose a multilevel low biased estimate for the price of the option. It turns out that the resulting complexity gain can be of order eps^{-1} with eps denoting the desired precision. The performance of the proposed multilevel algorithms is illustrated by a numerical example.

Belomestny, D. and Kraetschmer, V. (2015). Optimal Stopping Under Model Uncertainty: Randomized Stopping Times Approach, Annals of Applied Probability
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In this work we consider optimal stopping problems with conditional convex risk measures without assuming any kind of time-consistency. In particular, we generalize the additive dual representation of Rogers (2002) to the case of optimal stopping under uncertainty. Finally, we develop several Monte Carlo algorithms and illustrate their power for optimal stopping under Average Value at Risk.

Belomestny, D. and Spokoiny, V. (2014). Concentration inequalities for smooth random fields, Probability Theory and Its Applications, 59(4), 314-323
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In this note we derive a sharp concentration inequality for the supremum of a smooth random field over a finite dimensional set. It is shown that this supremum can be bounded with high probability by the value of the field at some deterministic point plus an intrinsic dimension of the optimisation problem. As an application we prove the exponential inequality for a function of the maximal eigenvalue of a random matrix is proved.

Belomestny, D. and Panov, V. (2013). Estimation of the activity of jumps in time-changed Levy models, Electronic Journal of Statistics, 7(1), 2970-3003
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In this paper we consider a class of time-changed Levy processes that can be represented in the form $Y_s=X_{T(s)}$, where X is a Levy process and T is a non-negative and non-decreasing stochastic process independent of X. The aim of this work is to infer on the Blumenthal-Getoor index of the process X from low-frequency observations of the time-changed Levy process Y. We propose a consistent estimator for this index, derive the minimax rates of convergence and show that these rates can not be improved in general. The performance of the estimator is illustrated by numerical examples.

Belomestny, D., Schoenmakers, J. and Dickmann, F. (2013). Multilevel dual approach for pricing American type derivatives, Finance and Stochastics, 17(4), 717-742
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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&Broadie that is, regarding complexity, virtually equivalent to a non-nested algorithm. The performance of this multilevel algorithm is illustrated by a numerical example.

Belomestny, D. (2013). Solving optimal stopping problems via empirical dual optimization, Annals of Applied Probability, 23(5), 1988-2019
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In this paper we consider a method of solving optimal stopping problems in discrete and continuous time based on their dual representation. A novel and generic simulation-based optimization algorithm not involving nested simulations is proposed and studied. The algorithm involves the optimization of a genuinely penalized dual objective functional over a class of adapted martingales. We prove the convergence of the proposed algorithm and demonstrate its eciency for optimal stopping problems arising in option pricing.

Belomestny, D. and Panov, V. (2012). Abelian theorems for stochastic volatility models with application to the estimation of jump activity, Stochastic Processes and Their Applications, 123(1), 15-44
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In this paper, we prove a kind of Abelian theorem for a class of stochastic volatility models (X,V) where both the state process X and the volatility process V may have jumps. Our results relate the asymptotic behavior of the characteristic function of X in a stationary regime to the Blumenthal-Getoor indexes of the Levy processes driving the jumps in X and V . The results obtained are used to construct consistent estimators for the above Blumenthal-Getoor indexes based on low-frequency observations of the state process X. We derive convergence rates for the corresponding estimator and show that these rates cannot be improved in general.

Belomestny, D. and Kraetschmer, V. (2012). Central limit theorems for law-invariant coherent risk measures, Journal of Applied Probability, 49(1), 1-21.
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In this paper we study the asymptotic properties of the canonical plugin estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent and identically distributed data, and then extend it to the case of weakly dependent data. Finally, a number of illustrating examples is presented

Belomestny, D. (2011). On the rates of convergence of simulation-based optimization algorithms for optimal stopping problems, Annals of Applied Probability, 21(1), 215-239.
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In this paper we study simulation-based optimization algorithms for solving discrete time optimal stopping problems. Using large deviation theory for the increments of empirical processes, we derive optimal convergence rates for the value function estimate and show that they can not be improved in general. The rates derived provide a guide to the choice of the number of simulated paths needed in optimization step, which is crucial for the good performance of any simulation-based optimization algorithm. Finally, we present a numerical example of solving optimal stopping problem arising in finance that illustrates our theoretical findings.

Belomestny, D. (2011). Pricing Bermudan options using regression: optimal rates of convergence for lower estimates, Finance and Stochastics, 15(4), 655-683.
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The problem of pricing Bermudan options using simulations and nonparametric regression is considered. We derive optimal nonasymptotic bounds for the low biased estimate based on a suboptimal stopping rule constructed from some estimates of the optimal continuation values. These estimates may be of different nature, local or global, with the only requirement being that the deviations of these estimates from the true continuation values can be uniformly bounded in probability. As an illustration, we discuss a class of local polynomial estimates which, under some regularity conditions, yield continuation values estimates possessing the required property.

Belomestny, D. (2011). Spectral estimation of the Levy density in partially observed affine models, Stochastic Processes and Their Applications, 121(1), 1217-1244.
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The problem of separating the jump part of a multidimensional regular affine process from its continuous part is considered. In particular, we present an algorithm for a nonparametric estimation of the jump distribution under the presence of a nonzero diffusion component. An estimation methodology is proposed which is based on the log-affine representation of the conditional characteristic function of a regular affine process and employees a smoothed in time version of the empirical characteristic function in order to estimate the derivatives of the conditional characteristic function. We derive almost sure uniform rates of convergence for the estimated Levy density and prove that these rates are optimal in the minimax sense. Finally, the performance of the estimation algorithm is illustrated in the case of the Bates stochastic volatility model.

Belomestny, D. (2011). Statistical inference for time-changed Levy processes via composite characteristic function estimation, Annals of Statistics, 39(4), 2205-2242.
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In this article the problem of semi-parametric inference on the parameters of a multidimensional Levy process L_t with independent components based on the low-frequency observations of the corresponding time-changed Levy process L_{mathcal{T}(t)}, where mathcal{T} is a non-negative, non-decreasing real-valued process independent of L_t, is studied. We show that this problem is closely related to the problem of composite function estimation that has recently got much attention in statistical literature. Under suitable identifiability conditions we propose a consistent estimate for the Levy density of L_t and derive the uniform as well as the pointwise convergence rates of the estimate proposed. Moreover, we prove that the rates obtained are optimal in a minimax sense over suitable classes of time-changed Levy models. Finally, we present a simulation study showing the performance of our estimation algorithm in the case of time-changed Normal Inverse Gaussian (NIG) Levy processes.

Belomestny, D. (2010). Spectral estimation of the fractional order of a Levy process, Annals of Statistics, 38(1), 317-351.
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We consider the problem of estimating the fractional order of a Levy process from low frequency historical and options data. An estimation methodology is developed which allows us to treat both estimation and cal- ibration problems in a unified way. The corresponding procedure consists of two steps: the estimation of a conditional characteristic function and the weighted least squares estimation of the fractional order in spectral domain. While the second step is identical for both calibration and estimation, the first one depends on the problem at hand. Minimax rates of convergence for the fractional order estimate are derived, the asymptotic normality is proved and a data-driven algorithm based on aggregation is proposed. The performance of the estimator in both estimation and calibration setups is illustrated by a simulation study.

Belomestny, D., Kolodko, A. and Schoenmakers, J. (2010). Regression methods for stochastic control problems and their convergence analysis, SIAM Journal on Control and Optimization, 48(5), 3562-3588.
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In this paper we develop several regression algorithms for solving general stochastic optimal control problems via Monte Carlo. This type of algorithm is particularly useful for problems with a high-dimensional state space and complex dependence structure of the underlying Markov process with respect to some control. The main idea behind the algorithms is to simulate a set of trajectories under some reference measure and to use the Bellman principle combined with fast methods for approximating conditional expectations and functional optimization. Theoretical properties of the presented algorithms are investigated, and the convergence to the optimal solution is proved under some assumptions. Finally, the presented methods are applied in a numerical example of a high-dimensional controlled Bermudan basket option in a financial market with a large investor.

Belomestny, D. and Gapeev, P. (2010). An iterative procedure for solving integral equations related to optimal stopping problem, Stochastics, 82(4), 365-380.
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We present an iterative algorithm for computing values of optimal stopping problems for one-dimensional diffusions on finite time intervals. The method is based on a time discretization of the initial model and a construction of discretized analogues of the associated integral equation for the value function. The proposed iterative procedure converges in a finite number of steps and delivers in each step a lower or an upper bound for the discretized value function on the whole time interval. We also give remarks on applications of the method for solving the integral equations related to several optimal stopping problems.

Belomestny, D., Bender, Ch. and Schoenmakers, J. (2009). True upper bounds for Bermudan products via non-nested Monte Carlo., Mathematical Finance, 19(1), 53-71.
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We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic "true" stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so con structed martingale can be employed for computing tight dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator. Numerical experiments indicate the efficiency of the proposed algorithms.

Belomestny, D., Kampen, J. and Schoenmakers, J. (2009). Holomorphic transforms with application to affine processes, Journal of Functional Analysis, 257(4), 1222-1250.
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In a rather general setting of Ito-Levy processes we study a class of transforms (Fourier for example) of the state variable of a process which are holomorphic in some disc around time zero in the complex plane. We show that such transforms are related to a system of analytic vectors for the generator of the process, and we state conditions which allow for holomorphic extension of these transforms into a strip which contains the positive real axis. Based on these extensions we develop a functional series expansion of these transforms in terms of the constituents of the generator. As application, we show that for multi-dimensional affine Ito-Levy processes with state dependent jump part the Fourier transform is holomorphic in a time strip under some stationarity conditions, and give log-affine series representations for the transform.

Belomestny, D. and Spokoiny, V. (2007). Spatial aggregation of local likelihood estimates with applications to classification, Annals of Statistics, 35(5), 2287-2311.
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This paper presents a new method for spatially adaptive local (constant) likelihood estimation which applies to a broad class of nonparametric mod els, including the Gaussian, Poisson and binary response models. The main idea of the method is, given a sequence of local likelihood estimates ("weak" estimates), to construct a new aggregated estimate whose pointwise risk is of order of the smallest risk among all "weak" estimates. We also propose a new approach toward selecting the parameters of the procedure by providing the prescribed behavior of the resulting estimate in the simple parametric situation. We establish a number of important theoretical results concerning the optimality of the aggregated estimate. In particular, our "oracle" result claims that its risk is, up to some logarithmic multiplier, equal to the smallest risk for the given family of estimates. The performance of the procedure is illustrated by application to the classification problem. A numerical study demonstrates its reasonable performance in simulated and real-life examples.

Belomestny, D. and Reiss, M. (2006). Spectral calibration of exponential Levy models, Finance and Stochastics, 10(4), 449-474.
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We investigate the problem of calibrating an exponential Levy model based on market prices of vanilla options. We show that this inverse problem is in general severely ill-posed and we derive exact minimax rates of convergence. The estimation procedure we propose is based on the explicit inversion of the option price formula in the spectral domain and a cut-off scheme for high frequencies as regularisation.