Constructive Quantization and Multilevel Algorithms for Quadrature of Stochastic Differential Equations

Altmayer, Martin; Dereich, Steffen; Li, Sangmeng; Müller-Gronbach, Thomas; Neuenkirch, Andreas; Ritter, Klaus; Yaroslavtseva, L.N.

doi:10.1007/978-3-319-08159-5_6

Cited by 3 publications

(6 citation statements)

References 30 publications

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“…and show that it can be either quite large or small. Let σ : R d → R d be the projection defined by σx := (x (1) , · · · , x (d−1) , 0).…”

Section: Remark 17mentioning

confidence: 99%

“…We derive now the upper bound in (1). Fix a large M > 0 and observe that the identity K = #{i : r i > 2M/n} + #{i : r i ≤ 2M/n} =:…”

Section: Dimension D =mentioning

confidence: 99%

“…This results in the larger bound. Proof: Consider the following collection of boxes: 1) , with k m ∈ {0, . .…”

Section: ℓ 1 -Normmentioning

confidence: 99%

“…attaining values in some function spaces, therefore called functional quantiation, see e.g. [10,7,8,9,14,15,5,12,1] and references therein for a selection, and their applications to numerical probability, see e.g. [16].…”

Section: Introductionmentioning

confidence: 99%

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How complex is a random picture?

Аурзада

Lifshits

2019

Journal of Complexity

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We study the amount of information that is contained in "random pictures", by which we mean the sample sets of a Boolean model. To quantify the notion "amount of information", two closely connected questions are investigated: on the one hand, we study the probability that a large number of balls is needed for a full reconstruction of a Boolean model sample set. On the other hand, we study the quantization error of the Boolean model w.r.t. the Hausdorff distance as a distortion measure. Keywords: Boolean model; functional quantization; high resolution quantization; information based complexity; metric entropy. 2010 Mathematics Subject Classification: 94A29, 60D05, secondary: 52A22.problem, the following random variable is crucial. Moreover, we believe that it may be of independent interest. Define the effective number of balls visible in the picture asIn other words, with K balls one can reproduce the black picture S exactly as with the original N balls.We are interested in the upper tail of K, i.e. P[K ≥ n] when n → ∞. This means we study the probability that one needs many balls in order to reconstruct the picture S. In particular, we would like to understand when one can "save balls" w.r.t. the original Poisson number of balls N . To make this more precise, note that clearly K ≤ N , and soWe would like to show that the upper tail of K is thinner, i.e. for some a > 1 P[K ≥ n] = exp (−a · n log n · (1 + o(1))) , n → ∞.It turns out that this question is non-trivial and interesting. The answer depends on the dimension d, on the type of norm used, and on the distribution of the radii L(R 1 ). Boolean models are fundamental objects in stochastic geometry and have a large range of applications, [4,17]. However, to the knowledge of the authors, until recently mostly the average of observables of Boolean models are studied. Often this plays a role when estimating parameters of the model in applications. On the contrary, the present paper deals with rare events, i.e. with large deviation probabilities.As mentioned above, the upper tail of the random variable K is an essential ingredient for solving the so-called quantization problem, which we recall now. Let an arbitrary norm ||.|| be fixed on R d . Let d H denote the corresponding Hausdorff distance between the closed subsets of R d . We define the respective quantization error for pictures byHere, the sets C are called codebooks and the upper index (q) stands for "quantization". The idea is that the "analog" signal S should be encoded by an element A ∈ C. This incurs an error, d H (A, S), measured in Hausdorff distance. Losely speaking, D (q) is then the minimal average error over all codebooks C of a size not exceeding e r . We are interested in letting r → ∞, that is, the size of the codebooks grows; and we would like to understand the rate of decay of the corresponding quantization error.

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“…and show that it can be either quite large or small. Let σ : R d → R d be the projection defined by σx := (x (1) , · · · , x (d−1) , 0).…”

Section: Remark 17mentioning

confidence: 99%

“…We derive now the upper bound in (1). Fix a large M > 0 and observe that the identity K = #{i : r i > 2M/n} + #{i : r i ≤ 2M/n} =:…”

Section: Dimension D =mentioning

confidence: 99%

“…This results in the larger bound. Proof: Consider the following collection of boxes: 1) , with k m ∈ {0, . .…”

Section: ℓ 1 -Normmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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How complex is a random picture?

Аурзада

Lifshits

2019

Journal of Complexity

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“…We note that we only chose the random variable W to be 1-dimensional for simplicity and refer to Section 2 for the case when W is a possibly high-dimensional random variable. Our first step to derive the proposed approximation algorithm is to recall standard MC approximations for the parametric expectation in (1). Let M ∈ N, let W m,m : Ω → R, m, m ∈ N 0 , be i.i.d.…”

Section: Introductionmentioning

confidence: 99%

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

Becker¹,

Jentzen²,

Müller³

et al. 2022

Preprint

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Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

Becker,

Jentzen,

Müller

et al. 2023

Mathematical Finance

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In financial engineering, prices of financial products are computed approximately many times each trading day with (slightly) different parameters in each calculation. In many financial models such prices can be approximated by means of Monte Carlo (MC) simulations. To obtain a good approximation the MC sample size usually needs to be considerably large resulting in a long computing time to obtain a single approximation. A natural deep learning approach to reduce the computation time when new prices need to be calculated as quickly as possible would be to train an artificial neural network (ANN) to learn the function which maps parameters of the model and of the financial product to the price of the financial product. However, empirically it turns out that this approach leads to approximations with unacceptably high errors, in particular when the error is measured in the ‐norm, and it seems that ANNs are not capable to closely approximate prices of financial products in dependence on the model and product parameters in real life applications. This is not entirely surprising given the high‐dimensional nature of the problem and the fact that it has recently been proved for a large class of algorithms, including the deep learning approach outlined above, that such methods are in general not capable to overcome the curse of dimensionality for such approximation problems in the ‐norm. In this article we introduce a new numerical approximation strategy for parametric approximation problems including the parametric financial pricing problems described above and we illustrate by means of several numerical experiments that the introduced approximation strategy achieves a very high accuracy for a variety of high‐dimensional parametric approximation problems, even in the ‐norm. A central aspect of the approximation strategy proposed in this article is to combine MC algorithms with machine learning techniques to, roughly speaking, learn the random variables (LRV) in MC simulations. In other words, we employ stochastic gradient descent (SGD) optimization methods not to train parameters of standard ANNs but instead to learn random variables appearing in MC approximations. In that sense, the proposed LRV strategy has strong links to Quasi‐Monte Carlo (QMC) methods as well as to the field of algorithm learning. Our numerical simulations strongly indicate that the LRV strategy might indeed be capable to overcome the curse of dimensionality in the ‐norm in several cases where the standard deep learning approach has been proven not to be able to do so. This is not a contradiction to the established lower bounds mentioned above because this new LRV strategy is outside of the class of algorithms for which lower bounds have been established in the scientific literature. The proposed LRV strategy is of general nature and not only restricted to the parametric financial pricing problems described above, but applicable to a large class of approximation problems. In this article we numerically test the LRV strategy in the case of the pricing of European call options in the Black‐Scholes model with one underlying asset, in the case of the pricing of European worst‐of basket put options in the Black‐Scholes model with three underlying assets, in the case of the pricing of European average put options in the Black‐Scholes model with three underlying assets and knock‐in barriers, as well as in the case of stochastic Lorentz equations. For these examples the LRV strategy produces highly convincing numerical results when compared with standard MC simulations, QMC simulations using Sobol sequences, SGD‐trained shallow ANNs, and SGD‐trained deep ANNs.

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Constructive Quantization and Multilevel Algorithms for Quadrature of Stochastic Differential Equations

Cited by 3 publications

References 30 publications

How complex is a random picture?

How complex is a random picture?

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

Learning the random variables in Monte Carlo simulations with stochastic gradient descent: Machine learning for parametric PDEs and financial derivative pricing

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