Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Home
/
Papers
/
Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Posted Content•

Local Lipschitz continuity in the initial value and strong completeness for nonlinear stochastic differential equations

Sonja Cox¹, Martin Hutzenthaler², Arnulf Jentzen³•Institutions (3)

University of Amsterdam¹, University of Duisburg-Essen², University of Münster³

22 Sep 2013-arXiv: Probability-

TL;DR: In this paper, the authors provided sufficient conditions on the coefficien t functions of the SDE and on p ∈ (0, ∞) that ensure local Lipschitz continuity in the strong L p -sense with respect to the initial value.

read less

Abstract: Recently, Hairer et. al [14] showed that there exist SDEs with infinitely often differentiable and globally bounded coefficient functions whose solutions fail to be locally Lipschitz continuous in the strong L p -sense with respect to the initial value for every p ∈ [1,∞]. In this article we provide sufficient conditions on the coefficien t functions of the SDE and on p ∈ (0,∞] which ensure local Lipschitz continuity in the strong L p -sense with respect to the initial value and we establish explicit estimates for the local Lipschitz continuity constants. In particular, we prove local Lipschitz continuity in the initial value for several nonlinear SDEs from the literature such as the stochastic van der Pol oscillator, Brownian dynamics, the Cox-Ingersoll-Ross processes and the Cahn-Hilliard-Cook equation. As an application of our estimates, we obtain strong completeness for several nonlinear SDEs.

...read moreread less

Content maybe subject to copyright Report

Citations

PDF

Open Access

More filters

Posted Content•

A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

[...]

Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger

07 Sep 2018-arXiv: Numerical Analysis

TL;DR: It is proved, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.

...read moreread less

Abstract: Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, such numerical simulations indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are also a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of these mathematical results prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy $\varepsilon > 0$ and the PDE dimension $d \in \mathbb{N}$ and we thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.

...read moreread less

126 citations

Journal Article•DOI•

On a perturbation theory and on strong convergence rates for stochastic ordinary and partial differential equations with non-globally monotone coefficients

[...]

Martin Hutzenthaler, Arnulf Jentzen

01 Jan 2014-arXiv: Probability

TL;DR: In this article, a perturbation theory for stochastic differential equations (SDEs) was developed, by which they mean both stochastastic ordinary differential equations and stochastically partial differential equations.

...read moreread less

Abstract: We develope a perturbation theory for stochastic differential equations (SDEs) by which we mean both stochastic ordinary differential equations (SODEs) and stochastic partial differential equations (SPDEs). In particular, we estimate the $ L^p $-distance between the solution process of an SDE and an arbitrary Ito process, which we view as a perturbation of the solution process of the SDE, by the $ L^q $-distances of the differences of the local characteristics for suitable $ p, q > 0 $. As application of our perturbation theory, we establish strong convergence rates for numerical approximations of a class of SODEs with non-globally monotone coefficients. As another application of our perturbation theory, we prove strong convergence rates for spectral Galerkin approximations of solutions of semilinear SPDEs with non-globally monotone nonlinearities including Cahn-Hilliard-Cook type equations and stochastic Burgers equations. Further applications of the perturbation theory include the regularity of solutions of SDEs with respect to the initial values and small-noise analysis for ordinary and partial differential equations.

...read moreread less

103 citations

Cites background from "Local Lipschitz continuity in the i..."

...[14] together with assumption (80) and the assumption that E [ eU0(X0) + eÛ0(Y0) ] <∞ prove that ∥∥∥∥exp(T ∫ 0 [ 〈Ys−PXs,Pμ(Ys)−Pμ(PXs)〉H+ (p−1) (1+ε) 2 ‖Pσ(Ys)−Pσ(PXs)‖2HS(U,H) ‖Ys−PXs‖2H + χs ]+ ds )∥∥∥∥ Lq(Ω;R)...
[...]
...[14]), then the right-hand side of (79) can be further estimated in an appropriate way....
[...]
...[14] and can be used to study the regularity of solutions of SDEs in the initial value....
[...]
...[28] Hutzenthaler, M., Jentzen, A., and Kloeden, P. E. Strong convergence of an explicit numerical method for SDEs with non-globally Lipschitz continuous coefficients....
[...]
...[27] Hutzenthaler, M., Jentzen, A., and Kloeden, P. E. Strong and weak divergence in finite time of Euler’s method for stochastic differential equations with non-globally Lipschitz continuous coefficients....
[...]

Posted Content•

Solving stochastic differential equations and Kolmogorov equations by means of deep learning

[...]

Christian Beck, Sebastian Becker, Philipp Grohs, Nor Jaafari, Arnulf Jentzen - Show less +1 more

01 Jun 2018-arXiv: Numerical Analysis

TL;DR: In this article, a numerical approximation of the Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse of dimensionality is presented.

...read moreread less

Abstract: Stochastic differential equations (SDEs) and the Kolmogorov partial differential equations (PDEs) associated to them have been widely used in models from engineering, finance, and the natural sciences. In particular, SDEs and Kolmogorov PDEs, respectively, are highly employed in models for the approximative pricing of financial derivatives. Kolmogorov PDEs and SDEs, respectively, can typically not be solved explicitly and it has been and still is an active topic of research to design and analyze numerical methods which are able to approximately solve Kolmogorov PDEs and SDEs, respectively. Nearly all approximation methods for Kolmogorov PDEs in the literature suffer under the curse of dimensionality or only provide approximations of the solution of the PDE at a single fixed space-time point. In this paper we derive and propose a numerical approximation method which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region $[a,b]^d$ without suffering from the curse of dimensionality. Numerical results on examples including the heat equation, the Black-Scholes model, the stochastic Lorenz equation, and the Heston model suggest that the proposed approximation algorithm is quite effective in high dimensions in terms of both accuracy and speed.

...read moreread less

92 citations

Journal Article•

A proof that artificial neural networks overcome the curse of dimensionality in the numerical approximation of Black-Scholes partial differential equations

[...]

Philipp Grohs, Fabian Hornung, Arnulf Jentzen, Philippe von Wurstemberger

01 Sep 2018-The annual research report

TL;DR: In this paper, the authors showed that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs, and they showed that the number of required parameters of an ANN to approximate the solution of the BlackScholes PSDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy and the PDE dimension.

...read moreread less

83 citations

Journal Article•

Solving stochastic differential equations and Kolmogorov equations by means of deep learning.

[...]

Christian Beck, Sebastian Becker, Philipp Grohs, Nor Jaafari, Arnulf Jentzen - Show less +1 more

01 Jun 2018-The annual research report

TL;DR: A numerical approximation method is derived and proposed which aims to overcome both of the above mentioned drawbacks and intends to deliver a numerical approximation of the Kolmogorov PDE on an entire region without suffering from the curse of dimensionality.

...read moreread less

75 citations

1
2
3
4
…
5
6
7
8
9
10
11
12
13
14

Collapse

References

PDF

Open Access

More filters

Journal Article•DOI•

Deterministic nonperiodic flow

[...]

Edward N. Lorenz¹•Institutions (1)

Massachusetts Institute of Technology¹

01 Mar 1963-Journal of the Atmospheric Sciences

TL;DR: In this paper, it was shown that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states, and systems with bounded solutions are shown to possess bounded numerical solutions.

...read moreread less

Abstract: Finite systems of deterministic ordinary nonlinear differential equations may be designed to represent forced dissipative hydrodynamic flow. Solutions of these equations can be identified with trajectories in phase space For those systems with bounded solutions, it is found that nonperiodic solutions are ordinarily unstable with respect to small modifications, so that slightly differing initial states can evolve into considerably different states. Systems with bounded solutions are shown to possess bounded numerical solutions.

...read moreread less

16,554 citations

"Local Lipschitz continuity in the i..." refers methods in this paper

...In particular, Lemma 2.3 in Zhang [54] yields local Lipschitz continuity in the initial value for sufficiently small positive time points for the stochastic van der Pol oscillator in the case of globally bounded noise (Subsection 4.1), for the stochastic Duffing-van der Pol oscillator in the case of globally bounded noise (see Subsection 4.2), for the stochastic Lorenz equation with additive noise (see Subsection 4.3), for the Langevin dynamics under certain assumptions (see Subsection 4.4), for a model from experimental psychology (see Subsection 4.7) and for the stochastic Brusselator under certain assumptions (see Subsection 4.8)....
[...]
...Lorenz [32] suggested a three-dimensional ordinary differential equation as a simplified model of convection rolls in the atmosphere....
[...]
...3 Stochastic Lorenz equation with additive noise Lorenz [32] suggested a three-dimensional ordinary differential equation as a simplified model of convection rolls in the atmosphere....
[...]
...Another way for establishing strong completeness for the stochastic Lorenz equation (151) in the case of additive noise is to substract the noise process and then to solve the resulting random orindary differential equations for every continuous trajectory of the driving noise process (cf. the remarks in Subsection 3.3)....
[...]
...4.1 Stochastic van der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Stochastic Duffing-van der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Stochastic Lorenz equation with additive noise . . . . . . . . . . . . . . . . . . . . 35 4.4 Langevin dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.5 Brownian dynamics (Overdamped Langevin dynamics) . . . . . . . . . . . . . . . 37 4.6 Stochastic SIR model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.7 Experimental psychology model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.8 Stochastic Brusselator in the well-stirred case . . . . . . . . . . . . . . . . . . . . 41 4.9 Stochastic volatility processes and interest rate models (CIR, Ait-Sahalia, 3/2- model, CEV) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.10 Wright-Fisher diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.11 Stochastic Burgers equation with a globally bounded diffusion coefficient and trace class noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.12 Cahn-Hilliard-Cook equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.12.1 Globally bounded diffusion coefficient and trace class noise . . . . . . . . . 49 4.12.2 Space-time white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50...
[...]

Book•

Singular Integrals and Differentiability Properties of Functions.

[...]

Elias M. Stein

01 Feb 1971

TL;DR: Stein's seminal work Real Analysis as mentioned in this paper is considered the most influential mathematics text in the last thirty-five years and has been widely used as a reference for many applications in the field of analysis.

...read moreread less

Abstract: Singular integrals are among the most interesting and important objects of study in analysis, one of the three main branches of mathematics. They deal with real and complex numbers and their functions. In this book, Princeton professor Elias Stein, a leading mathematical innovator as well as a gifted expositor, produced what has been called the most influential mathematics text in the last thirty-five years. One reason for its success as a text is its almost legendary presentation: Stein takes arcane material, previously understood only by specialists, and makes it accessible even to beginning graduate students. Readers have reflected that when you read this book, not only do you see that the greats of the past have done exciting work, but you also feel inspired that you can master the subject and contribute to it yourself. Singular integrals were known to only a few specialists when Stein's book was first published. Over time, however, the book has inspired a whole generation of researchers to apply its methods to a broad range of problems in many disciplines, including engineering, biology, and finance. Stein has received numerous awards for his research, including the Wolf Prize of Israel, the Steele Prize, and the National Medal of Science. He has published eight books with Princeton, including Real Analysis in 2005.

...read moreread less

9,595 citations

Book•

Continuous martingales and Brownian motion

[...]

Daniel Revuz, Marc Yor

01 Jan 1990

TL;DR: In this article, the authors present a comprehensive survey of the literature on limit theorems in distribution in function spaces, including Girsanov's Theorem, Bessel Processes, and Ray-Knight Theorem.

...read moreread less

Abstract: 0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.- VIII. Girsanov's Theorem and First Applications.- IX. Stochastic Differential Equations.- X. Additive Functionals of Brownian Motion.- XI. Bessel Processes and Ray-Knight Theorems.- XII. Excursions.- XIII. Limit Theorems in Distribution.- 1. Gronwall's Lemma.- 2. Distributions.- 3. Convex Functions.- 4. Hausdorff Measures and Dimension.- 5. Ergodic Theory.- 6. Probabilities on Function Spaces.- 7. Bessel Functions.- 8. Sturm-Liouville Equation.- Index of Notation.- Index of Terms.- Catalogue.

...read moreread less

7,338 citations

Book•

Numerical Solution of Stochastic Differential Equations

[...]

Peter E. Kloeden, Eckhard Platen

01 Jun 1992

TL;DR: In this article, a time-discrete approximation of deterministic Differential Equations is proposed for the stochastic calculus, based on Strong Taylor Expansions and Strong Taylor Approximations.

...read moreread less

Abstract: 1 Probability and Statistics- 2 Probability and Stochastic Processes- 3 Ito Stochastic Calculus- 4 Stochastic Differential Equations- 5 Stochastic Taylor Expansions- 6 Modelling with Stochastic Differential Equations- 7 Applications of Stochastic Differential Equations- 8 Time Discrete Approximation of Deterministic Differential Equations- 9 Introduction to Stochastic Time Discrete Approximation- 10 Strong Taylor Approximations- 11 Explicit Strong Approximations- 12 Implicit Strong Approximations- 13 Selected Applications of Strong Approximations- 14 Weak Taylor Approximations- 15 Explicit and Implicit Weak Approximations- 16 Variance Reduction Methods- 17 Selected Applications of Weak Approximations- Solutions of Exercises- Bibliographical Notes

...read moreread less

6,284 citations

"Local Lipschitz continuity in the i..." refers background in this paper

...2 in Kloeden & Platen [26] for details)....
[...]

Journal Article•DOI•

Singular integrals and differentiability properties of functions

[...]

J. L. B. Cooper

01 Mar 1973-Bulletin of The London Mathematical Society

5,201 citations

"Local Lipschitz continuity in the i..." refers background in this paper

...1] can be obtained from an extension result that can be found in the book of Stein [47]....
[...]
...The Kolmogorov-Chentsov theorem as provided in [35, Theorem 2.1] can be obtained from an extension result that can be found in the book of Stein [47]....
[...]