Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

Cox, Sonja; Hutzenthaler, Martin; Jentzen, Arnulf; Neerven, Jan van; Welti, Timo

doi:10.1093/imanum/drz063

Cited by 17 publications

(26 citation statements)

References 34 publications

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“…Proof. By uniform continuity of F on [0, T ], the sequence (F n ) converges uniformly to F for d H , and F n dH F dH for all n. Furthermore, we have N α,T (F n ) N α,T (F ) for all n, see [9].…”

Section: Proposition 317 (Continuity Of the Indefinite Aumannmentioning

confidence: 91%

On a Set-Valued Young Integral with Applications to Differential Inclusions

Coutin,

Marie,

de Fitte

2021

Preprint

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We present a new Aumann-like integral for a Hölder multifunction with respect to a Hölder signal, based on the Young integral of a particular set of Hölder selections. This restricted Aumann integral has continuity properties that allow for numerical approximation as well as an existence theorem for an abstract stochastic differential inclusion. This is applied to concrete examples of first order and second order stochastic differential inclusions directed by fractional Brownian motion.

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Section: Proposition 317 (Continuity Of the Indefinite Aumannmentioning

confidence: 91%

On a Set-Valued Young Integral with Applications to Differential Inclusions

Coutin,

Marie,

de Fitte

2021

Preprint

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“…In particular, under suitable assumptions, item (i) in Lemma 3.1 proves that the SFPE in (32) below has a unique solution within the set of functions which grow at most like a certain (32)…”

Section: Error Analysis For Multi-grid Approximationsmentioning

confidence: 99%

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

et al. 2020

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For a long time it has been well-known that high-dimensional linear parabolic partial differential equations (PDEs) can be approximated by Monte Carlo methods with a computational effort which grows polynomially both in the dimension and in the reciprocal of the prescribed accuracy. In other words, linear PDEs do not suffer from the curse of dimensionality. For general semilinear PDEs with Lipschitz coefficients, however, it remained an open question whether these suffer from the curse of dimensionality. In this paper we partially solve this open problem. More precisely, we prove in the case of semilinear heat equations with gradient-independent and globally Lipschitz continuous nonlinearities that the computational effort of a variant of the recently introduced multilevel Picard approximations grows at most polynomially both in the dimension and in the reciprocal of the required accuracy.

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“…In the latter reference, convergence rates of order n −ν are obtained under the assumption that the solution u belong to C ν ([0, T ]; L 2 ( ; V )) ∩ L 2 ( ; L ∞ (0, T ; V )). Results on uniform convergence in time (and sometimes even convergence in Hölder norms in time) for schemes involving space and time discretisation can be found in many papers, including [14][15][16]36,39,53,80,109]. Results concerning uniform convergence in case of white noise and discretisation in time only can be found in [3,4,40,41].…”

Section: Applications To Spdementioning

confidence: 99%

Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

Neerven

Veraar

2021

Stoch PDE: Anal Comp

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We prove a new Burkholder–Rosenthal type inequality for discrete-time processes taking values in a 2-smooth Banach space. As a first application we prove that if $$(S(t,s))_{0\leqslant s\le t\leqslant T}$$ ( S ( t , s ) ) 0 ⩽ s ≤ t ⩽ T is a $$C_0$$ C 0 -evolution family of contractions on a 2-smooth Banach space X and $$(W_t)_{t\in [0,T]}$$ ( W t ) t ∈ [ 0 , T ] is a cylindrical Brownian motion on a probability space $$(\Omega ,{\mathbb {P}})$$ ( Ω , P ) adapted to some given filtration, then for every $$0<p<\infty $$ 0 < p < ∞ there exists a constant $$C_{p,X}$$ C p , X such that for all progressively measurable processes $$g: [0,T]\times \Omega \rightarrow X$$ g : [ 0 , T ] × Ω → X the process $$(\int _0^t S(t,s)g_s\,\mathrm{d} W_s)_{t\in [0,T]}$$ ( ∫ 0 t S ( t , s ) g s d W s ) t ∈ [ 0 , T ] has a continuous modification and $$\begin{aligned} {\mathbb {E}}\sup _{t\in [0,T]}\Big \Vert \int _0^t S(t,s)g_s\,\mathrm{d} W_s \Big \Vert ^p\leqslant C_{p,X}^p {\mathbb {E}} \Bigl (\int _0^T \Vert g_t\Vert ^2_{\gamma (H,X)}\,\mathrm{d} t\Bigr )^{p/2}. \end{aligned}$$ E sup t ∈ [ 0 , T ] ‖ ∫ 0 t S ( t , s ) g s d W s ‖ p ⩽ C p , X p E ( ∫ 0 T ‖ g t ‖ γ ( H , X ) 2 d t ) p / 2 . Moreover, for $$2\leqslant p<\infty $$ 2 ⩽ p < ∞ one may take $$C_{p,X} = 10 D \sqrt{p},$$ C p , X = 10 D p , where D is the constant in the definition of 2-smoothness for X. The order $$O(\sqrt{p})$$ O ( p ) coincides with that of Burkholder’s inequality and is therefore optimal as $$p\rightarrow \infty $$ p → ∞ . Our result improves and unifies several existing maximal estimates and is even new in case X is a Hilbert space. Similar results are obtained if the driving martingale $$g_t\,\mathrm{d} W_t$$ g t d W t is replaced by more general X-valued martingales $$\,\mathrm{d} M_t$$ d M t . Moreover, our methods allow for random evolution systems, a setting which appears to be completely new as far as maximal inequalities are concerned. As a second application, for a large class of time discretisation schemes (including splitting, implicit Euler, Crank-Nicholson, and other rational schemes) we obtain stability and pathwise uniform convergence of time discretisation schemes for solutions of linear SPDEs $$\begin{aligned} \,\mathrm{d} u_t = A(t)u_t\,\mathrm{d} t + g_t\,\mathrm{d} W_t, \quad u_0 = 0, \end{aligned}$$ d u t = A ( t ) u t d t + g t d W t , u 0 = 0 , where the family $$(A(t))_{t\in [0,T]}$$ ( A ( t ) ) t ∈ [ 0 , T ] is assumed to generate a $$C_0$$ C 0 -evolution family $$(S(t,s))_{0\leqslant s\leqslant t\leqslant T}$$ ( S ( t , s ) ) 0 ⩽ s ⩽ t ⩽ T of contractions on a 2-smooth Banach spaces X. Under spatial smoothness assumptions on the inhomogeneity g, contractivity is not needed and explicit decay rates are obtained. In the parabolic setting this sharpens several know estimates in the literature; beyond the parabolic setting this seems to provide the first systematic approach to pathwise uniform convergence to time discretisation schemes.

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Convergence in Hölder norms with applications to Monte Carlo methods in infinite dimensions

Cited by 17 publications

References 34 publications

On a Set-Valued Young Integral with Applications to Differential Inclusions

On a Set-Valued Young Integral with Applications to Differential Inclusions

Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

Maximal inequalities for stochastic convolutions and pathwise uniform convergence of time discretisation schemes

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