A mild Itô formula for SPDEs

By Giuseppe Da Prato, Arnulf Jentzen, M. Roeckner. Transactions of the American Mathematical Society, 2010

65 citations recorded by the source corpus.

This article introduces a certain class of stochastic processes, which we suggest calling mild Itô processes, and a new, somehow mild, Itô-type formula for such processes. Examples of mild Itô processes are mild solutions of stochastic partial differential equations (SPDEs) and their numerical approximation processes. We illustrate the capacity of the mild Itô formula by several applications. In particular, we illustrate how the mild Itô formula can be used to derive improved a priori bounds for SPDEs, we demonstrate how the mild Itô formula can be employed to establish improved Hölder continuity properties for solutions of Kolmogorov partial differential equations (PDEs) in Hilbert spaces, and we illustrate how the mild Itô formula can be used to solve the weak convergence problem for numerical approximations of SPDEs with nonlinear diffusion coefficients.

Subfields: Applied Mathematics, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/021cd0e13a3dea7f90cee0f104e318293307ee11

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Arnaud Debussche
2. Arnulf Jentzen
3. Charles-Edouard Br'ehier
4. Martina Hofmanov'a
5. Martin Hairer
6. Franco Flandoli
7. Panagiotis E. Souganidis
8. Benjamin Gess

In Finance

1. Charles-Edouard Br'ehier
2. Arnaud Debussche
3. Daniel Conus
4. Arnulf Jentzen
5. Peter E. Kloeden
6. Michael Röckner
7. Martina Hofmanov'a
8. Xuerong Mao

In Mathematical Physics

1. Charles-Edouard Br'ehier
2. Arnaud Debussche
3. Arnulf Jentzen
4. Fabian Hornung

Citations in the corpus

• Weak convergence rates of spectral Galerkin approximations for SPDEs with nonlinear diffusion coefficients (2014, background): “[8] to the second summand on the right hand side of (39).”
• Kolmogorov equations and weak order analysis for SPDEs with nonlinear diffusion coefficient (2017, background): “Another approach, using the concept of mild Itô processes, see [14], [17], has been recently studied to provide weak convergence rates for SPDEs (1) with multiplicative noise, for several examples of numerical schemes: see [13], [28], [29], [30].”