Loss of regularity for Kolmogorov equations

By Martin Hairer, Martin Hutzenthaler, Arnulf Jentzen. 2012

101 citations recorded by the source corpus.

The celebrated Hormander condition is a sucient (and nearly necessary) condition for a second-order linear Kolmogorov partial dierential equation (PDE) with smooth coecients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coecients of the PDE are smooth and satisfy Hormander's condition even if the initial function is only continuous but not dierentiable. First-order linear Kolmogorov PDEs with smooth coecients do not have this smoothing eect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth. In this article, we consider the intermediate regime of non-hypoelliptic second-order Kolmogorov PDEs with smooth coecients. The main observation of this article is that there exist counterexamples to regularity preservation in that case. More precisely, we give an example of a second-order linear Kolmogorov PDE with globally bounded and smooth coecients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally Holder continuous and, thereby, we disprove the existence of globally bounded classical solutions of this PDE. From the perspective of probability theory, this observation has the consequence that there exists a stochastic dierential equation (SDE) with globally bounded and smooth coecients and a smooth function with compact support which is mapped by the transition semigroup of the SDE to a non-locally Holder continuous function. In other words, degenerate noise can have a roughening eect. A further implication of this loss of regularity phenomenon is that numerical approximations may convergence slower than any arbitrarily small polynomial rate of convergence to the true solution of the SDE. More precisely, we prove for an example SDE with globally bounded and smooth coecients that the standard Euler approximations converge to the exact solution of the SDE in the strong and numerically weak sense slower than any arbitrarily small polynomial rate of convergence.

Subfields: Control and Systems Engineering, Finance, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/08152a2862be9bb1f914029835d0ee7bea850331

How researchers see this paper

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

In Control and Systems Engineering

1. Sonja G. Cox
2. Martin Hutzenthaler
3. Arnulf Jentzen
4. Peter E. Kloeden

In Finance

1. Larisa Yaroslavtseva
2. Arnulf Jentzen
3. Martin Hutzenthaler
4. Thomas Muller-Gronbach
5. Sonja G. Cox
6. Xuerong Mao
7. Sotirios Sabanis
8. Peter E. Kloeden

In Modeling and Simulation

1. Arnulf Jentzen

Citations in the corpus

• On the performance of the Euler–Maruyama scheme for SDEs with discontinuous drift coefficient (2018, background): “Note that the recent literature on numerical approximation of SDEs contains a number of examples of SDEs with coefficients that are not Lipschitz continuous and such that the EulerMaruyama scheme at the final time does not achieve an Lp-error rate of 1/2, see [2, 6, 9, 11, 12, 22, 29].”
• Local Lipschitz Continuity in the Initial Value and Strong Completeness for Nonlinear Stochastic Differential Equations (2013, background): “[14] implies that this loss of regularity phenomenon can happen even in the case of globally bounded and smooth coefficients.”