Humboldt-Universität zu Berlin - Faculty of Mathematics and Natural Sciences - RTG1845

Topic I: One-dimensional Reflecting SDEs

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1. The Skorokhod problem. Existence and uniqueness theorem.
2. SDEs with reflection. Existence and uniqueness theorem.
3. Skorokhod's reflection term and a local time:
   • Definition of a local time
   • Relation between Skorokhod's reflection term and a local time. Expectation of a local time (relation with a transition density).
   • Example: reflecting Brownian motion. Explicit formula. Transition density. Equality in a distribution with the absolute value of a Brownian motion.
4. Approximation by SDEs without reflection. Penalization method. (partial proofs).
5. Moments estimate. Continuous dependence on initial conditions and coefficients of the equation. Existence of continuous modification. Markov property. (Partial proofs. Gronwall’s inequality and Kolmogorov’s theorem without proof).
6. Relation between reflecting SDEs and parabolic equations with Neumann boundary conditions (partial proofs).