"Diffusions with reflection"
Topic I: One-dimensional Reflecting SDEs
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- The Skorokhod problem. Existence and uniqueness theorem.
- SDEs with reflection. Existence and uniqueness theorem.
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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.
- Approximation by SDEs without reflection. Penalization method. (partial proofs).
- 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).
- Relation between reflecting SDEs and parabolic equations with Neumann boundary conditions (partial proofs).