Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients
2. Article Structure and Notations
3. New Regularity Results
3.1. Regularity Estimate for Linear Parabolic Equations with Weight in Time Derivative Term
- is a bounded open set in , , is a (possibly nonsymmetric) matrix of functions on U that is uniformly strictly elliptic and bounded, i.e., there exist constants , , such that, for all , , it holdswith , , , and there exists , such that on U, and finally
3.2. Elliptic Hölder Regularity and Estimate
4. -Generator and Its Strong Feller Semigroup
- Operator on defined by
- generates a sub-Markovian -semigroup of contractions on .
- Let be a family of bounded open subsets of satisfying and . Then in , for all and .
- and it holds
4.3. Existence of Infinitesimally Invariant Measure and Strong Feller Properties
- is fixed, and is a symmetric matrix of functions that are locally uniformly strictly elliptic on , such that for all . is a positive function, such that and is a Borel measurable vector field on satisfying .
- with . Fix such that .
- has a locally Hölder continuous μ-versionIn particular, Equation (23) extends by linearity to all , i.e., is -strong Feller.
- has a continuous μ-versionIn particular, Equation (24) extends by linearity to all , i.e., is -strong Feller.Finally, for any ,
5.1. Weak Existence
- (A4) , where s is as in (A2).
- Condition (A4) is not necessary to get a Hunt process (and consequently a weak solution to the corresponding SDE for merely quasi-every starting point) as in the following proposition.
- Assume Conditions(A1),(A2),(A3), and . Then, for any bounded open subset V of , it holds that
- Two simple examples where Conditions(A1),(A2),(A3), and(A4)are satisfied are given as follows: for the first example, let A, ψ satisfy the assumptions of(A1), , , and ; for the second, let A, ψ satisfy the assumptions of(A1), , and . In both cases, can be chosen to be arbitrarily small.
- , for some ,
- , for some ,
- , on and for some , where so that .
5.2. Uniqueness in Law
- (A1) holds with , (A2) holds with some , is fixed, such that , and .
- is a filtered probability space, satisfying the usual conditions,
- is an -adapted continuous -valued stochastic process,
- is a standard m-dimensional -Brownian motion starting from zero,
- for the (real-valued) Borel measurable functions , , with σ is as in Theorem 8, it holds
- In Definition 1, the (real-valued) Borel measurable functions are fixed. In particular, the solution and the integrals involving the solution in Equation (28) may depend on the versions that we choose. When we fix the Borel measurable version with for all , as in Definition 1, we always consider corresponding extended Borel measurable function ψ defined byThus, the choice of the special version for ψ depends on the previously chosen Borel measurable version .
- If of Theorem 8 is nonexplosive (has infinite lifetime for any starting point), then it is a weak solution to Equation (28). Thus, a weak solution to Equation (28) exists just under Assumptions(A1),(A2),(A3), and(A4), and a suitable growth condition (cf. Remark 2) on the coefficients. For this special weak solution, we know that integrals involving the solution do not depend on the chosen Borel versions. This follows similarly to  (Lemma 3.14(i)).
- for all .
- For each and it holdswhere ψ denotes the extended Borel measurable version as explained in Remark 3(i). Moreover, Equation (5) is equivalent to Equation (29).
Conflicts of Interest
Appendix A. Proofs and Auxiliary Statements
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Lee, H.; Trutnau, G. Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients. Symmetry 2020, 12, 570. https://doi.org/10.3390/sym12040570
Lee H, Trutnau G. Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients. Symmetry. 2020; 12(4):570. https://doi.org/10.3390/sym12040570Chicago/Turabian Style
Lee, Haesung, and Gerald Trutnau. 2020. "Well-Posedness for a Class of Degenerate Itô Stochastic Differential Equations with Fully Discontinuous Coefficients" Symmetry 12, no. 4: 570. https://doi.org/10.3390/sym12040570