We consider singular SDEs like dZ_t = b(t, Z_t )dt + AZ_t dt + σ(t)dL_t , t ∈ [0, T ], Z_0 = x ∈ R^n , where A is a real n × n matrix, i.e., A ∈ R^n ⊗ R^n , b is bounded and Hölder continuous, σ : [0, ∞) → R^n ⊗ R^d is a locally bounded function and L = (L_t ) is an R^d -valued Lévy process, 1 ≤ d ≤ n. We show that strong existence and uniqueness together with L_p -Lipschitz dependence on the initial condition x imply Davie’s uniqueness or path by path uniqueness. This extends a result of [E. Priola, AIHP, 2018] proved for (1) when n = d, A = 0 and σ(t) ≡ I. We apply the result to some singular degenerate SDEs associated to the kinetic transport operator when n = 2d and L is an R^d -valued Wiener process. For such equations strong existence and uniqueness are known under Hölder type conditions on b. We show that in addition also Davie’s uniqueness holds.
On Davie's uniqueness for some degenerate SDEs
enrico priola
2020-01-01
Abstract
We consider singular SDEs like dZ_t = b(t, Z_t )dt + AZ_t dt + σ(t)dL_t , t ∈ [0, T ], Z_0 = x ∈ R^n , where A is a real n × n matrix, i.e., A ∈ R^n ⊗ R^n , b is bounded and Hölder continuous, σ : [0, ∞) → R^n ⊗ R^d is a locally bounded function and L = (L_t ) is an R^d -valued Lévy process, 1 ≤ d ≤ n. We show that strong existence and uniqueness together with L_p -Lipschitz dependence on the initial condition x imply Davie’s uniqueness or path by path uniqueness. This extends a result of [E. Priola, AIHP, 2018] proved for (1) when n = d, A = 0 and σ(t) ≡ I. We apply the result to some singular degenerate SDEs associated to the kinetic transport operator when n = 2d and L is an R^d -valued Wiener process. For such equations strong existence and uniqueness are known under Hölder type conditions on b. We show that in addition also Davie’s uniqueness holds.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
