On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes

Engelbert, H. J.; Kurenok, V. P.; Zălinescu, Adrian

doi:10.1007/978-3-540-30788-4_11

Cited by 7 publications

(5 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…So we have for α ∈ [1, 2), u 0 = ∞ a.s., in other words, for any t > 0, the original height process Y can reach level ye t with probability 1. When α ∈ (0, 1), the proof of [7,Theorem 4.9] illustrates that P z (γ t = ∞) > 0 for every t > 0. This proves the lemma.…”

Section: Derivative Estimatesmentioning

confidence: 99%

Schramm–Loewner Equations Driven by Symmetric Stable Processes

Chen

Rohde

2008

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

Section: Derivative Estimatesmentioning

confidence: 99%

Schramm–Loewner Equations Driven by Symmetric Stable Processes

Chen

Rohde

2008

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…Using simple time change arguments (cf. [5]), we can conclude that there exists a symmetric stable process Z defined on the same probability space such that…”

Section: The Time Change Methodsmentioning

confidence: 92%

“…After the time change t → T −1 t in (1.2), we obtain L t =Z T −1 t + t. Once again, by standard time change arguments (see [5]), there is a symmetric stable processZ of the same index α such thatZ…”

Section: Proposition 32 In Addition To the Assumptions Of Propositionmentioning

confidence: 89%

The Time Change Method and SDEs with Nonnegative Drift

Kurenok¹

2010

Can. math. bull.

View full text Add to dashboard Cite

Using the time change method we show how to construct a solution to the stochastic equation dXt = b(Xt–)dZt + a(Xt )dt with a nonnegative drift a provided there exists a solution to the auxililary equation dLt = [a–1/αb](Lt–) + dt where Z, ¯ are two symmetric stable processes of the same index α ∈ (0, 2]. This approach allows us to prove the existence of solutions for both stochastic equations for the values 0 < α < 1 and only measurable coefficients a and b satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.

show abstract

“…The corresponding integral estimates were proven by H. Pragarauskas in [8]. Some other sufficient existence conditions for the time-dependent equation without drift different from those in [9] and with 0 < α < 2 were found in [4].…”

Section: Introductionmentioning

confidence: 92%

On solutions of equations with measurable coefficients driven by α- stable processes

Kurenok

2019

Stochastics

View full text Add to dashboard Cite

We prove the existence of solutions for the stochastic differential equation dX t = b(t, X t− )dZ t +a(t, X t )dt, X 0 ∈ IR, t ≥ 0, with only measurable coefficients a and b satisfying the condition 0 < µ ≤ |b(t, x)| ≤ ν and |a(t, x)| ≤ K for all t ≥ 0, x ∈ IR where µ, ν, and K are some constants. The driving process Z is a symmetric stable process of index 1 < α < 2. This generalizes the result of N. V. Krylov [5] for the case of α = 2, that is when Z is a Brownian motion. The proof is based on integral estimates of Krylov type for the given equation which are also derived in the note and are of independent interest. Moreover, unlike in [5], we use a different approach to derive the corresponding integral estimates.

show abstract

On Existence and Uniqueness of Reflected Solutions of Stochastic Equations Driven by Symmetric Stable Processes

Cited by 7 publications

References 14 publications

Schramm–Loewner Equations Driven by Symmetric Stable Processes

Schramm–Loewner Equations Driven by Symmetric Stable Processes

The Time Change Method and SDEs with Nonnegative Drift

On solutions of equations with measurable coefficients driven by α- stable processes

Contact Info

Product

Resources

About