On One-Dimensional Stochastic Equations Driven by Symmetric Stable Processes

2002 ◽  
pp. 89-118
Keyword(s):  
Stochastic Equations ◽  
Stable Processes ◽  
Symmetric Stable Processes ◽  
One Dimensional

The Time Change Method and SDEs with Nonnegative Drift

2010 ◽  
Vol 53 (3) ◽  
pp. 503-515
Author(s):  
V. P. Kurenok
Keyword(s):  
Stochastic Equation ◽  
Existence Of Solutions ◽  
Stochastic Equations ◽  
Time Change ◽  
Stable Processes ◽  
Existence Proof ◽  
Symmetric Stable Processes ◽  
Measurable Coefficients

AbstractUsing 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.

scholarly journals Regularity of solutions to anisotropic nonlocal equations

2020 ◽  
Vol 296 (3-4) ◽  
pp. 1135-1155 ◽  
Author(s):  
Jamil Chaker
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Harmonic Functions ◽  
Regularity Of Solutions ◽  
Hölder Regularity ◽  
Stable Processes ◽  
Symmetric Stable Processes ◽  
One Dimensional ◽  
Nonlocal Equations ◽  
Bounded Harmonic Functions

Abstract We study harmonic functions associated to systems of stochastic differential equations of the form $$dX_t^i=A_{i1}(X_{t-})dZ_t^1+\cdots +A_{id}(X_{t-})dZ_t^d$$ d X t i = A i 1 ( X t - ) d Z t 1 + ⋯ + A id ( X t - ) d Z t d , $$i\in \{1,\dots ,d\}$$ i ∈ { 1 , ⋯ , d } , where $$Z_t^j$$ Z t j are independent one-dimensional symmetric stable processes of order $$\alpha _j\in (0,2)$$ α j ∈ ( 0 , 2 ) , $$j\in \{1,\dots ,d\}$$ j ∈ { 1 , ⋯ , d } . In this article we prove Hölder regularity of bounded harmonic functions with respect to solutions to such systems.

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