Mixed stochastic delay differential equations
Анотація
We consider a stochastic delay differential equation driven by a Hölder continuous process <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a Wiener process. Under fairly general assumptions on coefficients of the equation, we prove that it has a unique solution. We also give a sufficient condition for finiteness of moments of the solution and prove that the solution depends on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z"> <mml:semantics> <mml:mi>Z</mml:mi> <mml:annotation encoding="application/x-tex">Z</mml:annotation> </mml:semantics> </mml:math> </inline-formula> continuously.
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