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2017 On uniqueness and blowup properties for a class of second order SDEs
Alejandro Gomez, Jong Jun Lee, Carl Mueller, Eyal Neuman, Michael Salins
Electron. J. Probab. 22: 1-17 (2017). DOI: 10.1214/17-EJP95

Abstract

As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution $(X_t,Y_t)$ of the equations $dX_t= Y_tdt$, $dY_t = |X_t|^\alpha dB_t$, $(X_0,Y_0)=(x_0,y_0)$. In particular, we prove that solutions are nonunique if $0<\alpha <1$ and $(x_0,y_0)=(0,0)$ and unique if $1/2<\alpha $ and $(x_0,y_0)\neq (0,0)$. We also show that blowup in finite time holds if $\alpha >1$ and $(x_0,y_0)\neq (0,0)$.

Citation

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Alejandro Gomez. Jong Jun Lee. Carl Mueller. Eyal Neuman. Michael Salins. "On uniqueness and blowup properties for a class of second order SDEs." Electron. J. Probab. 22 1 - 17, 2017. https://doi.org/10.1214/17-EJP95

Information

Received: 7 March 2017; Accepted: 16 August 2017; Published: 2017
First available in Project Euclid: 13 September 2017

zbMATH: 06797882
MathSciNet: MR3698741
Digital Object Identifier: 10.1214/17-EJP95

Subjects:
Primary: 60H10
Secondary: 60H15

JOURNAL ARTICLE
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