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On driftless one-dimensional SDEs with time-dependent diffusion coefficients. (English) Zbl 0946.60061

The one-dimensional stochastic differential equation \(X_t=x_0+ \int^t_0 b(s,X_s) dB_s\) is considered where \(B\) is a one-dimensional Brownian motion and \(b:R_+\times R\to R\) is a measurable, locally square integrable function such that \[ \{x\in R_+\times R:\int_Ub^{-2} (z)dz= +\infty,\text{ for all open }U\text{ with }x\in U\}\subset \{x\in R_+\times R:b(x)=0\}. \] It is shown that there exists a weak solution (possibly exploding) which has the representation property. The author’s results improve those by T. Senf [Stochastics Stochastics Rep. 43, No. 3/4, 199-220 (1993; Zbl 0786.60077)] and by A. Rozkosz and L. Słomiński [ibid. 42, No. 3/4, 199-208 (1993; Zbl 0814.60051)]. The time change method is applied here as by the preceding authors. Apart from degenerate diffusion, the main difference is that monotone approximation for the solution of the associated time changed equation, in contrast to weak convergence, is systematically exploited.
Reviewer: R.Buckdahn (Brest)

MSC:

60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H05 Stochastic integrals
34F05 Ordinary differential equations and systems with randomness
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