On one-dimensional stochastic differential equations without drift and with time-dependent diffusion coefficients. (English) Zbl 0786.60077

Senf, Torsten

Stochastics Stochastics Rep. 43, No. 3-4, 199-220 (1993).

The one-dimensional stochastic differential equation \[ X_ t=x_ 0+\int^ t_ 0b(s,X_ s)dB_ s \tag{1} \] is considered where \(B\) is Brownian motion and \(b:[0,\infty) \times \mathbb{R} \to \mathbb{R}\) is a measurable function. It is shown that under the conditions
(i) \(\int^ t_ 0 \int^ m_{-m}b^ 2(s,y)dyds<\infty\) for all \(t \geq 0\), \(m\in\mathbb{N},\)
(ii) \(\int^ t_ 0\int^ m_{-m}b^{-2}(s,y)dyds<\infty\) for all \(t\geq 0\), \(m\in\mathbb{N}\),
for each \(x_ 0 \in \mathbb{R}\) there exists a solution of (1) which may explode. If moreover the condition
(iii) Lebesgue measure of \(\{x \in \mathbb{R}:\sup_{t \leq m}b^ 2(t,x)<\infty\}\) for all \(m \in \mathbb{N}\)
is satisfied, then for each \(x_ 0 \in \mathbb{R}\) there exists a nonexploding solution of (1).

Reviewer: D.Jarušková (Praha)

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J65	Brownian motion
60H05	Stochastic integrals
60B10	Convergence of probability measures

Keywords:

continuous local martingales; weak convergence; stochastic differential equation; Brownian motion