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A note on one-dimensional stochastic equations

Hans-Jürgen Engelbert — 2001

Czechoslovak Mathematical Journal

We consider the stochastic equation X t = x 0 + 0 t b ( u , X u ) d B u , t 0 , where B is a one-dimensional Brownian motion, x 0 is the initial value, and b [ 0 , ) × is a time-dependent diffusion coefficient. While the existence of solutions is well-studied for only measurable diffusion coefficients b , beyond the homogeneous case there is no general result on the uniqueness in law of the solution. The purpose of the present note is to give conditions on b ensuring the existence as well as the uniqueness in law of the solution.

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