Extension of the method of quasilinearization for stochastic initial value problems
Consider the stochastic initial value problem \[ u'(t;\omega) = f(t,u (t;\omega); \omega) + g(t,u(t;\omega); \omega) \text{ a.e. for }t \in [0,T], \quad u(0;\omega) = u_ 0(\omega),\tag{\(*\)} \] where \(f : [0,T] \times \mathbb{R} \times \Omega \to \mathbb{R}\) and \(g : [0,T] \times \mathbb{R} \times \Omega \to \mathbb{R}\) \(((\Omega, {\mathcal A}, P)\) be a probability measure space) satisfy: (i) \(f(t, u; \cdot)\) and \(g(t, u; \cdot)\) are measurable for all \((t,u)\), (ii) \(f(\cdot, u; \cdot)\) and \(g(\cdot, u; \cdot)\) are measurable for every \(u\), (iii) \(f(t, \cdot; \omega)\) and \(g(t, \cdot; \omega)\) are continuous for all \((t;\omega)\) and (iv) \(| f(t,x;\omega)| \leq K(t;\omega)\) on \([0,T] \times \mathbb{R} \times \Omega\), where \(K : [0,T] \times \Omega \to \mathbb{R}_ +\) is measurable in \(t\) and \(\int^ T_ 0 K(s; \omega) ds < \infty\) on \(\Omega\). The authors prove the existence of a unique sample solution for the initial value problem \((*)\) and that the convergence is quadratic.
scientific article; zbMATH DE number 757816