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Stochastic integral equations without probability. (English) Zbl 0963.60060
L. C. Young has constructed an extension of the Riemann-Stieltjes integral \(\int_a^b f dh\), where the functions \(f,h:[a,b]\to{\mathbb R}\) have finite \(p\)-variation and finite \(q\)-variation, respectively, with \(p^{-1}+q^{-1}>1\). Motivated by the fact that certain stochastic processes which are not semimartingales may have a bounded \(p\)-variation for some \(p<2\) (like e.g. a large family of Lévy processes, fractional Brownian motions, …), the authors investigate some integral equations such as \[ F(y)=F(a)+\int_{a}^{y}F df+g(y)-g(a)\quad \text{and}\quad u(t) =u(0)-\beta\int_{0}^{t}u(s) ds+g(t) -g(0) , \] where the functions \(f,g\) have bounded \(p\)-variation for some \(0<p<2\). Results on the existence and uniqueness of the solution are given.

60H20 Stochastic integral equations
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