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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.

##### MSC:
 60H20 Stochastic integral equations
##### Keywords:
$$p$$-variation; chain rule; stochastic integral equation
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