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The existence and uniqueness of the solution of an integral equation driven by a \(p\)-semimartingale of special type. (English) Zbl 1059.60068

Existence and uniqueness of adapted solution with almost all sample paths in the space of continuous functions with a bounded \(q\)-variation is proved for the equation \[ X_t = \xi + \int _0^t f(X_s)\,dW_s + \int _0^t g(X_s)\,dB^H_s,\quad 0\leq t\leq T, \] where \(W\) and \(B^H\) denote the standard Brownian motion and the fractional Brownian motion with the Hurst parameter \(H\in (1/2,1)\), respectively, \(f\) is a Lipschitz function on \(\mathbb R\) and \(g\in C^{1+\alpha }(\mathbb R)\), if \(0<\alpha <1,\;1/H<p<2\) are such that \(\alpha /q + 1/p >1\).

MSC:

60H05 Stochastic integrals
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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