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Stochastic Integration with Respect to fBm and Related Topics

Part of the Lecture Notes in Mathematics book series (LNM, volume 1929)

Pathwise Stochastic Integration in the Fractional Sobolev-type Spaces

In this subsection we consider pathwise integrals \(\int\limits_0^T {f\left( t \right)dB_t^H } \) for processes f from the fractional Sobolev type spaces \(I_{a + }^\alpha \left( {L^p } \right)\) for some p > 1. This approach was developed by Zähle (Zah98), (Zah99), (Zah01).

Consider two nonrandom functions f and g defined on some interval \(\left[ {a,b} \right] \subset {\rm R}\) and suppose that the limits \(f\left( {u + } \right): = \lim _{\delta \downarrow 0} f\left( {u + \delta } \right)\) and \(g\left( {u - } \right): = \lim _{\delta \downarrow 0} g\left( {u - \delta } \right),a \le u \le b,\) exist. Put \(f_{a + } \left( x \right): = \left( {f\left( x \right) - f\left( {a + } \right)} \right)1_{\left( {a,b} \right)} \left( x \right),g_{b - } \left( x \right): = \left( {g\left( {b - } \right) - g\left( x \right)} \right)1_{\left( {a,b} \right)} \left( x \right).\) Suppose also that \(f_{a + } \in I_{a + }^\alpha \left( {L_p \left[ {a,b} \right]} \right),g_{b - } \in I_{b - }^{1 - \alpha } \left( {L_p \left[ {a,b} \right]} \right)\) for some \(p \ge 1,q \ge 1,1/p + 1/q \le 1,0 \le \alpha \le 1.\) Then evidently, \(D_{a + }^\alpha f_{a + } \in L_p \left[ {a,b} \right],D_{b - }^{1 - \alpha } g_{b - } \in L_q \left[ {a,b} \right].\)

Keywords

Related Topic Fractional Brownian Motion Standard Brownian Motion Stochastic Integration Predictable Process 
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