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# Stochastic Integration with Respect to fBm and Related Topics

Chapter
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
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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