The classical Lebesgue–Stieltjes integral ∫
 b
 a
 
fdg of real or complex-valued functions on a finite interval (a,b) is extended to a large class of integrands f and integrators g of unbounded variation. The key is to use composition formulas and integration-by-part rules for fractional integrals and Weyl derivatives. In the special case of Holder continuous functions f and g of summed order greater than 1 convergence of the corresponding Riemann–Stieltjes sums is proved. The results are applied to stochastic integrals where g is replaced by the Wiener process and f by adapted as well as anticipating random functions. In the anticipating case we work within Slobodeckij spaces and introduce a stochastic integral for which the classical Ito formula remains valid. Moreover, this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional Brownian motion.

Integration with respect to fractal functions and stochastic calculus. I

Lectures on stochastic differential equations and Malliavin calculus

For every value of the Hurst index H ∈(0,1) we define a stochastic integral with respect to fractional Brownian motion of index  H . We do so by approximating fractional Brownian motion by semi-martingales. Then, for H >1/6, we establish an Ito's change of variables formula, which is more precise than Privault's Ito formula (1998) (established for every H >0), since it only involves anticipating integrals with respect to a driving Brownian motion.

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Stochastic integration with respect to fractional brownian motion

We develop a stochastic calculus for the fractional Brownian motion with Hurst parameter H > 1/2 using the techniques of the Malliavin calculus. We establish estimates in L p , maximal inequalities and a continuity criterion for the stochastic integral. Finally, we derive an Ito's formula for integral processes.

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Stochastic integration with respect to the fractional Brownian motion

In recent years fractional Brownian motion has been suggested to replace the classical Brownian motion as driving process in the modelling of many real world phenomena, including stock price modelling. In several papers seemingly contradictory results on the existence or absence of a riskless gain (arbitrage) in such stock models have been stated. This survey tries to clarify this issue by pointing to the importance of the chosen class of admissible trading strategies.

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Arbitrage with fractional Brownian motion

We define three types of non causal stochastic integrals: forward, backward and symmetric. Our approach consists in approximating the integrator. Two optics are considered: the first one is based on traditional usual stochastic calculus and the second one on Wiener distributions.

Forward, backward and symmetric stochastic integration

We introduce two classes of random variablesV such that the Brownian local time process at timeV is distributed as a 0 or 2 dimensional Bessel bridge. Moreover we obtain new decompositions of the Brownian path on the interval [0,V], which generalize Williams' results.

Une extension des théorèmes de Ray et Knight sur les temps locaux Browniens

This article develops a framework of stochastic calculus with respect to a c` adl` ag finite quadratic variation process. We apply it to the study of a generalization of a semi- martingale driven SDE studied by Kurtz, Pardoux and Protter (KPP). We prove an Itˆ o's formula for functions f( X)of a semimartingale with jumps when f has weak smoothness properties. Examples of X for which this formula is valid are time reversible semimartin- gales and solutions of (KPP) equations driven by L´ evy processes, provided the sum of the absolute values of the jumps, raised to the power 1 + λ, is a.s. finite, where λ takes values between 0 and 1.

Pierre Vallois

Papers

Forward, backward and symmetric stochastic integration

Une extension des théorèmes de Ray et Knight sur les temps locaux Browniens

o's formula for C 1,λ -functions of a c`