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For an interpolation pair $(E_0,E_1)$ of Banach spaces with $E_1 \hookrightarrow E_0$ we use vectors $b_1,b_2,\ldots \in E_1$ that satisfy an extremal property with respect to the $J$- and $K$-functional to construct sub-spaces that are isometric to $\ell_q^{(\theta)}$. The construction is based on a randomisation using independent Rademacher variables. We verify that systems obtained by re-scaling a function with a certain periodicity property share this extreme property. This implies the sharpness of natural embeddings of H\"older spaces obtained by the real interpolation into the corresponding Gaussian Besov spaces.

We discuss in a stochastic framework the interplay between Riemann-Liouville type operators applied to stochastic processes, real interpolation, bounded mean oscillation, and an approximation problem for stochastic integrals. We provide upper and lower bounds for gradient processes on the L\'evy-It\^o space, which arise in the special case of the Wiener space from the Feynman-Kac theory for parabolic PDEs. The upper bounds are formulated in terms of BMO-conditions on the fractional integrated gradient, the lower bounds in terms of oscillatory quantities. On the general L\'evy-It\^o space we are concerned with gradient processes with values in a Hilbert space, where the regularity depends on the direction in this Hilbert space. We discuss two applications of our techniques: on the Wiener space an approximation problem for H\"older functionals and on the L\'evy-It\^o space an orthogonal decomposition of H\"older functionals into a sum of stochastic integrals with a control of the corresponding integrands.

In this paper, we study in the Markovian case the rate of convergence in the Wasserstein distance of an approximation of the solution to a BSDE given by a BSDE which is driven by a scaled random walk as introduced in Briand, Delyon and M{\'e}min (Electron. Comm. Probab. 6(2001),1-14).

We consider decoupling inequalities for random variables taking values in a Banach space $X$. We restrict the class of distributions that appear as conditional distributions while decoupling and show that each adapted process can be approximated by a Haar type expansion in which only the same conditional distributions appear. Moreover, we show that in our framework a progressive enlargement of the underlying filtration does not effect the decoupling properties (e.g., the constants involved). As special case we deal with one-sided moment inequalities when decoupling dyadic (i.e., Paley-Walsh) martingales. We establish the decoupling constant of $\mathbb{R}^d$ with the $l^{\infty}$-norm. As an example of an application, we demonstrate that Burkholder-Davis-Gundy type inequalities for stochastic integrals of $X$-valued processes can be obtained from decoupling inequalities for $X$-valued dyadic martingales.

We deduce conditional $L_p$-estimates for the variation of a solution of a BSDE. Both quadratic and sub-quadratic types of BSDEs are considered, and using the theory of weighted bounded mean oscillation we deduce new tail estimates for the solution $(Y,Z)$ on subintervals of $[0,T]$. Some new results for the decoupling technique introduced in \cite{jossain} are obtained as well and some applications of the tail estimates are given.

We introduce a decoupling method on the Wiener space to define a wide class of an\-iso\-tro\-pic Besov spaces. The decoupling method is based on a general distributional approach and not restricted to the Wiener space. The class of Besov spaces we introduce contains the traditional isotropic Besov spaces obtained by the real interpolation method, but also new spaces that are designed to investigate backwards stochastic differential equations (BSDEs). As examples we discuss the Besov regularity (in the sense of our spaces) of forward diffusions and local times. It is shown that among our newly introduced Besov spaces there are spaces that characterize quantitative properties of directional derivatives in the Malliavin sense without computing or accessing these Malliavin derivatives explicitly. Regarding BSDEs, we deduce regularity properties of the solution processes from the Besov regularity of the initial data, in particular upper bounds for their $L_p$-variation, where the generator might be of quadratic type and where no structural assumptions, for example in terms of a forward diffusion, are assumed. As an example we treat sub-quadratic BSDEs with unbounded terminal conditions. Among other tools, we use methods from harmonic analysis. As a by-product, we improve the asymptotic behaviour of the multiplicative constant in a generalized Fefferman inequality and verify the optimality of the bound we established.

We establish general moment estimates for the discrete and continuous exit times of a general It\^o process in terms of the distance to the boundary. These estimates serve as intermediate steps to obtain strong convergence results for the approximation of a continuous exit time by a discrete counterpart, computed on a grid. In particular, we prove that the discrete exit time of the Euler scheme of a diffusion converges in the L1 norm with an order 1/2 with respect to the mesh size.

Let $v:[0,T]\times \R^d \to \R$ be the solution of the parabolic backward equation $ \partial_t v + (1/2) \sum_{i,l} [\sigma \sigma^\perp]_{il} \partial_{x_i \partial_{x_l} v + \sum_{i} b_i \partial_{x_i}v + kv =0$ with terminal condition $g$, where the coefficients are time- and state-dependent, and satisfy certain regularity assumptions. Let $X=(X_t)_{t\in [0,T]}$ be the associated $\R^d$-valued diffusion process on some appropriate $(\Omega,\cF,\Q)$. For $p\in [2,\infty)$ and a measure $d\P=\lambda_T d\Q$, where $\lambda_T$ satisfies the Muckenhoupt condition $A_\alpha$ for $\alpha \in (1,p)$, we relate the behavior of $\|g(X_T)-\ept g(X_T) \|_{L_p(\P)}$, $\|\nabla v(t,X_t) \|_{L_p(\P)}$ and $\|D^2 v(t,X_t) \|_{L_p(\P)}$ to each other, where $D^2v:=(\partial_{x_i \partial_{x_l}v)_{i,l}$ is the Hessian matrix.

We consider Gaussian Besov spaces obtained by real interpolation and Riemann-Liouville operators of fractional integration on the Gaussian space and relate the fractional smoothness of a functional to the regularity of its heat extension. The results are applied to study an approximation problem in $L_p$ for $2\le p<\infty$ for stochastic integrals with respect to the $d$-dimensional (geometric) Brownian motion., Comment: Published in at http://dx.doi.org/10.1214/13-AOP884 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Assume a L\'evy process $X$ on the time interval $[0,1]$ that is an $L_2$-martingale and let $Y$ be either its stochastic exponential or $X$ itself. We consider Riemann-approximations of certain stochastic integrals driven by $Y$ and relate the $L_2$-approximation rates to the Malliavin fractional smoothness of the integral to be approximated. The Malliavin fractional smoothness is described by Besov spaces generated with the real interpolation method.

We relate the $L_p$-variation, $2\le p < \infty$, of a solution of a backward stochastic differential equation with a path-dependent terminal condition to a generalized notion of fractional smoothness. This concept of fractional smoothness takes into account the quantitative propagation of singularities in time.

This overview article concerns the notion of fractional smoothness of random variables of the form $g(X_T)$, where $X=(X_t)_{t\in [0,T]}$ is a certain diffusion process. We review the connection to the real interpolation theory, give examples and applications of this concept. The applications in stochastic finance mainly concern the analysis of discrete time hedging errors. We close the review by indicating some further developments., Comment: Chapter of AMAMEF book. 20 pages.

We determine the sub-collections of the dyadic intervals that are able to detect the Haar type of a Banach space. The underlying dichotomy is expressed in terms of the Carleson packing condition.

We study the extrapolation properties of vector valued rearrangement operators acting on the normalized Haar basis in $L^p_X .$

We consider weak convergence of the rescaled error processes arising from Riemann discretizations of certain stochastic integrals and relate the $L_p$-integrability of the weak limit to the fractional smoothness in the Malliavin sense of the stochastic integral., Comment: Published in at http://dx.doi.org/10.3150/09-BEJ197 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Let $\on_{k \in \nz}$ be an orthonormal system on some $\sigma$-finite measure space $(\Om,p)$. We study the notion of cotype with respect to $\Phi$ for an operator $T$ between two Banach spaces $X$ and $Y$, defined by $\fco T := \inf$ $c$ such that \[ \Tfmm \pl \le \pl c \pll \gmm \hspace{.7cm}\mbox{for all}\hspace{.7cm} (x_k)\subset X \pl,\] where $(g_k)_{k\in \nz}$ is a sequence of independent and normalized gaussian variables. It is shown that this $\Phi$-cotype coincides with the usual notion of cotype $2$ iff \linebreak $\fco {I_{\lin}} \sim \sqrt{\frac{n}{\log (n+1)}}$ uniformly in $n$ iff there is a positive $\eta>0$ such that for all $n \in \nz$ one can find an orthonormal $\Psi = (\psi_l)_1^n \subset {\rm span}\{ \phi_k \p|\p k \in \nz\}$ and a sequence of disjoint measurable sets $(A_l)_1^n \subset \Om$ with \[ \int\limits_{A_l} \bet \psi_l\rag^2 d p \pl \ge \pl \eta \quad \mbox{for all}\quad l=1,...,n \pl. \] A similar result holds for the type situation. The study of type and cotype with respect to orthonormal systems of a given length provides the appropriate approach to this result. We intend to give a quite complete picture for orthonormal systems in measure space with few atoms.

For a Banach space X we define RUMD_n(X) to be the infimum of all c>0 such that (AVE_{\epsilon_k =\pm 1} || \sum_1^n epsilon_k (M_k - M_{k-1} )||_{L_2^X}^2 )^{1/2} <= c || M_n ||_{L_2^X} holds for all Walsh-Paley martingales {M_k}_0^n subset L_2^X with M_0 =0. We relate the asymptotic behaviour of the sequence {RUMD(X)}_{n=1}^{infinity} to geometrical properties of the Banach space X such as K-convexity and superreflexivity.