We combine Stein's method with Malliavin calculus in order to obtain explicit bounds in the multidimensional normal approximation (in the Wasserstein distance) of functionals of Gaussian fields. Our results generalize and refine the main findings by Peccati and Tudor (2005), Nualart and Ortiz-Latorre (2007), Peccati (2007) and Peccati (2007b, 2008); in particular, they apply to approximations by means of Gaussian vectors with an arbitrary, positive definite covariance matrix. Among several examples, we provide an application to a functional version of the Breuer-Major CLT for fields subordinated to a fractional Brownian motion.

We derive the asymptotic behavior of weighted quadratic variations of fractional Brownian motion B with Hurst index H = 1/4. This completes the only missing case in a very recent work by I. Nourdin, D. Nualart and C. A. Tudor. Moreover, as an application, we solve a recent conjecture of K. Burdzy and J. Swanson on the asymptotic behavior of the Riemann sums with alternating signs associated to B.

We introduce a new class of Backward Stochastic Differential Equations in which the T -terminal value Y T of the solution (Y, Z) is not fixed as a random variable, but only satisfies a weak constraint of the form E[Ψ(Y T )] ≥ m, for some (possibly random) non-decreasing map Ψ and some threshold m. We name them BSDEs with weak terminal condition and obtain a representation of the minimal time t-values Y t such that (Y, Z) is a supersolution of the BSDE with weak terminal condition. It provides a non-Markovian BSDE formulation of the PDE characterization obtained for Markovian stochastic target problems under controlled loss in Bouchard, Elie and Touzi [2]. We then study the main properties of this minimal value. In particular, we analyze its continuity and convexity with respect to the m-parameter appearing in the weak terminal condition, and show how it can be related to a dual optimal control problem in Meyer form. These last properties generalize to a non Markovian framework previous results on quantile hedging and hedging under loss constraints obtained in Föllmer and Leukert [6,7], and in Bouchard, Elie and Touzi [2].

In this paper we deal with the utility maximization problem with a general utility function. We derive a new approach in which we reduce the utility maximization problem with general utility to the study of a fully-coupled Forward-Backward Stochastic Differential Equation (FBSDE).

Let B be a fractional Brownian motion with Hurst parameter H = 1/6. It is known that the symmetric Stratonovich-style Riemann sums for g(B(s)) dB(s) do not, in general, converge in probability. We show, however, that they do converge in law in the Skorohod space of càdlàg functions. Moreover, we show that the resulting stochastic integral satisfies a change of variable formula with a correction term that is an ordinary Itô integral with respect to a Brownian motion that is independent of B.as the mesh of the partition {t j } goes to zero. Typically, we regard (1.1) as a process in t, and require that it converges uniformly on compacts in probability (ucp). * Supported in part by the (french) ANR grant 'Exploration des Chemins Rugueux'. † Supported by DFG research center Matheon project E2. ‡ Supported in part by NSA grant H98230-09-1-0079.This is closely related to the so-called symmetric integral, denoted by t 0 X(s) d • Y (s), which is the ucp limit, if it exists, of2) as ε → 0. The symmetric integral is an example of the regularization procedure, introduced by Russo and Vallois, and on which there is a wide body of literature. For further details on stochastic calculus via regularization, see the excellent survey article [13] and the many references therein. A special case of interest that has received considerable attention in the literature is when Y = B H , a fractional Brownian motion with Hurst parameter H. It has been shown independently in [2] and [5] that when Y = B H and X = g(B H ) for a sufficiently differentiable function g(x), the symmetric integral exists for all H > 1/6. Moreover, in this case, the symmetric integral satisfies the classical Stratonovich change of variable formula,g(B H (t)) = g(B H (0)) + t 0 h n (x) = (−1) n e x 2 /2 d n dx n (e −x 2 /2 ). (2.1) Note that the first few Hermite polynomials are h 0 (x) = 1, h 1 (x) = x, h 2 (x) = x 2 − 1, and h 3 (x) = x 3 − 3x. The following orthogonality property is well-known: if U and V are jointly normal with E(U) = E(V ) = 0 and E(U 2 ) = E(V 2 ) = 1, then E[h p (U)h q (V )] = q!(E[UV ]) q if p = q, 0 otherwise. (2.2)