On the hedging of American options in discrete time markets with proportional transaction costs.
On the infinite time horizon linear-quadratic regulator problem under a fractional brownian perturbation
In this paper we solve the basic fractional analogue of the classical infinite time horizon linear-quadratic gaussian regulator problem. For a completely observable controlled linear system driven by a fractional brownian motion, we describe explicitely the optimal control policy which minimizes an asymptotic quadratic performance criterion.
On the infinite time horizon linear-quadratic regulator problem under a fractional Brownian perturbation
In this paper we solve the basic fractional analogue of the classical infinite time horizon linear-quadratic Gaussian regulator problem. For a completely observable controlled linear system driven by a fractional Brownian motion, we describe explicitely the optimal control policy which minimizes an asymptotic quadratic performance criterion.
On the innovations conjecture of nonlinear filtering with dependent data.
On the integrability of Banach space valued Walsh polynomials
On the integrability of the ergodic maximal function
On the invariant measure of the random difference equation Xn = AnXn−1 + Bn in the critical case
We consider the autoregressive model on ℝd defined by the stochastic recursion Xn = AnXn−1 + Bn, where {(Bn, An)} are i.i.d. random variables valued in ℝd× ℝ+. The critical case, when , was studied by Babillot, Bougerol and Elie, who proved that there exists a unique invariant Radon measureν for the Markov chain {Xn}. In the present paper we prove that the weak limit of properly dilated measure ν exists and defines a homogeneous measure on ℝd ∖ {0}.
On the Kaczmarz algorithm of approximation in infinite-dimensional spaces
The Kaczmarz algorithm of successive projections suggests the following concept. A sequence of unit vectors in a Hilbert space is said to be effective if for each vector x in the space the sequence (xₙ) converges to x where (xₙ) is defined inductively: x₀ = 0 and , where . We prove the effectivity of some sequences in Hilbert spaces. We generalize the concept of effectivity to sequences of vectors in Banach spaces and we prove some results for this more general concept.
On the Karhunen-Loeve expansion for transformed processes.
We discuss the influence of the transformation {X(t)} → {f(t) X(τ(t))} on the Karhunen-Loève expansion of {X(t)}. Our main result is that, in general, the Karhunen-Loève expansion of {X(t)} with respect to Lebesgue's measure is transformed in the Karhunen-Loève expansion of {f(t) X(τ(t))} with respect to the measure f-2(t)dτ(t). Applications of this result are given in the case of Wiener process, Brownian bridge, and Ornstein-Uhlenbeck process.
On the law of large numbers for continuous-time martingales and applications to statistics.
In order to develop a general criterion for proving strong consistency of estimators in Statistics of stochastic processes, we study an extension, to the continuous-time case, of the strong law of large numbers for discrete time square integrable martingales (e.g. Neveu, 1965, 1972). Applications to estimation in diffusion models are given.
On the law of the iterated logarithm for increments of sums of independent random variables.
On the level crossing of multi-dimensional delayed renewal processes.
On the Lévy transformation of brownian motions and continuous martingales
On the limit behaviour of sums of a random number of independent random variables
On the limit distribution of the well-distribution measure of random binary sequences
We prove the existence of a limit distribution of the normalized well-distribution measure (as ) for random binary sequences , by this means solving a problem posed by Alon, Kohayakawa, Mauduit, Moreira and Rödl.
On the limit distributions of kth order statistics for semi-pareto processes
Asymptotic properties of the kth largest values for semi-Pareto processes are investigated. Conditions for convergence in distribution of the kth largest values are given. The obtained limit laws are represented in terms of a compound Poisson distribution.
On the limiting velocity of random walks in mixing random environment
We consider random walks in strong-mixing random Gibbsian environments in , . Based on regeneration arguments, we will first provide an alternative proof of Rassoul-Agha’s conditional law of large numbers (CLLN) for mixing environment (Electron. Commun. Probab.10(2005) 36–44). Then, using coupling techniques, we show that there is at most one nonzero limiting velocity in high dimensions ().
On the linear prediction problem of certain non-stationary stochastic processes.
On the local time of sub-fractional Brownian motion
be a sub-fractional Brownian motion with . We establish the existence, the joint continuity and the Hölder regularity of the local time of . We will also give Chung’s form of the law of iterated logarithm for . This results are obtained with the decomposition of the sub-fractional Brownian motion into the sum of fractional Brownian motion plus a stochastic process with absolutely continuous trajectories. This decomposition is given by Ruiz de Chavez and Tudor [10].
On the lower bound for the number of real roots of a random algebraic equation.