Daniel Ocone

A generalized clark representation formula, with application to optimal portfolios

A modification of J. M. C. Clarks formula is established for the stochastic integral representation of Wiener functionals under an equivalent (Girsanov) change of probability measure. It is shown how this modified Clark formula leads to the representation of optimal portfolios fora variety of situations in the modern theory of financial economics.

Malliavin's calculus and stochastic integral representations of functional of diffusion processes †

If F is a Frechet differentiable functional on is a Brownian motion, and clarks formula states that where is the measure defining the Frechet derivative of F at b.In this paper we extend Clarks formula to the more general class of weakly H-differentiablefunctionals, and we give a simple proff based on Malliavins calculus. again using Malliavin calculus techniques, we also derive Haussmanns stochastic integral representation of a function F(y) of the diffusion process In doing this, we show that is weakly H-differentiable if m and have bounded, continuous, first derivatives in y.

Asymptotic Stability of the Optimal Filter with Respect toIts Initial Condition

Consider the problem of estimation of a diffusion signal observed in additive white noise. If the solution to the filtering equations, initialized with an incorrect prior distribution, approaches the true conditional distribution asymptotically in time, then the filter is said to be asymptotically stable with respect to perturbations of the initial condition. This paper presents asymptotic stability results for linear filtering problems and for signals with limiting ergodic behavior. For the linear case, stability of the Riccati equation of Kalman filtering is used to derive almost sure asymptotic stability of linear filters for possibly non-Gaussian initial conditions. In the nonlinear case, asymptotic stability in a weak convergence sense is shown for filters of signal diffusions which converge in law to an invariant distribution.

Linear stochastic differential equations with boundary conditions

SummaryWe study linear stochastic differential equations with affine boundary conditions. The equation is linear in the sense that both the drift and the diffusion coefficient are affine functions of the solution. The solution is not adapted to the driving Brownian motion, and we use the extended stochastic calculus of Nualart and Pardoux [16] to analyse them. We give analytical necessary and sufficient conditions for existence and uniqueness of a solution, we establish sufficient conditions for the existence of probability densities using both the Malliavin calculus and the co-aera formula, and give sufficient conditions that the solution be either a Markov process or a Markov field.

Exponential stability of discrete-time filters for bounded observation noise

This paper proves exponential asymptotic stability of discrete-time filters for the estimation of solutions to stochastic difference equations, when the observation noise is bounded. No assumption is made on the ergodicity of the signal. The proof uses the Hilbert projective metric, introduced into filter stability analysis by Atar and Zeitouni [1,2]. It is shown that when the signal noise is sufficiently regular, boundedness of the observation noise implies that the filter update operation is, on average, a strict contraction with respect to the Hilbert metric. Asymptotic stability then follows.

Stochastic calculus of variations for stochastic partial differential equations

Abstract This paper develops the stochastic calculus of variations for Hilbert space-valued solutions to stochastic evolution equations whose operators satisfy a coercivity condition. An application is made to the solutions of a class of stochastic pdes which includes the Zakai equation of nonlinear filtering. In particular, a Lie algebraic criterion is presented that implies that all finite-dimensional projections of the solution define random variables which admit a density. This criterion generalizes hypoellipticity-type conditions for existence and regularity of densities for finite-dimensional stochastic differential equations.

Multiple Integral Expansions for Nonlinear Filtering

Multiple stochastic integral expansions are applied to the problem of filtering a signal observed in additive noise. It is shown that the optimal mean-square estimate may be represented as a ratio of two multiple integral series. A formula for expanding the product of two multiple integrals is developed and applied to deriving equations for the kernels of best, finite expansion approximations to the optimal filter. These equations are studied in detail in the quadratic case.

Exponential stability in discrete-time filtering for non-ergodic signals

In this paper we prove exponential asymptotic stability for discrete-time filters for signals arising as solutions of d-dimensional stochastic difference equations. The observation process is the signal corrupted by an additive white noise of sufficiently small variance. The model for the signal admits non-ergodic processes. We show that almost surely, the total variation distance between the optimal filter and an incorrectly initialized filter converges to 0 exponentially fast as time approaches [infinity].

Finite-fuel singular control with discretionary stopping

We discuss the finite-fuel, singular stochastic control problem of optimally tracking the standard Brownian motion started at , by an adapted process of bounded total variation , so as to minimize the total expected discounted cost over such processes and stopping times τ. Here , and are given real numbers. In its form this problem goes back to the seminal paper of Bene[sbreve], Shepp and Witsenhausen (1980). For fixed α>0 and δ>0 we characterize explicitly the optimal policy in the case λ>αδ (of the “act-or-stop” type, since the continuation cost is relatively large), and in the case with (of the “act, stop, or wait” type, since the relative continuation cost is relatively small). In the latter case, an associated free-boundary problem is solved exactly. The case , of “moderate” relative continuation cost, is suggested as an open question

An extension of clark' formula

The representation formula of Clark (1970) and Haussmann (1979) is established for Brownian functional in the space D1.1