Interdisciplinary Mathematical SciencesStochastic Analysis and Applications to Finance, pp. 225-242 (2012) No AccessSome Results on Backward Stochastic Differential Equations Driven by Fractional Brownian MotionsYaozhong Hu, Daniel Ocone, and Jian SongYaozhong HuDepartment of Mathematics, University of Kansas, Lawrence, Kansas, 66045, USA, Daniel OconeDepartment of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USA, and Jian SongDepartment of Mathematics, Rutgers University, Piscataway, NJ 08854-8019, USAhttps://doi.org/10.1142/9789814383585_0012Cited by:3 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: This paper deals with backward stochastic differential equations (bsde) driven by fractional Brownian motions with a more general terminal condition. A relation between fractional bsde and partial differential equation (pde) of mixed type is established. A comparison result for fractional bsde is also obtained. FiguresReferencesRelatedDetailsCited By 3Density Estimates for the Solutions of Backward Stochastic Differential Equations Driven by Gaussian ProcessesXiliang Fan and Jiang-Lun Wu29 February 2020 | Potential Analysis, Vol. 54, No. 3Linear backward stochastic differential equations with Gaussian Volterra processesHabiba Knani and Marco Dozzi3 December 2020 | Modern Stochastics: Theory and Applications, Vol. 75Comparison theorem, Feynman–Kac formula and Girsanov transformation for BSDEs driven by G -Brownian motionMingshang Hu, Shaolin Ji, Shige Peng and Yongsheng Song1 Feb 2014 | Stochastic Processes and their Applications, Vol. 124, No. 2 Stochastic Analysis and Applications to FinanceMetrics History PDF download

We consider an infinite horizon stochastic control problem with discretionary stopping. The state process is given by a one dimensional stochastic differential equation. The diffusion coefficient is chosen by an adaptive choice of the controller and it is allowed to take the value zero. The controller also chooses the quitting time to stop the system. Here we develop a martingale characterization of the value function and use it and the principle of smooth fit to derive an explicit optimal strategy when the drift coefficient of the state process is of the form b(x)=−θx where θ>0 is a constant.

Page 1. http://www.elsevier.com/locate/jcss Journal of Computer and System Sciences 68 (2004) 862 Author Index for Volume 68 A Achlioptas, Dimitris, 238 Ambainis, Andris, 398 Anceaume, E., 123 Anderson, James H., 157 B Bar-Yossef, Ziv, 702 Beame, Paul, 238 Becchetti, Luca, 80 Bshouty, Nader H., 205 C Charikar, Moses, 417 Chazelle, Bernard, 269 Chen, Bing-Chang, 598 Choi, Kwok Pui, 22 Cohen, Edith, 701 D de Alfaro, Luca, 374 Downey, Rod G., 96 Drewes, Frank, 611 Dunagan, John, 335 $Duri$s, Pavol, 675 E Engelfriet, Joost, 611 F Feldman, Jon, 733 Fern!andez, A., 123 Fischer, Eldar, 753 Franceschini, Gianni, 788 G Goemans, Michel X., 442 Grandjean, Etienne, 546 Grohe, Martin, 285 Grossi, Roberto, 788 Guruswami, Venkatesan, 701 H Hirschfeldt, Denis R., 96 Homan, Christopher M., 657 Hromkovi$c, Juraj, 675 I Inoue, Katsushi, 675 J Jackson, Jeffrey C., 205 Jayram, TS, 702 …

This paper considers a modification of a PAC learning theory problem in which each instance of the training data is supplemented with side information. In this case, a transformation, given by a side-information map, of the training instance is also classified. However, the learning algorithm needs only to classify a new instance, not the instance and its value under the side information map. Side information can improve general learning rates, but not always. This paper shows that side information leads to the improvement of standard PAC learning theory rate bounds, under restrictions on the probable overlap between concepts and their images under the side information map.

We formulate a stochastic control problem with a general information structure, and show that an optimal law exists and is characterized as the unique solution of a recursive stochastic equation. For a special information structure of the "signal-plus-noise" type and with quadratic cost-functions, this recursive equation is solved for the value function of the control problem. This value function is then shown to satisfy the Mortensen equation of Dynamic Programming in function space.

This paper takes a computational learning theory approach to a problem of linear systems identification. It is assumed that inputs are generated randomly from a known class consisting of linear combinations of k sinusoidals. The output of the system is classified at some single instant of time. The main result establishes that the number of samples needed for identification with small error and high probability, independently from the distribution of inputs, scales polynomially with n, the system dimension, and logarithmically with k.

We study the problem of stationary control by adaptive choice of the diffusion coefficient in the case that control degeneracy is allowed and the drift admits a unique, asymptotically stable equilibrium point. We characterize the optimal value and obtain it as an Abelian limit of optimal discounted values and as a limiting average of finite horizon optimal values, and we also characterize the optimal stationary strategy. In the case of linear drift, the optimal stationary value is expressed in terms of the solution of an optimal stopping problem. We generalize the above results to allow unbounded cost functions.

We explore a new learning setting, in which each randomly generated sample gives rise to an additional deterministic sample, called side information, that is also classified by the oracle. Hence a learning algorithm utilizing side information chooses from a smaller and more accurate set of concepts, and is expected to operate more efficiently. In general, side information learning utilizes dependent data and the training space differs from the evaluation space, as the the output of the algorithm will only need to classify a typical observation. We analyze a simple problem of learning open intervals and compare exact learning rates with and without side information. Many times side information yields exponentially better learning rates, but in some cases this improvement may vanish.

The identification of continuous-time control systems is considered as a learning problem. As the general problem is too rich to be learnable, input signals are assumed to have only a finite number k of frequency components. When the system being identified has a dimension at most n, the required sample size scales polynomially with n and logarithmically with k.

This paper studies bounded-velocity control of a Brownian motion when discretionary stopping, or 'leaving', is allowed. The goal is to choose a control law and a stopping time in order to minimize the expected sum of a running and a termination cost, when both costs increase as a function of distance from the origin. There are two versions of this problem: the fully observed case, in which the control multiplies a known gain, and the partially observed case, in which the gain is random and unknown. Without the extra feature of stopping, the fully observed problem originates with Benes (Stochastic Process, Appl. 2 (1974) 127-140), who showed that the optimal control takes the 'bang-bang' form of pushing with maximum velocity toward the origin. We show here that this same control is optimal in the case of discretionary stopping; in the case of power-law costs, we solve the variational equation for the value function and explicitly determine the optimal stopping policy.We also discuss qualitative features of the solution for more general cost structures. When no discretionary stopping is allowed, the partially observed case has been solved by Bene et al. (Stochastics Monographs, Vol. 5, Gordon & Breach, New York and London, pp. 121-156) and Karatzas and Ocone (Stochastic Anal. Appl. 11 (1993) 569-605). When stopping is allowed, we obtain lower bounds on the optimal stopping region using stopping regions of related, fully observed problems. (C) 2002 Elsevier Science B.V. All rights reserved.

Consider an Itô equation for a scalar-valued process that is controlled through a dynamic and adaptive choice of its diffusion coefficient. Such a control is called a variance control and is said to degenerate when it becomes zero. We consider the problem of choosing a control to minimize a discounted, infinite-horizon cost that penalizes state values close to an equilibrium point of the drift and also imposes a control cost. Admissible controls are required to take values in the closed, bounded interval $[0,\sigma_0]$, where $\sigma_0>0$; in particular, the control can be degenerate. In general, there will be a bang-bang optimal control that takes the value $\sigma_0$ in some open set and is zero otherwise. We discuss the existence and properties of solutions to stochastic differential equations with such controls and characterize the value function and optimal control in more detail, in the case of both linear and nonlinear drift. Employing the Hamilton--Jacobi--Bellman equation and results of [N. V. Krylov, Theory Probab. Appl., 17 (1973), pp. 114--131] and [P.-L. Lions, Comm. Pure Appl. Math., 34 (1981), pp. 121--147], we derive sufficient conditions for the existence of single-region optimal controls, construct examples of multiple-region controls, and provide bounds on the number and size of the regions in which the optimal control is positive.

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

We first consider system classification as a learning problem and study a class realized by continuous-time linear control systems. The difficulty of learning is characterized by the Vapnik-Chervonenkis (VC) dimension of the class of such mappings with binary output classifications. We provide upper and lower bounds for the VC-dimension in terms of the system dimensions and constants related to controls. Also, pseudo-dimension bounds are given for studying the input-output behavior without classification. For systems with bounded controls and parameters, fat-shattering bounds are proved in terms of the dimension of the parameter set and the Lipschitz constant associated to the systems. A control application motivates the problem of learning with side information in which each random sample x gives rise to a translate s(x), where s is a known side information mapping. Both x and s(x) are classified for training, but the aim is to classify correctly only a future unseen x-sample. The learning utilizes non-i.i.d. data and the training and evaluation spaces are different. First we consider a simple problem pointing to phenomena that hold more generally: we calculate exact learning rates for a fixed algorithm for learning an interval on the unit circle under the uniform distribution and reflection as side information. Typically, learning rates with side information correspond to traditional learning with twice as large a sample, but this may fail for some targets. In general, the advantage of side information depends on the distribution, learning algorithm and due to non-i.i.d. data there is an interaction between the target and the side information mapping. We incorporate side information in the analysis of uniform convergence of empirical probabilities. Two convergence bounds are analyzed and the exponential improvement in the convergence rate is indicated. As a new technique the bound utilizing Hoeffding's inequality is studied in the large deviations setting. The best improvement doubles the sample size in a bound for consistent algorithms and the sample size is multiplied by 4 in a general convergence bound. However, there are cases in which the exponential improvement due to side information fails.

This paper considers the relative entropy between the conditional distribution and an incorrectly initialized filter for the estimation of one component of a Markov process given observations of the second component. Using the Markov property, we first establish a decomposition of the relative entropy between the measures on observation path space associated to different initial conditions. Using this decomposition, it is shown that the relative entropy of the optimal filter relative to an incorrectly initialized filter is a positive supermartingale. By applying the decomposition to signals observed in additive, white noise, a relative entropy bound is obtained on the integrated, expected, mean square difference between the optimal and incorrectly initialized estimates of the observation function.

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 ∞.

Suppose that the nonlinear filtering equations are solved using an incorrect initial condition. It is known that the relative entropy of the actual conditional distribution with respect to this incorrectly initialized filter is a positive supermartingale. In this paper, we study the filtering of diffusion signals. Using the Kushner-Stratonovich equations, we decompose the relative entropy supermartingale into decreasing and local martingale terms, and we derive an entropy bound on information and error measures of the difference between conditional distribution and incorrectly initialized filter.

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.

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.

Stochastic evolution equations. Linear Theory and Applications to Nonlinear Filtering, by B. L, Rozovskii. Kluwer, Dordrecht/Boston/London, March 1991. 336 pp., $129. ISBN 0-7923-00370-8. Stochastic equations in infinite dimensions Stochastic equations in infinite dimensions, by G. Da Prato & J. Zabczyk. Cambridge University Press, Port Chester/New York, March 1993. 454 pp., $89.95. ISBN 0-541-38529-6

A Bayesian adaptive control problem with several interesting features, due to Benes and Rishel, was treated as a stochastic control problem with partial observations - and on an infinite horizon with discounting - in the papers [2] and [10]. We discuss here in full detail the finite-horizon version of that problem, by solving fairly explicitly the associated, fully nonlinear and degenerate, Hamilton-Jacobi-Bellman equation of parabolic type.