Large Deviation Principle for Optimal Filtering

Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.     

Exponential Convergence for a System of Conuclear Space-Valued Diffusions with Mean-Field Interaction

We consider an interacting system of n diffusion processes X ⁿ _j (t): t∈[0,1] , j=1,2,. . ., n , taking values in a conuclear space Φ' . Let ζ ⁿ _t =(1/n)Σ ⁿ _j=1 δ _Xnj(t) be the empirical process. It has been proved that ζ ⁿ , as n→∞ , converges to a deterministic measure-valued process which is the unique solution of a nonlinear differential equation. In this paper we show that, under suitable conditions, ζ ⁿ converges to ζ at an exponential rate. Accepted 20 October 1997     

Coupling and exponential ergodicity for stochastic differential equations driven by Lévy processes

We present a novel idea for a coupling of solutions of stochastic differential equations driven by Lévy noise, inspired by some results from the optimal transportation theory. Then we use this coupling to obtain exponential contractivity of the semigroups associated with these solutions with respect to an appropriately chosen Kantorovich distance. As a corollary, we obtain exponential convergence rates in the total variation and standard $L^1$-Wasserstein distances.     

Second order backward stochastic differential equations with quadratic growth

We extend the well posedness results for second order backward stochastic differential equations introduced by Soner, Touzi and Zhang (2012)  [31] to the case of a bounded terminal condition and a generator with quadratic growth in the z$z$ variable. More precisely, we obtain uniqueness through a representation of the solution inspired by stochastic control theory, and we obtain two existence results using two different methods. In particular, we obtain the existence of the simplest purely quadratic 2BSDEs through the classical exponential change, which allows us to introduce a quasi-sure version of the entropic risk measure. As an application, we also study robust risk-sensitive control problems. Finally, we prove a Feynman–Kac formula and a probabilistic representation for fully non-linear PDEs in this setting.     

Harnack and super poincaré inequalities for generalized Cox-Ingersoll-Ross model

Abstract

In this paper, Wang’s Harnack inequalities and super Poincaré inequality for generalized Cox-Ingersoll-Ross model are obtained. Since the noise is degenerate, the intrinsic metric has been introduced to construct the coupling by change of measure. By using isoperimetric constant, some optimal estimate of the rate function in the super Poincaré inequality for the associated Dirichlet form is also obtained.     

Exponential Formula for the Reachable Sets of Quantum Stochastic Differential Inclusions

We establish an exponential formula for the reachable sets of quantum stochastic differential inclusions (QSDI) which are locally Lipschitzian with convex values. Our main results partially rely on an auxilliary result concerning the density, in the topology of the locally convex space of solutions, of the set of trajectories whose matrix elements are continuously differentiable. By applying the exponential formula, we obtain results concerning convergence of the discrete approximations of the reachable set of the QSDI. This extends similar results of Wolenski[20] Wolenski, P.R. 1990. The exponential formula for the reachable set of a Lipschitz differential inclusion. SIAM J. Control Optim., 28(5): 1148–1161.  [Google Scholar] for classical differential inclusions to the present noncommutative quantum setting.     

Smooth Solutions to Optimal Investment Models with Stochastic Volatilities and Portfolio Constraints

   Abstract. This paper deals with an extension of Merton's optimal investment problem to a multidimensional model with stochastic volatility and portfolio constraints. The classical dynamic programming approach leads to a characterization of the value function as a viscosity solution of the highly nonlinear associated Bellman equation. A logarithmic transformation expresses the value function in terms of the solution to a semilinear parabolic equation with quadratic growth on the derivative term. Using a stochastic control representation and some approximations, we prove the existence of a smooth solution to this semilinear equation. An optimal portfolio is shown to exist, and is expressed in terms of the classical solution to this semilinear equation. This reduction is useful for studying numerical schemes for both the value function and the optimal portfolio. We illustrate our results with several examples of stochastic volatility models popular in the financial literature.     

Conditional large deviations for a sequence of words

Cut an i.i.d. sequence (_Xi)$\left(X_i\right)$ of ‘letters’ into ‘words’ according to an independent renewal process. Then one obtains an i.i.d. sequence of words, and thus the level 3 large deviation behaviour of this sequence of words is governed by the specific relative entropy. We consider the corresponding problem for the conditional   empirical process of words, where one conditions on a typical underlying (_Xi)$\left(X_i\right)$. We find that if the tails of the word lengths decay exponentially, the large deviations under the conditional distribution are almost surely again governed by the specific relative entropy, but the set of attainable limits is restricted.     

Exit time and invariant measure asymptotics for small noise constrained diffusions

Constrained diffusions, with diffusion matrix scaled by small ?>0, in a convex polyhedral cone G⊂R^k, are considered. Under suitable stability assumptions small noise asymptotic properties of invariant measures and exit times from domains are studied. Let B⊂G be a bounded domain. Under conditions, an “exponential leveling” property that says that, as ?→0, the moments of functionals of exit location from B, corresponding to distinct initial conditions, coalesce asymptotically at an exponential rate, is established. It is shown that, with appropriate conditions, difference of moments of a typical exit time functional with a sub-logarithmic growth, for distinct initial conditions in suitable compact subsets of B, is asymptotically bounded. Furthermore, as initial conditions approach 0 at a rate ?² these moments are shown to asymptotically coalesce at an exponential rate.     

Tail estimates for exponential functionals and applications to SDEs

In this paper, based on techniques of Malliavin calculus, we obtain an explicit bound for tail probabilities of a general class of exponential functionals. We apply the obtained results to derive asymptotic behaviors for the tail of the exponential functional of stochastic differential equations.     

Existence of Optimal Controls for Semilinear Parabolic Equations without Cesari-Type Conditions

   Abstract. Optimal control problems governed by semilinear parabolic partial differential equations are considered. No Cesari-type conditions are assumed. By proving the existence theorem and the Pontryagin maximum principle of optimal ``state-control" pairs for the corresponding relaxed problems, an existence theorem of optimal pairs for the original problem is established.     

Nonlinear Filtering of Diffusion Processes in Correlated Noise: Analysis by Separation of Variables

   Abstract. An approximation to the solution of a stochastic parabolic equation is constructed using the Galerkin approximation followed by the Wiener chaos decomposition. The result is applied to the nonlinear filtering problem for the time-homogeneous diffusion model with correlated noise. An algorithm is proposed for computing recursive approximations of the unnormalized filtering density and filter, and the errors of the approximations are estimated. Unlike most existing algorithms for nonlinear filtering, the real-time part of the algorithm does not require solving partial differential equations or evaluating integrals. The algorithm can be used for both continuous and discrete time observations. \par     

On a “lack of memory” property

For two independent nonnegative random variablesX andY we say thatX is ageless relative toY if the conditional probability P[X> Y+x|X>Y] is defined and is equal to P[X>x] for allx>0. Suppose thatX is ageless relative to a nonlatticeY with P[Y=0]<P [Y<X]. We show that the only suchX is the exponential variable. As a corollary it follows that exponential variable is the only one which possesses the ageless property relative to a continuous variable. Research partially supported by NRC of Canada grants #A8057 and #T0500. Work partially completed while on leave at Division of Math. Stat., C.S.I.R.O., Australia.     

Optimal vector quantization in terms of Wasserstein distance

The optimal quantizer in memory-size constrained vector quantization induces a quantization error which is equal to a Wasserstein distortion. However, for the optimal (Shannon-)entropy constrained quantization error a proof for a similar identity is still missing. Relying on principal results of the optimal mass transportation theory, we will prove that the optimal quantization error is equal to a Wasserstein distance. Since we will state the quantization problem in a very general setting, our approach includes the Rényi-α-entropy as a complexity constraint, which includes the special case of (Shannon-)entropy constrained (α=1) and memory-size constrained (α=0) quantization. Additionally, we will derive for certain distance functions codecell convexity for quantizers with a finite codebook. Using other methods, this regularity in codecell geometry has already been proved earlier by György and Linder (2002, 2003) [11] and [12].     

Shape Optimization in Viscous Compressible Fluids

   Abstract. We present a method for solving the optimal shape problems for profiles surrounded by viscous compressible fluids. The class of admissible profiles is quite general including the minimal volume condition and a constraint on the thickness of the boundary. The fluid flow is modelled by the Navier—Stokes system for a general viscous barotropic fluid.     

On the conjecture of Kochar and Korwar

In this article, we partially solve a conjecture by Kochar and Korwar (1996) [9] in relation to the normalized spacings of the order statistics of a sample of independent exponential random variables with different scale parameters. In the case of a sample of size n=3, they proved the ordering of the normalized spacings and conjectured that result holds for all n. We prove this conjecture for n=4 for both spacings and normalized spacings and generalize some results to n>4.     

On the joint distribution of studentized order statistics

In this paper, the joint distribution of some special linear combinations of the (internally) studentized order statistics are derived for both normal and exponential populations; the exact relationship between their pdf's is also obtained. The exact sampling distributions of studentized extreme deviation statistic, which has been proposed by Pearson and Chandra Sekar (1936,Biometrika,28, 308–320), are derived for these two populations. An application to the most powerful location and scale invariant test is discussed briefly.     

An Extended Faber Operator

   Abstract. One of the basic tools in the theory of polynomial approximation in the uniform norm on compact plane sets is the Faber operator. Usually, the Faber operator is viewed as an operator acting on functions in the disk algebra, that is, functions which are holomorphic in the open unit disk D and continuous on D. We consider an extended Faber operator acting on arbitrary functions continuous on ; D.     

An Obstacle Control Problem with a Source Term

   Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.