New explicit filters and smoothers for diffusions with nonlinear drift and measurements

The optimal least-squares filtering of a diffusion x(t) from its noisy measurements {y(τ); 0 τ t} is given by the conditional mean E[x(t)|y(τ); 0 τ t]. When x(t) satisfies the stochastic diffusion equation dx(t) = f(x(t)) dt + dw(t) and y(t) = ∫₀^tx(s) ds + b(t), where f(·) is a global solution of the Riccati equation /xf(x) + f(x)² = f(x)² = αx² + βx + γ, for some , and w(·), b(·) are independent Brownian motions, Benes gave an explicit formula for computing the conditional mean. This paper extends Benes results to measurements y(t) = ∫₀^tx(s) ds + ∫₀^t dx(s) + b(t) (and its multidimensional version) without imposing additional conditions on f(·). Analogous results are also derived for the optimal least-squares smoothed estimate E[x(s)|y(τ); 0 τ t], s < t. The methodology relies on Girsanov's measure transformations, gauge transformations, function space integrations, Lie algebras, and the Duncan-Mortensen-Zakai equation.     

On the non-existence of stationary diffusions which satisfy the Bene condition

The one-dimensional diffusion x_t satisfying dx_t = f(x_t)dt + dw_t, where w_t is a standard Brownian motion and f(x) satisfies the Bene condition f′(x) + f²(x) = ax² + bx + c for all real x, is considered. It is shown that this diffusion does not admit a stationary probability measure except for the linear case f(x) = αx + β, α < 0.     

Exponential stability in mean square of neutral stochastic differential functional equations

The aim of this paper is to investigate the exponential stability in mean square for a neutral stochastic differential functional equation of the form d[x(t) − G(x_t)] = [f(t,x(t)) + g(t, x_t)]dt + σ(t, x_t)dw(t), where x_t = {x(t + s): − τ s 0}, with τ > 0, is the past history of the solution. Several interesting examples are a given for illustration.     

On the joint nonlinear filtering-smoothing of diffusion processes

The smoothing of diffusions dx_t = f(x_t) dt + σ(x_t) dw_t, measured by a noisy sensor dy_t = h(x_t) dt + dv_t, where w_t and v_t are independent Wiener processes, is considered in this paper. By focussing our attention on the joint p.d.f. of (x_τ x_t), 0 ≤ τ < t, conditioned on the observation path {y_s, 0 ≤ s ≤ t}, the smoothing problem is represented as a solution of an appropriate joint filtering problem of the process, together with its random initial conditions. The filtering problem thus obtained possesses a solution represented by a Zakai-type forward equation. This solution of the smoothing problem differs from the common approach where, by concentrating on the conditional p.d.f. of x_τ alone, a set of ‘forward and reverse’ equations needs to be solved.     

A note on two-sided stochastic control problems

We consider a class of two-sided stochastic control problems. For each continuous process π_t = π_t⁺ − π_t⁻ with bounded variation, the state process (x_t) is defined by x_t = B_t + f₀^t I(x_s - a)^d_πs+ − f₀^t I(x_s a)^dπ_s−, where a is a positive constant and (B_t) is a standard Brownian motion. We show the existence of an optimal policy so as to minimize the cost function J(π) = E [f₀^∞ e^−αsX_s² ds], with discount rate α > 0, associated with π.     

Stochastic suppression and stabilization of delay differential systems

In this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential system ${\dot{x}}(t)=f(x(t),x(t-\delta(t)),t)In this paper, we investigate stochastic suppression and stabilization for nonlinear delay differential system ${\dot{x}}(t)=f(x(t),x(t-\delta(t)),t)$, where δ(t) is the variable delay and f satisfies the one‐sided polynomial growth condition. Since f may defy the linear growth condition or the one‐sided linear growth condition, this system may explode in a finite time. To stabilize this system by Brownian noises, we stochastically perturb this system into the nonlinear stochastic differential system dx(t)=f(x(t), x(t?δ(t)), t)dt+qx(t)dw₁(t)+σ|x(t)|^βx(t)dw₂(t) by introducing two independent Brownian motions w₁(t) and w₂(t). This paper shows that the Brownian motion w₂(t) may suppress the potential explosion of the solution of this stochastic system for appropriate choice of β under the condition σ≠0. Moreover, for sufficiently large q, the Brownian motion w₁(t) may exponentially stabilize this system. Copyright © 2010 John Wiley & Sons, Ltd.     

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b₁(τ), …, B_n(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)εΩ-Γ and x<λ(τ)y at time τ and the probability that D(τ) Ω given that x < Λ(τ) y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.     

Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt=f(t,x_t), we stochastically perturb it into a stochastic functional differential equation , where Σ is a matrix and B(t) a Brownian motion while X_t={X(t+θ):-τθ0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,x_t) may be unstable.     

Inclusion properties for random relations under the hypotheses of stochastic independence and non-interactivity

This paper investigates whether random set inclusion is preserved by non-interactivity and by stochastic independence. Let (𝒳₁, x ₁), (𝒳₂, x ₂) be two random sets on U ₁ and U ₂, respectively, and let (𝒴₁, y ₁), (𝒴₂, y ₂) be two consonant inclusions of theirs. Let (𝒵₁, z ₁) be the random relation on U ₁ × U ₂ obtained from (𝒳₁, x ₁) and (𝒳₂, x ₂) under the hypothesis of stochastic independence, and let (𝒵₂, z ₂) ((𝒵₃, z ₃), respectively) be the random relation on U ₁ × U ₂ obtained from (𝒴₁, y ₁), (𝒴₂, y ₂) under the hypothesis of non-interactivity (stochastic independence, respectively). We prove that these hypotheses do not imply that (𝒵₁, z ₁) ? (𝒵₂, z ₂), but imply that (𝒵₁, z ₁) ? (𝒵₃, z ₃).     

On input-output linearization of discrete-time nonlinear systems

We consider a nonlinear discrete-time system of the form Σ: x(t+1)=f(x(t), u(t)), y(t) =h(x(t)), where x ε R^N, u ε R^m, y ε R^q and f and h are analytic. Necessary and sufficient conditions for local input-output linearizability are given. We show that these conditions are also sufficient for a formal solution to the global input-output linearization problem. Finally, we show that zeros at infinity of ε can be obtained by the structure algorithm for locally input-output linearizable systems.     

System Reduction and Solution Algorithms for Singular Linear Difference Systems under Rational Expectations

A first-order linear difference system under rational expectations is, AEy_t+1|I_t=By_t+C(F)Ex_t|I_t , where y_t is a vector of endogenous variables;x_t is a vector ofexogenous variables; Ey_t+1|I_t is the expectation ofy_t+1 givendate t information; and C(F)Ex_t|I_t =C₀x_t+C₁Ex_t+1|I_t+dot;s +C_nEx_t+n|I_t. If the model issolvable, then y_t can be decomposed into two sets of variables:dynamicvariables d_t that evolve according toEd_t+1|I_t = Wd_t + ¶si_d(F)Ex_t|I_t and other variables thatobey the dynamicidentities f_t =–Kd_t–¶si_f(F)Ex_t|I_t. We developan algorithm for carrying out this decomposition and for constructing theimplied dynamic system. We also provide algorithms for (i) computing perfectforesight solutions and Markov decision rules; and (ii) solutions to relatedmodels that involve informational subperiods.     

Stochastic stabilization and destabilization

It is shown in this paper that any nonlinear systems in ^d can be stabilized by Brownian motion provided |ƒ(x,t)| ≤ K|x| for some K > 0. On the other hand, this system can also be destabilized by Brownian motion if the dimension d ≥ 2. Similar results are also obtained for any given stochastic differential equation dx(t) = ƒ(x(t), t) + g(x(t), t) dW(t).     

Embedding a fault-free hamiltonian cycle in a class of faulty generalized honeycomb tori

Generalized honeycomb torus (GHT) is recognized as an attractive alternative to existing torus interconnection networks in parallel computing systems. Assume that m and d are integers with m ? 2 and d ? 8. This paper addresses the fault-tolerant hamiltonicity of GHT(m, 2d, d) with fault set F = {(w, y), (x, y)}, where w < x, w + y is even and x + y is odd. We show that such a faulty GHT is hamiltonian by presenting a systematic method for constructing a fault-free hamiltonian cycle. This result reveals another appealing feature of GHTs.     

Sequential monte carlo techniques for the solution of linear systems

Given alinear system Ax=b, wherex is anm-vector,direct numerical methods, such as Gaussian elimination, take timeO(m ³) to findx. Iterative numerical methods, such as the Gauss-Seidel method or SOR, reduce the system to the formx=a+Hx, whence and then apply the iterationsx ₀=a,x _s+1=a+Hx _s , until sufficient accuracy is achieved; this takes timeO(m ²) per iteration. They generate the truncated sums The usualplain Monte Carlo approach uses independent random walks, to give an approximation to the truncated sumx _s , taking timeO(m) per random step. Unfortunately, millions of random steps are typically needed to achieve reasonable accuracy (say, 1% r.m.s. error). Nevertheless, this is what has had to be done, ifm is itself of the order of a million or more.The alternative presented here, is to apply a sequential Monte Carlo method, in which the sampling scheme is iteratively improved. Simply put, ifx=y+z, wherey is a current estimate ofx, then its correction,z, satisfiesz=d+Hz, whered=a+Hy–y. At each stage, one uses plain Monte Carlo to estimatez, and so, the new estimatey. If the sequential computation ofd is itself approximated, numerically or stochastically, then the expected time for this process to reach a given accuracy is againO(m) per random step; but the number of steps is dramatically reduced [improvement factors of about 5,000, 26,000, 550, and 1,500 have been obtained in preliminary tests].     

Diagonal polynomials and diagonal orders on multidimensional lattices

HereR andN denote the real numbers and the nonnegative integers, respectively. Alsos(x)=x ₁+···+x _n whenx=(x ₁, …,x _n) inR ⁿ. A mapf:R ⁿ →R is call adiagonal function of dimensionn iff|N ⁿ is a bijection ontoN and, for allx, y inN ⁿ, f(x)<f(y) whens(x)<s(y). Morales and Lew [6] constructed 2 ⁿ⁻² inequivalent diagonal polynomial functions of dimensionn for eachn>1. Here we use new combinatorial ideas to show that numberd _n of such functions is much greater than 2 ⁿ⁻² forn>3. These combinatorial ideas also give an inductive procedure to constructd _n+1 diagonal orderings of {1, …,n}.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

Generalized dilations and the stabilization of uncertain systems via measurement feedback

We study the problem of semiglobally stabilizing uncertain nonlinear system

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, with (A,B) in Brunowski form. We prove that if p₁(z,u,t)u and p₂(z,u,t)u are of order greater than 1 and 0, respectively, with “generalized” dilation δ_l(z,u)=(l¹⁻ⁿz₁,…,l⁻¹z_n−1,z_n,lu) and uniformly with respect to t, where z_i is the ith component of z, then we can achieve semiglobal stabilization via arbitrarily bounded linear measurement feedback.     

A note on a problem on ω-limit sets of N–dimensional skew-product maps

In 2003, Balibrea et al. stated the problem of finding a skew-product map G on 𝕀³ holding ω _G ={0}×𝕀²=ω _G (x, y, z) for any (x, y, z)∈𝕀³, x≠0. We present a method for constructing skew-product maps F on 𝕀 ⁿ⁺¹ holding ω _F ={0}×𝕀 ⁿ =ω _F (x ₁, x ₂, …, x _n+1), (x ₁, x ₂, …, x _n+1)∈𝕀 ⁿ⁺¹, x ₁≠0.     

Existence of solutions to singular integral equations

Nonnegative solutions are established for singular integral equations of the form y(t) = h(t) + ∫^T₀ k(t, s)f(s, y(s)) ds for t ∈ [0, T]. Here f may be singular at y = 0.     

Stochastic adaptive pole-zero assignment with convergence analysis

The adaptive control u_n is designed for the stochastic system A(z)y_n+1 = B(z)u_n+C(z)w_n+1 with unknown constant matrix coefficients in the polynomials A(z), B(z) and C(z) in the shift-back operator with the purposes that (1) the unknown matrices are strongly consistently estimated and (2) the poles and zeros are replaced in such a way that the system itself is transferred to A⁰(z)y_n+1 = B⁰(z)u_n⁰+_n+1 with given A⁰(z), B⁰(z) and u_n⁰ so that the pole-zero assignment error {_n+1} is minimized. The problem of adaptive pole-zero assignment combined with tracking is also considered in this paper. Conditions used are imposed only on A(z), B(z) and C(z).