Standard Wiener Process Research Articles

On Statistical Control of Stochastic Servo-Systems: Performance-Measure Statistics and State-Feedback Paradigm

Abstract This paper provides a concise and up-to-date analysis of the foundations of performance robustness of a linear-quadratic class of servo-systems with respect to variability in a stochastic environment. The dynamics of servo-systems are corrupted by a standard stationary Wiener process and include input functions that are controlled by controllers. Basic assumptions will be that controllers have access to the current value of the states of the systems and would like to learn about performance uncertainty of the systems that is now affected by other non-cooperative learners, i.e. model deviations and environmental disturbances named Nature. The controller considered here optimizes a multi-objective criterion over time where optimization takes place with high regard for possible random sample realizations by Nature who may more likely not be acting in concert. It is found that the optimal servo in the finite horizon case is a novel two-degrees-of-freedom controller with: one, a feedback controller with state measurements that is robust against performance uncertainty; two, a model-following controller that minimizes the difference between the reference model and the system outputs.

Filtering of Stochastic Nonlinear Differential Systems via a Carleman Approximation Approach

This paper deals with the state estimation problem for stochastic nonlinear differential systems, driven by standard Wiener processes, and presents a filter that is a generalization of the classical extended kalman-bucy filter (EKBF). While the EKBF is designed on the basis of a first order approximation of the system around the current estimate, the proposed filter exploits a Carleman-like approximation of a chosen degree v ges 1. The approximation procedure, applied to both the state and the measurement equations, allows to define an approximate representation of the system by means of a bilinear system, for which a filtering algorithm is available from the literature. Numerical simulations on an example show the improvement, in terms of sample error covariance, of the filter based on the first-order, second-order and third-order system approximations (v = 1,2,3).

PRV property and the ϕ-asymptotic behavior of solutions of stochastic differential equations

In this paper, we investigate the a.s. asymptotic behavior of the solution of the stochastic differential equation dX(t) = g(X(t)) dt + σ(X(t))dW(t), X(0) ≢ 1, where g(·) and σ(·) are positive continuous functions, and W(·) is a standard Wiener process. By means of the theory of PRV functions we find conditions on g(·), σ(·), and ϕ(·) under which ϕ(X(·)) may be approximated a.s. by ϕ(μ(·)) on {X(t) → ∞}, where μ(·) is the solution of the ordinary differential equation dμ(t) = g(μ(t)) dt with μ(0) = 1.

Optimal Approximation of the Second Iterated Integral of Brownian Motion

In this article, a theorem is proved that describes the optimal approximation (in the L 2(ℙ)-sense) of the second iterated integral of a standard two-dimensional Wiener process, W, by a function of finitely many elements of the Gaussian Hilbert space generated by W. This theorem has some interesting corollaries: First of all, it implies that Euler's method has the optimal rate of strong convergence among all algorithms that depend solely on linear functionals of the Wiener process, W; second, it shows that the approximation of the second iterated integral based on Karhunen–Loève expansion of the Brownian bridge is asymptotically optimal.

Relativistic diffusion processes and random walk models

The nonrelativistic standard model for a continuous, one-parameter diffusion process in position space is the Wiener process. As is well known, the Gaussian transition probability density function (PDF) of this process is in conflict with special relativity, as it permits particles to propagate faster than the speed of light. A frequently considered alternative is provided by the telegraph equation, whose solutions avoid superluminal propagation speeds but suffer from singular (noncontinuous) diffusion fronts on the light cone, which are unlikely to exist for massive particles. It is therefore advisable to explore other alternatives as well. In this paper, a generalized Wiener process is proposed that is continuous, avoids superluminal propagation, and reduces to the standard Wiener process in the nonrelativistic limit. The corresponding relativistic diffusion propagator is obtained directly from the nonrelativistic Wiener propagator, by rewriting the latter in terms of an integral over actions. The resulting relativistic process is non-Markovian, in accordance with the known fact that nontrivial continuous, relativistic Markov processes in position space cannot exist. Hence, the proposed process defines a consistent relativistic diffusion model for massive particles and provides a viable alternative to the solutions of the telegraph equation.

On parameter estimation of stochastic delay differential equations with guaranteed accuracy by noisy observations

Let ( X ( t ) , t ⩾ - 1 ) and ( Y ( t ) , t ⩾ 0 ) be stochastic processes satisfying d X ( t ) = aX ( t ) d t + bX ( t - 1 ) d t + d W ( t ) and d Y ( t ) = X ( t ) d t + d V ( t ) , respectively. Here ( W ( t ) , t ⩾ 0 ) and ( V ( t ) , t ⩾ 0 ) are independent standard Wiener processes and ϑ = ( a , b ) ′ is assumed to be an unknown parameter from some subset Θ of R 2 . The aim here is to estimate the parameter ϑ based on continuous observation of ( Y ( t ) , t ⩾ 0 ) . Sequential estimation plans for ϑ with preassigned mean square accuracy ε are constructed using the so-called correlation method. The limit behaviour of the duration of the estimation procedure is studied if ε tends to zero.

On the stability of some stochastic integro partial differential equations

Stochastic integro partial differential equations of the form; du(x, t )= n i=1 ∂ 2 u(x, t) ∂x 2 dt + F (u(x, t) ,x , t)dt + t 0 K(t − θ)u(x, θ)dθdt +[ f (t)u(x, t )+ g(x, t)]dW (t), are considered, where {W (t ): t ≥ 0} is a standard one-dimensional Wiener process and the kernel K decreases to zero non-exponentially. The behavior of solutions and their convergence to zero are studied. It is proved under suitable conditions that lim t→∞ u(x, t) K(t) = ∞ , almost surely. The considered stochastic integro partial differential equations arise if we consider the Black-Scholes market consists of a bank account or a bond and a stock. These stochastic models can also applied to population dynamics in biology. Mathematics Subject Classifications: 34D20, 60H10, 65H20, 45N05

Euler's Approximations of Weak Solutions of Reflecting SDEs with Discontinuous Coefficients

Here X0 = x0 ∈ D = D ∪ ∂D, X is a re ecting process on D, K is a bounded variation process with variation |K| increasing only when Xt ∈ ∂D, W is a d-dimensional standard Wiener process and σ : D −→ R ⊗ R, b : D −→ R are measurable functions. This presentation will be devoted to show convergence in law as well as in L for the Euler and Euler-Peano schemes for SDEs (1). The coe cients are measurable, continuous almost everywhere with respect to the Lebesgue measure and the di usion coe cient may degenerate on some subsets of the domain. The proofs of theorems are based on some new generalized inequalities of Krylov's type for stochastic integrals. These results strengthen and extend these given in the papers [2],[3]. First, the assumption of uniform ellipticity of σσ∗ is considerably weaken, and secondly, theorems for D satisfying the conditions (A) and (B) are given. Joint work with Katarzyna Borkowska.

The Generalized Wiener Process for Colored Noise

We define the generalized Wiener process as the output of a first-order differential equation when the input is an arbitrary stochastic input. This is in contrast to the standard Wiener process, where the input is white Gaussian noise. We obtain a simple explicit result for any input wide sense stationary random process, namely, that the Wigner spectrum of the output random process is the product of the power spectrum of the input process times a simple universal function of time and frequency. We also obtain the impulse response function wherein the output of the generalized process is expressed as the impulse response function integrated with the input random process. Various limiting values are derived. We apply the method to the case where the driving stochastic process has an exponentially decaying autocorrelation function

Hausdorff–Besicovitch measure of fractal functional limit laws induced by Wiener process in Hölder norms

Let { W ( t ) : t ⩾ 0 } denote a standard Wiener process. In this paper, we first establish a de Acosta [A. de Acosta, On the functional form of Lévy's modulus of continuity for Brownian motion, Z. Wahr. Verw. Gebiete 69 (1985) 567–579] type strong law for a family of Hölder norms. More precisely, we obtain, for α ∈ ( 0 , 1 / 2 ) , the exact rate of convergence, as h ↓ 0 , of T α , f ( h ) : = inf 0 ⩽ t ⩽ 1 − h ‖ ( 2 h log ( 1 / h ) ) − 1 / 2 ( W ( t + h ⋅ ) − W ( t ) ) − f ‖ α when f ∈ S satisfies ∫ 0 1 { d d u f ( u ) } 2 d u < 1 , where S denotes the Strassen [V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahr. Verw. Gebiete 3 (1964) 211–226] set. In a second part we give some general technical tools for evaluating the upper and the lower critical functions of the Hausdorff–Besicovitch measures respectively for limsup random sets and for random Cantor type sets. As an application we deduce the Hausdorff dimension of the random fractal constituted of exceptional points in [ 0 , 1 ] where the previous rate is reached.

Reliability of difference analogues to preserve stability properties of stochastic Volterra integro-differential equations

We consider the reliability of some numerical methods in preserving the stability properties of the linear stochastic functional differential equation , where α, β, σ, τ ≥ 0 are real constants, and W(t) is a standard Wiener process. The areas of the regions of asymptotic stability for the class of methods considered, indicated by the sufficient conditions for the discrete system, are shown to be equal in size to each other and we show that an upper bound can be put on the time-step parameter for the numerical method for which the system is asymptotically mean-square stable. We illustrate our results by means of numerical experiments and various stability diagrams. We examine the extent to which the continuous system can tolerate stochastic perturbations before losing its stability properties and we illustrate how one may accurately choose a numerical method to preserve the stability properties of the original problem in the numerical solution. Our numerical experiments also indicate that the quality of the sufficient conditions is very high.

One-Dimensional global optimization for observations with noise

A problem of one-dimensional global optimization in the presence of noise is considered. The approach is based on modeling the objective function as a standard Wiener process which is observed with independent Gaussian noise. An asymptotic bound for the average error is estimated for the nonadaptive strategy defined by a uniform grid. Experimental results consistent with the asymptotic results are presented. An adaptive algorithm is proposed and experimentally compared with the nonadaptive strategy with respect to the average error.

Pricing Equity Swaps in an Economy with Jumps

This article considers the pricing of equity swaps with constant notional principal in an incomplete market. The market is driven by a general marked point process as well as by a standard multidimensional Wiener process. We derive the swap values using martingale methods and the technique of convexity correction. Our results are consistent with the results of Liao & Wang (2003), but achieved in a different way, which enables us to extend their generalized formula for pricing equity swaps to the point process case.

Towards a General Theory of Good Deal Bounds

We consider an incomplete market in the form of a multidimensional Markovian factor model, driven by a general marked point process (representing discrete jump events) as well as by a standard multidimensional Wiener process. Within this framework we study arbitrage free good deal pricing bounds for derivative assets along the lines of Cochrane and Saa-Requejo (2000), extending the results from their paper to the point process case. As a concrete application we present numerical results for the classic Merton jump-diffusion model. As a by product of the general theory we also extend the Hansen-Jagannathan bounds for the Sharpe Ratio to the point process setting.

BIFURCATIONS IN APPROXIMATE SOLUTIONS OF STOCHASTIC DELAY DIFFERENTIAL EQUATIONS

We consider stochastic delay differential equations of the form [Formula: see text] interpreted in the Itô sense, with Y(t)=Φ(t) for t∈[t0-τ,t0] (here, W(t) is a standard Wiener process and τ&gt;0 is the constant "lag", or "time-lag"). We are interested in bifurcations (that is, changes in the qualitative behavior of solutions of these equations) and we draw on insights from the related deterministic delay differential equation, for which there is a substantial body of known theory, and numerical results that enable us to discuss where changes occur in the behavior of the (exact and approximate) solutions of the equation. Rather diverse components of mathematical background are necessary to understand the questions of interest. In this paper we first review some deterministic results and some basic elements of the stochastic analysis that (i) suggests lines of investigation for the stochastic case and (ii) are expected to facilitate the theoretical investigation of the stochastic problem. We then present the results of numerical experiments that illustrate some of the complexities that arise when considering bifurcations in stochastic delay differential equations. They give prima facie evidence for certain convergence properties of the bifurcation points estimated using the Euler–Maruyama method for the equations considered. We conclude by drawing attention to a number of open questions in the field.

Global asymptotic stability of solutions of cubic stochastic difference equations

Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable martingale differences is proven under appropriate conditions in ℝ1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods with variable step sizes for ordinary stochastic differential equations driven by standard Wiener processes, is discussed.

Rényi test statistics for partially observed diffusion processes

Diffusion processes X( t) verifying the stochastic differential equation dX(t)=a dt+b dW(t) , X( t 0)= X 0, b>0, are considered for standard Wiener processes W( t) in 0< t 0< t< T. Problems of testing hypotheses about the parameters are analyzed when the stochastic process is partially observed. A family of test statistics is introduced on the basis of the Rényi divergence measure and their asymptotic distributions are obtained. To finish, a simulation study is given in order to compare powers of some of the introduced statistics.

A FUBINI THEOREM FOR GENERALIZED ANALYTIC FEYNMAN INTEGRALS AND FOURIER-FEYNMAN TRANSFORMS ON FUNCTION SPACE

In this paper we use a generalized Brownian motion process to define a generalized analytic Feynman integral. We then establish a Fubini theorem for the function space integral and generalized analytic Feynman integral of a functional F belonging to Banach algebra <TEX>$S(L^2_{a,b}[0,T])$</TEX> and we proceed to obtain several integration formulas. Finally, we use this Fubini theorem to obtain several Feynman integration formulas involving analytic generalized Fourier-Feynman transforms. These results subsume similar known results obtained by Huffman, Skoug and Storvick for the standard Wiener process.

Rezimi kursnog koridora u kontekstu valutnih kriza

One of the pillars of open-economy macro is the well-known axiom of "inconsistent trinity", which claims it is impossible to simultaneously achieve significant liberalisation of international capital movements, monetary policy independence and fixed exchange rate. Ignorance or inability to accept this identity, have recently placed many developing countries with fixed (yet adjustable) exchange rate regimes in the centre of fierce international financial crises with both monetary and real repercussions. Precautionary or after being burnt by crisis, majority of small open economies adopted the so-called bipolar view, i.e. one of exchange rate modalities which stretch along the wide axis between rigid fixation and free floating. After having dealt with elementary dynamics of crawling/monitoring band intermediate regime, this paper analyses its behaviour in the face of speculative attacks against domestic currency. In doing so, first, we reminded that exchange rate trajectory, even inside ("honeymoon"-free) currency band, does not follow Brownian motion (standard Wiener process). We showed, thereafter, that imperfectly-credible target zones could also have an asymptotically stabilizing effect on intramarginal exchange rate dynamics. This mean-reverting trait enables monetary authorities to control speculative volatility of exchange rate even in cases when they lack sufficient amount of foreign reserves, inasmuch as fundamental or globally spread crises need not be defended by means of open-market intervention, but by unanticipated, temporally irregular inside-the-band realignment or simply by sustainable widening of the band. Nevertheless, in principle, currency band should be shielded against non-fundamental market pressures. Therefore, we felt it was worthwhile calibrating the imminent danger of initial and cumulative attacks cum total currency crash, by means of introducing formal mathematical relationships among key variables which reflect both structural and behavioural features of the economy at hand (and, primus inter pares, its exchange rate regime), on the one hand, and on the other hand, objective functions of both policymaker and speculators (individually). At first armed with ad hoc log-nonlinear stochastic model of currency band, and later on by intertemporal optimisation modeling, we reconstructed preattack exchange rate trajectory, "sufficient" amount of foreign exchange reserves, exchange rate behaviour during the currency crisis and sequence of crisis events. At last, we carried out rigorous analysis of variables which influence width and/or location of ranges of effective commitment. Finally, we suggested alleys which may prove fruitful for further research efforts.

Asymptotic inference in time series regressions with a unit root and infinite variance errors

In this paper, we study the limiting distribution of the OLS estimators and t-statistics of the null hypothesis of a unit root in various regression models when the true generating mechanism is either a driftless random walk or a random walk with drift and the distribution of the error term belongs to the normal domain of attraction of a stable law with characteristic exponent less than two. We show that in the former data generating processes (DGP) the same functional form as in the finite variance case applies with the Lévy process replacing the standard Wiener process. On the other hand, under the random walk with drift DGP, we show that different limiting distributions are obtained according to the magnitude of the maximal moment exponent α. Finally, we investigate the consequences of a “local” departure from the finite variance setup and provide simulation evidence on the robustness of the limiting distributions (derived under finite variance) to heavy tails in finite samples.