Multidimensional Wiener Process Research Articles

Term Structure Models with Parallel and Proportional Shifts

The paper investigates the possibility of an arbitrage‐free model for the term structure of interest rates where the yield curve only changes through a parallel shift. HJM type forward rate models driven by a multidimensional Wiener process and by a general marked point process are considered. Within this general framework it is shown that there does indeed exist a large variety of nontrivial parallel shift term structure models, and we also describe these in detail. It is also shown that there exists no nontrivial flat term structure model. The same analysis is repeated for a similar case, in which the yield curve only changes through proportional shifts.

Classification of stochastic Runge–Kutta methods for the weak approximation of stochastic differential equations

In the present paper, a class of stochastic Runge–Kutta methods containing the second order stochastic Runge–Kutta scheme due to E. Platen for the weak approximation of Itô stochastic differential equation systems with a multi-dimensional Wiener process is considered. Order 1 and order 2 conditions for the coefficients of explicit stochastic Runge–Kutta methods are solved and the solution space of the possible coefficients is analyzed. A full classification of the coefficients for such stochastic Runge–Kutta schemes of order 1 and two with minimal stage numbers is calculated. Further, within the considered class of stochastic Runge–Kutta schemes coefficients for optimal schemes in the sense that additionally some higher order conditions are fulfilled are presented.

Continuous weak approximation for stochastic differential equations

A convergence theorem for the continuous weak approximation of the solution of stochastic differential equations (SDEs) by general one-step methods is proved, which is an extension of a theorem due to Milstein. As an application, uniform second order conditions for a class of continuous stochastic Runge–Kutta methods containing the continuous extension of the second order stochastic Runge–Kutta scheme due to Platen are derived. Further, some coefficients for optimal continuous schemes applicable to Itô SDEs with respect to a multi–dimensional Wiener process are presented.

Rate of Convergence of Wong–Zakai Approximations for Stochastic Partial Differential Equations

We investigate the rate of convergence of the Wong-Zakai approximations for second-order stochastic PDEs of parabolic type driven by a multi-dimensional Wiener process W.

Wiener Chaos and Nonlinear Filtering

We investigate the rate of convergence of the Wong-Zakai approximations for second-order stochastic PDEs of parabolic type driven by a multi-dimensional Wiener process W.

Multi-colored rooted tree analysis of the weak order conditions of a stochastic Runge–Kutta family

The aim of the present paper is to give a tractable way of seeking weak order conditions of a stochastic Runge–Kutta family for stochastic differential equations with a multi-dimensional Wiener process. This is accomplished by an extension of the rooted tree analysis for ordinary Runge–Kutta methods. As a result, weak order conditions can be obtained directly from diagrams of multi-colored rooted trees. This new methodology will be of benefit when high weak order conditions need to be sought.

Rooted Tree Analysis for Order Conditions of Stochastic Runge-Kutta Methods for the Weak Approximation of Stochastic Differential Equations

A general class of stochastic Runge-Kutta methods for the weak approximation of Itô and Stratonovich stochastic differential equations with a multi-dimensional Wiener process is introduced. Colored rooted trees are used to derive an expansion of the solution process and of the approximation process calculated with the stochastic Runge-Kutta method. A theorem on general order conditions for the coefficients and the random variables of the stochastic Runge-Kutta method is proved by rooted tree analysis. This theorem can be applied for the derivation of stochastic Runge-Kutta methods converging with an arbitrarily high order.

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.

Stochastic Taylor Expansions for the Expectation of Functionals of Diffusion Processes

Stochastic Taylor expansions of the expectation of functionals applied to diffusion processes which are solutions of stochastic differential equation systems are introduced. Taylor formulas w.r.t. increments of the time are presented for both, Itô and Stratonovich stochastic differential equation systems with multi-dimensional Wiener processes. Due to the very complex formulas arising for higher order expansions, an advantageous graphical representation by coloured trees is developed. The convergence of truncated formulas is analyzed and estimates for the truncation error are calculated. Finally, the stochastic Taylor formulas based on coloured trees turn out to be a generalization of the deterministic Taylor formulas using plain trees as recommended by Butcher for the solutions of ordinary differential equations.

On the construction of finite dimensional realizations for nonlinear forward rate models

We consider interest rate models of Heath-Jarrow-Morton type where the forward rates are driven by a multidimensional Wiener process, and where the volatility structure is allowed to be a smooth functional of the present forward rate curve. In a recent paper [3], Bjork and Svensson give necessary and sufficient conditions for the existence of a finite dimensional Markovian state space realization (FDR) for such a forward rate model, and in the present paper we provide a general method for the actual construction of an FDR.

On the Existence of Finite‐Dimensional Realizations for Nonlinear Forward Rate Models

We consider interest rate models of the Heath–Jarrow–Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite‐dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite‐dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.

A Probabilistic Representation for the Solution of One Problem of Mathematical Physics

We consider a multidimensional Wiener process with a semipermeable membrane located on a given hyperplane. The paths of this process are the solutions of a stochastic differential equation, which can be regarded as a generalization of the well-known Skorokhod equation for a diffusion process in a bounded domain with boundary conditions on the boundary. We randomly change the time in this process by using an additive functional of the local-time type. As a result, we obtain a probabilistic representation for solutions of one problem of mathematical physics.

Interest Rate Dynamics and Consistent Forward Rate Curves

We consider as given an arbitrage‐free interest rate model M, and a parametrized family of forward rate curves G. We study the question as to when the given family G is consistent with the dynamics of the interest rate model M, in the sense that M actually will produce forward rate curves belonging to G. We allow the interest rate model to be driven by a multidimensional Wiener process, as well as by a marked point process, and we give necessary and sufficient conditions for consistency. As test cases, we study some popular models, obtaining both positive and negative results about consistency. We also introduce a natural exponential‐polynomial family of forward rate curves, and for this family we give necessary and sufficient conditions for the existence of consistent interest rate models with deterministic volatility functions.

Minimal realizations of interest rate models

We consider interest rate models where the forward rates are allowed to be driven by a multidimensional Wiener process as well as by a marked point process. Assuming a deterministic volatility structure, and using ideas from systems and control theory, we investigate when the input-output map generated by such a model can be realized by a finite dimensional stochastic differential equation. We give necessary and sufficient conditions, in terms of the given volatility structure, for the existence of a finite dimensional realization and we provide a formula for the determination of the dimension of a minimal realization. The abstract state space for a minimal realization is shown to have an immediate economic interpretation in terms of a minimal set of benchmark forward rates, and we give explicit formulas for bond prices in terms of the benchmark rates as well as for the computation of derivative prices.

Asymptotic distribution of some quadratic functionals of linear stochastic evolution systems

A function space asymptotic distribution of quadratic functionals induced from an unknown system is obtained in terms of a multi-dimensional Wiener process where the control is a linear transformation of the state that depends smoothly on the unknown parameters. The result is easily specialized to the asymptotic distribution of the family of random variables formed as the upper limit of the integrals of the quadratic terms is varied. The result provides a measure of the dependence of such a quadratic functional on a family of strongly consistent estimates of the unknown parameters, and in some cases it provides an interesting contrast with the case of all known parameters. In this paper, it is shown that, for some linear stochastic evolution systems, there are special feedback control laws where the variance of the asymptotic normal distribution of the average costs is less for the control law based on the estimates of the parameters than for the control law based on the true parameter values. This phenomenon does not occur if the feedback control laws are optimal stationary controls.

Singular ergodic control for multidimensional Gaussian processes

A multidimensional Wiener process is controlled by an additive process of bounded variation. A convex nonnegative function measures the cost associated with the position of the state process, and the cost of controlling is proportional to the displacement induced. We minimize a limiting time-average expected (ergodic) criterion. Under reasonable assumptions, we prove that the optimal discounted cost converges to the optimal ergodic cost. Moreover, under some additional conditions there exists a convex Lipschitz continuous function solution to the corresponding Hamilton-Jacobi-Bellman equation which provides an optimal stationary feedback control.

On the Existence of Optimal Relaxed Controls of Stochastic Partial Differential Equations

This paper is concerned with control problem of systems governed by stochastic partial differential equations, the drift and diffusion terms of which are second- and first-order differential operators, respectively. The existence of an optimal relaxed control is studied in both cases where the systems are degenerate and nondegenerate. It is shown that the higher regularity conditions on the initial state, as required in the existing results, can be dispensed with if the Wiener process is one-dimensional. Some special cases of multidimensional Wiener process are also discussed, which in particular leads to an improvement of a recent result of Bensoussan and Nisio. The method is based on an analysis of the group generated by the first-order differential operator. As an application, an existence theorem of the optimal relaxed control is proved for partially observed diffusions with correlation between the controlled states and the observation noises.

Kolmogorov equation for solutions of Cauchy problems for a class of linear evolution equations

$S Here W(t) is a multidimensional Wiener process defined on .{fL St p} with mutually independent components w(t), wk(t), Wkj(t); D1(t), D2(t) are second-order differential operators with nonrandom coefficients a(t), ak(t), akj(t ) and ~(t), ~k(t), ~kj(t) belonging to the spaces Ll([s, T]) and L2([s, t]), respectively, -. OxhOxj n 0 n Oz O.~(t)dW(t) = o~(t)dw(t)+~ ~. (t) -~ dw.(t)+ + ~, =~j(t) dw.j (t). k-~'--I k-~l j~-k

Estimation of covariance matrices of vector wiener process by minque method

In this paper of vector random process Y(t)=W(t)+ε(t) is considered. The process W(t) is of multidimensional WIENER process, ε(t) is cleanly of random process. For these processes W(t) and ε(t) the estimators of covariance matrices by MINQUE method are obtained