D_∞-APPROXIMATION OF PRODUCT VARIATIONS OF TWO PARAMETER SMOOTH SEMIMARTINGALES

Let X be a two parameter smooth semimartingale and (?) be its process of the product variation. It is proved that (?) can be approximated as D_∞-limit of sums of its discrete product variations as the mesh of division tends to zero. Moreover, this result can be strengthen to yield the quasi sure convergence of sums by estimating the speed of the convergence.

Non-contact of Solutions to Stochastic Differential Equations Driven by Semimartingale with Non-Lipschitz Coefficients

A class of stochastic differential equations(SDEs) driven by semimartingale with non-Lipschitz coefficients was studied.By using Gronwall inequality,the non-confluence of solutions is proved under the general conditions.

On existence,uniqueness and convergence of multi-valued stochastic diferential equations driven by continuous semimartingales

In this paper we study the existence and uniqueness of solutions of multi-valued stochastic differen- tial equations driven by continuous semimartingales when the coefficients are stochastically Lipschitz continuous. We also show the convergence results when the random coefficients or the differentials converge.

RELATIONS BETWEEN SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY SEMIMARTINGALE WITH NON-LIPSCHITZ COEFFICIENTS

In this paper, a class of stochastic differential equations (SDEs) driven by semi-martingale with non-Lipschitz coefficients is studied. We investigate the dependence of solutions to SDEs on the initial value. To obtain a continuous version, we impose the conditions on the local characteristic of semimartingale. In this case, it gives rise to a flow of homeomorphisms if the local characteristic is compactly supported.

THE CHANGE OF VARIABLES FORMULA FOR THE LOCAL TIMES OF SEMIMARTINGALES

<正> Let (X_t) be a semimartingale. We denote by (L_t~α(X)) the local time at a of (X_t). If f is the difference of two convex functions on R, it is well known that f(X) is a semimartingale. The purpose of this note is to give the change of variables formula for the local times of semimartingales, that is, a formula expressing the local times of f(X) in terms of {L_t~α(X),α∈R}.

ESTIMATION OF THE NUISANCE PARAMETER FOR A SEMIMARTINGALE REGRESSION MODEL

A nuisance parameter is introduced to the semimartingale regression model proposed by Aalen(1980), and we construct two estimators for this nuisance parameter based on the results ofparametric estimation which were given by Mckeague (1986) using the method of sieves. Theconsistency of the estimators is also provided.

Semimartingale dynamics for a backward exchange rate process

Via a forward SDE solution(k_(t),t≥O)that captures money supply dynamics,a macroeconomic model known as the monetary model generates a backward exchange rate process(y_(t),t≥0).For any t≥0,y_(t)=k_(t)+α^(-1)μ_(t) where(μ_(t),t≥0)is a backward process andα>0 is a constant.Thus,(y_(t),t≥O)does not satisfy a conventional BSDE.Our paper proves(y_(t),t≥O)is a continuous semimartingale when restrictions on the SDE for(k_(t),t≥O)capture anti-inflationary initiatives.This new result in economic dynamics does not require the filtration to be the Brownian filtration.

Diffusion approximations for multiclass queueing networks under preemptive priority service discipline

We prove a heavy traffic limit theorem to justify diffusion approximations for multiclass queueing networks under preemptive priority service discipline and provide effective stochastic dynamical models for the systems. Such queueing networks appear typically in high-speed integrated services packet networks about telecommunication system. In the network, there is a number of packet traffic types. Each type needs a number of job classes （stages） of processing and each type of jobs is assigned the same priority rank at every station where it possibly receives service. Moreover, there is no inter-routing among different traffic types throughout the entire network.

Testing long memory based on a discretely observed process

In this paper we consider the problem of testing long memory for a continuous time process based on high frequency data. We provide two test statistics to distinguish between a semimartingale and a fractional integral process with jumps, where the integral is driven by a fractional Brownian motion with long memory. The small-sample performances of the statistics are evidenced by means of simulation studies. The real data analysis shows that the fractional integral process with jumps can capture the long memory of some financial data.

Uniform convergence rates for spot volatility estimation

This study presents the uniform convergence rate for spot volatility estimators based on delta sequences.Kernel and Fourier-based estimators are examples of this type of estimator.We also present the uniform convergence rates for kernel and Fourier-based estimators of spot volatility as applications of the main result.

Attraction and Stability for Neutral Stochastic Functional Differential Equations

The main aim of this paper is to establish several new criteria on the attractor for the solutions of neutral stochastic func- tional differential equations. A kind of ψ-function is introduced to our discussion, and some results on the attractor for the product of the ψ-function and the solutions are obtained. As a byproduct, a number of new criteria on asymptotic stability are also shown.

Stability of Doob-Meyer Decomposition Under Extended Convergence

In what follows, we consider the relation between Aldous's extended convergence and weak convergence of nitrations. We prove that, for a sequence (Xn) of J_t^n)-special semimartingales, with canonical decomposition Xn = Mn + An, if the extended convergence (Xn.Jrn)→ (X,F.) holds with a quasi-left continuous (Ft)-special semimartingale X = M + A, then, under an additional assumption of uniform integrability,we get the convergence in probability under the Skorokhod topology: Mn→M and An→A.

Affine processes under parameter uncertainty

We develop a one-dimensional notion of affine processes under parameter uncertainty,which we call nonlinear affine processes.This is done as follows:given a setof parameters for the process,we construct a corresponding nonlinear expectation on the path space of continuous processes.By a general dynamic programming principle,we link this nonlinear expectation to a variational form of the Kolmogorov equation,where the generator of a single affine process is replaced by the supremum over all corresponding generators of affine processes with parameters in.This nonlinear affine process yields a tractable model for Knightian uncertainty,especially for modelling interest rates under ambiguity.We then develop an appropriate Ito formula,the respective term-structure equations,and study the nonlinear versions of the Vasiˇcek and the Cox–Ingersoll–Ross(CIR)model.Thereafter,we introduce the nonlinear Vasicek–CIR model.This model is particularly suitable for modelling interest rates when one does not want to restrict the state space a priori and hence this approach solves the modelling issue arising with negative interest rates.

One-dimensional heat equation with discontinuous conductance

We study a second-order parabolic equation with divergence form elliptic operator,having a piecewise constant diffusion coefficient with two points of discontinuity.Such partial differential equations appear in the modelization of diffusion phenomena in medium consisting of three kinds of materials.Using probabilistic methods,we present an explicit expression of the fundamental solution under certain conditions.We also derive small-time asymptotic expansion of the PDE’s solutions in the general case.The obtained results are directly usable in applications.

A Useful Extension of It 's Formula with Applications to Optimal Stopping

Given a continuous semimartingale M = （Mt）t≥〉0 and a d-dimensional continuous process of locally bounded variation V = （V^1,……, V^d）, the multidimensional Ito Formula states that f（Mt, Vt） - f（M0, V0） = ∫[0, t] Dx0f（Ms, Vs）dMs＋∑i=1^d∫[0, t] Dxi F（Ms, Vs）dVs^i＋1/2∫[0, t] Dx0^2 f（Ms, Vs）d 〈M〉s if f（x0,……,xd） is of C^2-type with respect to x0 and of C^1-type with respect to the other arguments This formula is very useful when solving various optimal stopping problems based on Brownian motion. However, in such application the function f typically fails to satisfy the stated conditions in that its first partial derivative with respect to x0 is only absolutely continuous. We prove that the formula remains true for such functions and demonstrate its use with two examples from Mathematical Finance.

Stochastic regression and its application to hedging in finance

In this paper we investigate how to employ stochastic regression to hedge risks in finance,where the risk of a security is measured by its quadratic variation process.Mykland and Zhang used this technique to demonstrate how to reduce the risk of a given security by introducing another security.In this paper,we investigate how to further reduce the remaining unhedgable risk by adding more hedging securities.Some practical guidelines on how to choose those hedging securities in practice is also given.

Trading-flow assisted estimation of the jump activity index

Existing estimators for the jump activity index only made use of the price dynamics of assets.In this study,we incorporate trading information and propose a trading-flow-adjusted(TA)estimator for the jump activity index for pure-jump Ito semimartingales observed at high frequencies.We derive the central limit theorem of the estimator and perform simulation studies that justify the theory.The new estimator is shown to be more efficient in terms of the convergence rate as compared with the existing estimators,which use only the price information under some realistic conditions.Empirical analysis shows estimates with lower standard errors than those that do not incorporate the trading information.

DIFFUSION PROCESSES ON COMPLETE RIEMANNIAN MANIFOLDS

In this paper,a basic estimate for the conditional Riemannian Brownian motion on a complete manifold with non-negative Ricci curvature is established.Applying it to the heat kernel estimate of the operator 1/2△+b,we obtain the Aronson′s estimate for the operator 1/2△+b,which can be regarded as an extension of Peter Li and S.T.Yau's heat kernel estimate for the Laplace-Beltrami operator.

Information-based approach:Pricing of a credit risky asset in the presence of default time

We extend the information-based asset-pricing framework by Brody,Hughston&Macrina to incorporate a stochastic bankruptcy time for the writer of the asset.Our model introduces a non-defaultable cash flow Zr to be made at time T,alongside the time T of a possible bankruptcy of the writer of the asset are in line with the filtration generated by a Brownian random bridge with length v=T^T and pinning point ZT,where is a constant.Quantities Z and T are not necessarily independent.The model does not depend crucially on the interpretation of as a bankruptcy time.We derived the price process of the asset and compute the prices of associated options.The dynamics of the price process satisfy a diffusion equation.Employing the approach of P.-A.Meyer,we provide the explicit computation of the compensator of v.Leveraging special properties of the bridge process,we also provide the explicit expression of the compensator of Zr I(v,+o).The resulting conclusion highlights the totally inaccessible property of the stopping time v.This characteristic is particularly suitable for financial markets where the time of default of a writer cannot be predictable from any other signal in the system until default happens.