Showing papers on "Semimartingale published in 1998"

Asymptotic error distributions for the Euler method for stochastic differential equations

Abstract: We are interested in the rate of convergence of the Euler scheme approximation of the solution to a stochastic differential equation driven by a general (possibly discontinuous) semimartingale, and by the asymptotic behavior of the associated normalized error. It is well known that for Ito’s equations the rate is $1/ \sqrt{n}$; we provide a necessary and sufficient condition for this rate to be $1/ \sqrt{n}$ when the driving semimartingale is a continuous martingale, or a continuous semimartingale under a mild additional assumption; we also prove that in these cases the normalized error processes converge in law. The rate can also differ from $1/ \sqrt{n}$: this is the case for instance if the driving process is deterministic, or if it is a Levy process without a Brownian component. It is again $1/ \sqrt{n}$ when the driving process is Levy with a nonvanishing Brownian component, but then the normalized error processes converge in law in the finite-dimensional sense only, while the discretized normalized error processes converge in law in the Skorohod sense, and the limit is given an explicit form.

Mean‐Variance Hedging and Numéraire

Abstract: We consider the mean-variance hedging problem when the risky assets price process is a continuous semimartingale. The usual approach deals with self-financed portfolios with respect to the primitive assets family. By adding a numeraire as an asset to trade in, we show how self-financed portfolios may be expressed with respect to this extended assets family, without changing the set of attainable contingent claims. We introduce the hedging numeraire and relate it to the variance-optimal martingale measure. Using this numeraire both as a deflator and to extend the primitive assets family, we are able to transform the original mean-variance hedging problem into an equivalent and simpler one; this transformed quadratic optimization problem is solved by the Galtchouk–Kunita–Watanabe projection theorem under a martingale measure for the hedging numeraire extended assets family. This gives in turn an explicit description of the optimal hedging strategy for the original mean-variance hedging problem.

Asymptotic arbitrage in large financial markets

Abstract: A large financial market is described by a sequence of standard general models of continuous trading. It turns out that the absence of asymptotic arbitrage of the first kind is equivalent to the contiguity of sequence of objective probabilities with respect to the sequence of upper envelopes of equivalent martingale measures, while absence of asymptotic arbitrage of the second kind is equivalent to the contiguity of the sequence of lower envelopes of equivalent martingale measures with respect to the sequence of objective probabilities. We express criteria of contiguity in terms of the Hellinger processes. As examples, we study a large market with asset prices given by linear stochastic equations which may have random volatilities, the Ross Arbitrage Pricing Model, and a discrete-time model with two assets and infinite horizon. The suggested theory can be considered as a natural extension of Arbirage Pricing Theory covering the continuous as well as the discrete time case.

An invariance principle for semimartingale reflecting Brownian motions in an orthant

Abstract: Semimartingale reflecting Brownian motions in an orthant (SRBMs) are of interest in applied probability because of their role as heavy traffic approximations for open queueing networks. It is shown in this paper that a process which satisfies the definition of an SRBM, except that small random perturbations in the defining conditions are allowed, is close in distribution to an SRBM. This perturbation result is called an invariance principle by analogy with the invariance principle of Stroock and Varadhan for diffusions with boundary conditions. A crucial ingredient in the proof of this result is an oscillation inequality for solutions of a perturbed Skorokhod problem. In a subsequent paper, the invariance principle is used to give general conditions under which a heavy traffic limit theorem holds for open multiclass queueing networks.

A new look at the fundamental theorem of asset pricing

Abstract: In this paper we consider a security market whose asset price process is a vector semimartingale. The market is said to be fair if there exists an equivalent martingale measure for the price process, deflated by a numeraire asset. It is shown that the fairness of a market is invariant under the change of numeraire. As a consequence, we show that the characterization of the fairness of a market is reduced to the case where the deflated price process is bounded. In the latter case a theorem of Kreps (1981) has already solved the problem. By using a theorem of Delbaen and Schachermayer (1994) we obtain an intrinsic characterization of the fairness of a market, which is more intuitive than Kreps' theorem. It is shown that the arbitrage pricing of replicatable contingent claims is independent of the choice of numeraire and equivalent martingale measure. A sufficient condition for the fairness of a market, modeled by an Ito process, is given.

Stochastic area for Brownian motion on the Sierpinski gasket

Abstract: We construct a Levy stochastic area for Brownian motion on the Sierpinski gasket. The standard approach via Ito integrals fails because this diffusion has sample paths which are far rougher than those of semimartingales. We thus provide an example demonstrating the restrictions of the semimartingale framework. As a consequence of the existence of the area one has a stochastic calculus and can solve stochastic differential equations driven by Brownian motion on the Sierpinski gasket.

Closedness of some spaces of stochastic integrals

Abstract: We consider an R d -valued continous semimartingale (X t ) t∃[0, T] , the space of processes G p = {θ · X | θ · X is a semimartingale in S p } and the space of their terminal values G T P . We give necessary and sufficient conditions for completeness of G P in the norm ∥(θ · X)*∥ p and closedness of G T P in L p . These results are related to some problems in mathematical finance and have been given for p=2 in [DMSSS].

Spatial estimates for stochastic flows in euclidean space

Abstract: We study the behavior for large $|x|$ of Kunita-type stochastic flows $\phi(t, \omega x)$ on $R^d$, driven by continuous spatial semimartingales. For this class of flows we prove new spatial estimates for large $|x|$, under very mild regularity conditions on the driving semimartingale random field. It is expected that the results would be of interest for the theory of stochastic flows on noncompact manifolds as well as in the study of nonlinear filtering, stochastic functional and partial differential equations. Some examples and counterexamples are given.

Stochastic Functional Inclusion Driven by Semimartingale

Abstract: We define a set-valued stochastic process and an integral of a such process with respect to a semimartingale. Next we consider a stochastic integral inclusion. Using fixed point methods we prove some results on the existence of its solutions

Stability of stochastic differential equations in manifolds

Abstract: We extend the so-called topology of semimartingales to continuous semimartingales with values in a manifold and with lifetime, and prove that if the manifold is endowed with a connection ° then this topology and the topology of compact convergence in probability coincide on the set of continuous °-martingales. For the topology of manifold-valued semimartingales, we give results on differentiation with respect to a parameter for second order, Stratonovich and Ito stochastic differential equations and identify the equation solved by the derivative processes. In particular, we prove that both Stratonovich and Ito equations differentiate like equations involving smooth paths (for the Ito equation the tangent bundles must be endowed with the complete lifts of the connections on the manifolds). As applications, we prove that differentiation and antidevelopment of C1 families of semimartingales commute, and that a semimartingale with values in a tangent bundle is a martingale for the complete lift of a connection if and only if it is the derivative of a family of martingales in the manifold.

Nonstandard definition of the Stratonovich integral

Abstract: By using the relation between the Ito integral and the Stratonovich integral, a nonstandard definition of the Stratonovich integral is given.

Uniform convergence of semimartingales and minimum distance estimators

Abstract: We consider a sequence of special semimartingales depending on the parameter and give conditions, expressed in terms of the predictable characteristics of Xn,θ for the uniform in θ, weak convergence of semimartingales Xn,θ . Using results about uniform convergence we study the minimum distance estimators of θ for semimartingales

Backward self-similar stochastic processes in stochastic differential equations

Abstract: For the forward-backward semimartingale, we can define the backward semimartingale flow which is generated by the backward canonical stochastic differential equation. Therefore, we define the backward self-similar stochastic processes, and we study the backward self-similar stochastic flows through the canonical stochastic differential equations.