Minimal-Variance Hedging in Large Financial Markets: Random Fields Approach

We study a large financial market where the discounted asset prices are modeled by martingale random fields. This approach allows the treatment of both the cases of a market with a countable amount of assets and of a market with a continuum amount. We discuss conditions for these markets to be complete and we study the minimal variance hedging problem both in the case of full and partial information. An explicit representation of the minimal variance hedging portfolio is suggested. Techniques of stochastic differentiation are applied to achieve the main results. Examples of large market models with a countable number of assets are considered according to the literature and an example of market model with a continuum of assets is taken from the bond market.     

The strong predictable representation property in initially enlarged filtrations under the density hypothesis

We study the strong predictable representation property in filtrations initially enlarged with a random variable $L$. We prove that the strong predictable representation property can always be transferred to the enlarged filtration as long as the classical density hypothesis of Jacod (1985) holds. This generalizes the existing martingale representation results and does not rely on the equivalence between the conditional and the unconditional laws of $L$. Depending on the behavior of the density process at zero, different forms of martingale representation are established. The results are illustrated in the context of hedging contingent claims under insider information.     

General Linear Quadratic Optimal Stochastic Control Problem Driven by a Brownian Motion and a Poisson Random Martingale Measure with Random Coefficients

In this article, we consider a linear-quadratic optimal control problem (LQ problem) for a controlled linear stochastic differential equation driven by a multidimensional Browinan motion and a Poisson random martingale measure in the general case, where the coefficients are allowed to be predictable processes or random matrices. By the duality technique, the dual characterization of the optimal control is derived by the optimality system (so-called stochastic Hamilton system), which turns out to be a linear fully coupled forward-backward stochastic differential equation with jumps. Using a decoupling technique, the connection between the stochastic Hamilton system and the associated Riccati equation is established. As a result, the state feedback representation is obtained for the optimal control. As the coefficients for the LQ problem are random, here, the associated Riccati equation is a highly nonlinear backward stochastic differential equation (BSDE) with jumps, where the generator depends on the unknown variables K, L, and H in a quadratic way (see (5.9) herein). For the case where the generator is bounded and is linearly dependent on the unknown martingale terms L and H, the existence and uniqueness of the solution for the associated Riccati equation are established by Bellman's principle of quasi-linearization.     

Continuous local martingales and stochastic integration in UMD Banach spaces

Recently, van Neerven, Weis and the author, constructed a theory for stochastic integration of UMD Banach space valued processes. Here the authors use a (cylindrical) Brownian motion as an integrator. In this note we show how one can extend these results to the case where the integrator is an arbitrary real-valued continuous local martingale. We give several characterizations of integrability and prove a version of the Itô isometry, the Burkholder–Davis–Gundy inequality, the Itô formula and the martingale representation theorem.     

Nonparametric regression with martingale increment errors

We consider the problem of adaptive estimation of the regression function in a framework where we replace ergodicity assumptions (such as independence or mixing) by another structural assumption on the model. Namely, we propose adaptive upper bounds for kernel estimators with data-driven bandwidth (Lepski’s selection rule) in a regression model where the noise is an increment of martingale. It includes, as very particular cases, the usual i.i.d. regression and auto-regressive models. The cornerstone tool for this study is a new result for self-normalized martingales, called “stability”, which is of independent interest. In a first part, we only use the martingale increment structure of the noise. We give an adaptive upper bound using a random rate, that involves the occupation time near the estimation point. Thanks to this approach, the theoretical study of the statistical procedure is disconnected from usual ergodicity properties like mixing. Then, in a second part, we make a link with the usual minimax theory of deterministic rates. Under a β-mixing assumption on the covariates process, we prove that the random rate considered in the first part is equivalent, with large probability, to a deterministic rate which is the usual minimax adaptive one.     

Martingale-coboundary representation for a class of random fields

It is known that under some conditions, a stationary random sequence admits a representation as a sum of two sequences: one of them is a martingale difference sequence, and another one is a so-called coboundary. Such a representation can be used for proving some limit theorems by means of the martingale approximation. A multivariate version of such a decomposition is presented in the paper for a class of random fields generated by several commuting, noninvertible, probability preserving transformations In this representation, summands of mixed type appear, which behave with respect to some group of directions of the parameter space as reversed rnultiparameter martingale differences (in the sense of one of several known definitions), while they look as coboundaries relative to other directions. Applications to limit theorems will be published elsewhere. Bibliography: 14 titles.     

Stochastic Integrals of Continuous Local Martingales,I

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITǒ integral of X. Using the method of random time change systematically, in the present paper the representation problem for continuous local martingales is treated. We describe a class of martingales Y that can be represented as stochastic integral of X by probabilistic conditions. This leads to sufficient conditions for the representation property of X being true. Besides, an interesting characterization of continuous processes with independent increments is obtained. In part II. we proceed with general examples, applications to the n-dimensional case, and, in particular, to the n-dimensional time change of continuous local martingales with orthogonal components.     

Some properties of random fields related to stochastic integrals with respect to strong martingales

We investigate the properties of random fields related to stochastic intergals with respect to a two-parameter strong square-integrable martingale in the case where the martingale and the integrands depend on the limits of integration. We prove inequalities for the moments of the uniform norm and the modulus of continuity of trajectories of modifications of such fields. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 320, 2004, pp. 80–96.     

Spectral Representation of Multivariate Regularly Varying Lévy and CARMA Processes

A spectral representation for regularly varying Lévy processes with index between one and two is established and the properties of the resulting random noise are discussed in detail, giving also new insight in the L ²-case where the noise is a random orthogonal measure. This allows a spectral definition of multivariate regularly varying Lévy-driven continuous time autoregressive moving average (CARMA) processes. It is shown that they extend the well-studied case with finite second moments and coincide with definitions previously used in the infinite variance case when they apply.     

A New Martingale Representation Theorem (Discrete Time)

In this paper, we prove a new theorem concerning representation of discrete time martingales, which is based on solving a specially chosen problem of optimal control with random sequences. We establish a relationship between the solution of the Bellman equation and the martingale representation. We illustrate the results with an example of calculating a European type option.     

Utility maximization under risk constraints and incomplete information for a market with a change point

In this article, we consider an optimization problem of expected utility maximization of continuous-time trading in a financial market. This trading is constrained by a benchmark for a utility-based shortfall risk measure. The market consists of one asset whose price process is modelled by a Geometric Brownian motion where the market parameters change at a random time. The information flow is modelled by initially and progressively enlarged filtrations which represent the knowledge about the price process, the Brownian motion and the random time. We solve the maximization problem and give the optimal terminal wealth depending on these different filtrations for general utility functions by using martingale representation results for the corresponding filtration.     

Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function f, are semimartingales whose martingale terms are identified with integrals of f with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure-valued processes that we consider are solutions of stochastic differential equations in the space of L² (Ω)-valued vector measures.     

Information,no-arbitrage and completeness for asset price models with a change point

We consider a general class of continuous asset price models where the drift and the volatility functions, as well as the driving Brownian motions, change at a random time τ$\tau$. Under minimal assumptions on the random time and on the driving Brownian motions, we study the behavior of the model in all the filtrations which naturally arise in this setting, establishing martingale representation results and characterizing the validity of the NA1 and NFLVR no-arbitrage conditions.     

Stochastic Integrals of Continuous Local Martingales,II

Consider a continuous local martingale X. We say that X satisfies the representation property if any martingale Y of X can be represented as stochastic ITÔ integral of X. On the basis of part I of the present paper, in section 4 several general examples of continuous local martingales X satisfying the representation property are given: Stochastic continuous GAUSSian martingales, processes with conditionally independent increments, stopped continuous local martingales, random time change of WIENER processes, weak solutions of stochastic differential equations. Theorem 7 states that every (homogeneous) continuous strong MARKOV local martingale has the representation property. In section 5, the results of part I are applied to n-dimensional continuous local martingales and analogous representation results are obtained. In section 6, we consider an application of section 5 to the n-dimensional time change for reducing every n-dimensional continuous local martingale with orthogonal components to the WIENER process. This improves a theorem of F. B. KNIGHT and simplifies its proof considerably.     

Sur certaines martingales de Mandelbrot

We consider a non-negative martingale, defined by sums of product of non-negative random weights indexed by nodes of a Galton-Watson tree. In case the limit variable is not degenerate, we study the asymptotic behaviour at infinity of its distribution; in the contrary case, we prove that there is an associated natural martingale which converges to a non-negative random variable with infinite mean. The two limit variables satisfy the same distributional equation.     

Martingale expansion in mixed normal limit

The quasi-likelihood estimator and the Bayesian type estimator of the volatility parameter are in general asymptotically mixed normal. In case the limit is normal, the asymptotic expansion was derived by Yoshida [28] as an application of the martingale expansion. The expansion for the asymptotically mixed normal distribution is then indispensable to develop the higher-order approximation and inference for the volatility. The classical approaches in limit theorems, where the limit is a process with independent increments or a simple mixture, do not work. We present asymptotic expansion of a martingale with asymptotically mixed normal distribution. The expansion formula is expressed by the adjoint of a random symbol with coefficients described by the Malliavin calculus, differently from the standard invariance principle. Applications to a quadratic form of a diffusion process (“realized volatility”) are discussed.     

Stochastic integral representations,stochastic derivatives and minimal variance hedging

The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Lévy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

Global martingale solution for the stochastic Boussinesq system with zero dissipation

We study the two-dimensional stochastic Boussinesq system with zero dissipation and multiplicative noise. We show the existence of a martingale solution by a priori estimates using stochastic calculus, and applications of Prokhorov's, Skorokhod's, and martingale representation theorems. Due to the lack of dissipation, the proof requires higher regularity estimates, taking advantage of the structure of the nonlinear term. Moreover, we obtain the existence of the pressure term via an application of de Rham's theorem for processes.     

On the Positivity of the Stochastic Heat Equation

We study the positivity preserving properties of the heat equation with a white noise potential and random initial condition. Moreover, we find a generalized Feynman--Kac formula for the solution of the problem using methods from the white noise analysis. The initial condition can anticipate the driving white noise. We show that the solution is positive, when the random initial condition is positive. For the case of a time-dependent white noise potential, we give a special representation of the solution together with regularity results.