Seminar on Stochastic Analysis, Random Fields and Applications VI

We consider the heat semigroup \(e^{-\frac{t}{\hbar}H}, t > 0,\,\, {\rm on}\,\, \mathbb{R}^d\) with generator H corresponding to a potential growing polynomially at infinity. Its trace for positive times is represented as an analytically continued infinite-dimensional oscillatory integral. The asymptotics in the small parameter _ is exhibited by using Laplace’s method in infinite dimensions in the case of a degenerate phase (this corresponds to the limit from quantum mechanics to classical mechanics, in a situation where the Euclidean action functional has a degenerate critical point).

We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence result is known. We prove, under suitable assumptions, existence of a measure-valued solution, for the corresponding Fokker–Planck equation.

Assume there is a probability space on which a time homogeneous Poisson random measure \( \eta \)and a progressively measurable process\(\xi\) are given. Let us consider the law of the triplet \((\eta, \xi, { I}),\) where \({ I}\)is the Itˆo integral of \(\xi\) with respect to \( \eta \). Then we ask the question whether the law is unique.

The derivative of the log-likelihood function, known as score function, plays a central role in parametric statistical inference. It can be used to study the asymptotic behavior of likelihood and pseudo-likelihood estimators. For instance, one can deduce the local asymptotic normality property which leads to various asymptotic properties of these estimators. In this article we apply Malliavin Calculus to obtain the score function as a conditional expectation. We then show, through different examples, how this idea can be useful for asymptotic inference of stochastic processes. In particular, we consider situations where there are jumps driving the data process.

We characterize the solution of Navier-Stokes equation as a stochastic geodesic on the diffeomorphisms group, thus generalizing Arnold’s description of the Euler flow.

Some recent results on stochastic ergodic control problems in infinite- dimensional state spaces are reviewed, a special attention being paid to the ergodic control of stochastic semilinear reaction-diffusion equations. Earlier achievements obtained in this field (as well as some of those obtained for the discounted cost optimization problem) are summarized. Some of the recently obtained results that will appear in the forthcoming paper [15] are described in more detail.

The aim of this note is to present an elementary proof of a variation of Harris’ ergodic theorem of Markov chains.

It is often important, in applications of stochastic calculus to financial modelling, to know whether a given local martingale is a martingale or a strict local martingale. We address this problem in the context of a time-homogenous diffusion process with a finite lower boundary, presented as the solution of a driftless stochastic differential equation. Our main theorem demonstrates that the question of whether or not this process is a martingale may be decided simply by examining the slope of a certain increasing function. Further results establish the connection between our theorem and other results in the literature, while a number of examples are provided to illustrate the use of our criterion.

We provide a device, called the local predictor, which extends the idea of the predictable compensator. It is shown that a fBm with the Hurst index greater than 1/2 coincides with its local predictor while fBm with the Hurst index smaller than 1/2 does not admit any local predictor.

This paper considers a problem of controlling a stochastic Lagrangian systems so as to prevent it from leaving a prescribed set. In the absence of noise, the system is asymptotically stable; weak noise induces exits from the domain of attraction of the stable equilibrium with a non-zero probability. The paper suggests a control strategy aimed at building a controlled system with exit rate asymptotically independent of noise (in the small noise limit). The analysis employs previously found explicit asymptotics of the mean exit time for stochastic Lagrangian systems. A physically meaningful example illustrates the developed methodology.

In the framework of a probabilistic deformation of the classical calculus of variations, we consider the simplest problem of constraints, and solve it in two different ways. First by a pathwise argument in the line of Euclidean Quantum Mechanics. Second from an entropic (measure theoretic) perspective.

First we review types of path measures arising from various extensions of the Feynman-Kac formula. Then we consider more closely the case of Gibbs measures on Brownian paths with respect to densities dependent on double Itô integrals. We explain the framework of stochastic currents used in order to give a sensible meaning to Gibbs specifications. Exponential integrability and DLR consistence will be established by using rough paths techniques. Finally we show the results on existence, uniqueness, typical path behaviour and mixing properties that can be derived for limit Gibbs random fields.

We derive a family of series representations of the multiparameter fractional Brownian motion in the centred ball of radius R in theNdimensional space R ^N . Some known examples of series representations are shown to be the members of the family under consideration.

Under suitable assumptions of regularity and non-degeneracy on the covariance of the driving additive noise, any Markov solution to the stochastic Navier-Stokes equations has an associated generator of the diffusion and is the unique solution to the corresponding martingale problem. Some elementary examples are discussed to interpret these results.

We give an alternative representation of the Wasserstein Dirichlet form that was introduced by von Renesse and Sturm in [7]. Based on this alternative representation we improve and generalize the Poincaré and logarithmic Sobolev inequality obtained for the Wasserstein Dirichlet form in [3]. A simple two-dimensional generalization of the Wasserstein Dirichlet form is investigated. The associated process can be interpreted as the projective limit of reflecting lines diffusions.

We construct the entropic measure \(\mathbb{P}^\beta\) on compact manifolds of any dimension. It is defined as the push forward of the Dirichlet process (a random probability measure, well-known to exist on spaces of any dimension) under the conjugation map This conjugation map is a continuous involution. It can be regarded as the canonical extension to higher-dimensional spaces of a map between probability measures on 1-dimensional spaces characterized by the fact that the distribution functions of μ and C(μ) are inverse to each other.

We also present a heuristic interpretation of the entropic measure as

We establish properties of strong local nondeterminism for several classes of α-stable random fields such as harmonizable-type fractional stable fields with stationary increments, harmonizable and linear fractional stable sheets. We apply these properties to study existence and joint continuity of the local times of stable random fields.

When managing energy or weather related risk often only imperfect hedging instruments are available. In the first part we illustrate problems arising with imperfect hedging by studying a toy model. We consider an airline’s problem with covering income risk due to fluctuating kerosine prices by investing into futures written on heating oil with closely correlated price dynamics. In the second part we outline recent results on exponential utility based cross hedging concepts. They highlight in a generalization of the Black- Scholes delta hedge formula to incomplete markets. Its derivation is based on a purely stochastic approach of utility maximization. It interprets stochastic control problems in the BSDE language, and profits from the power of the stochastic calculus of variations.

We examine the auto-dependence structure of strictly stationary solutions of linear stochastic recurrence equations and of strictly stationary GARCH(1, 1) processes from the point of view of ordinary and generalized tail dependence coefficients. Since such processes can easily be of infinite variance, a substitute for the usual auto-correlation function is needed.

The goal of this paper is to propose an equilibrium model for the joint price formation of allowances issued by regulators in the framework of a cap-and-trade scheme and offset certificates such as CERs generated within the framework of the Clean Development Mechanism (CDM) or the Joint Implemenation (JI) of the Kyoto Protocol. Thereby we consider a system of cap-and-trade schemes, such as, e.g., the EUETS and a possible American Market (USETS) or Japan ETS, which are linked indirectly by the Clean Development Mechanism and for which banking is allowed. Besides deriving equilibrium price formulas for the joint price dynamics of these linked markets, the main thrust of the paper is to explain the spreads between European emission allowances EUAs and CERs as observed historically.

Optimal investment problems in an incomplete financial market with pure jump stock dynamics are studied. An investor with Constant Relative Risk Aversion (CRRA) preferences, including the logarithmic utility, wants to maximize her/his expected utility of terminal wealth by investing in a bond and in a risky asset. The risky asset price is modeled as a geometric marked point process, whose dynamics is driven by two independent Poisson processes, describing upwards and downwards jumps. A stochastic control approach allows us to provide optimal investment strategies and closed formulas for the value functions associated to the utility optimization problems. Moreover, the solution to the dual problems associated to the utility maximization problems are derived. The case when intermediate consumption is allowed is also discussed.

This paper provides a general framework for doubly stochastic term structure models for portfolio of credits, such as collateralized debt obligations (CDOs). We introduce the defaultable (T, x)-bonds, which pay one if the aggregated loss process in the underlying pool of the CDO has not exceededx at maturityT, and zero else. Necessary and sufficient conditions on the stochastic term structure movements for the absence of arbitrage are given. Moreover, we show that any exogenous specification of the forward rates and spreads volatility curve actually yields a consistent loss process and thus an arbitrage-free family of (T, x)-bond prices. For the sake of analytical and computational efficiency we then develop a tractable class of affine term structure models.

We consider the problem of minimizing the risk of a financial position (hedging) in an incomplete market. It is well known that the industry standard for risk measure, the Value-at-Risk, does not take into account the natural idea that risk should be minimized through diversification. This observation led to the recent theory of coherent and convex risk measures. But, as a theory on bounded financial positions, it is not ideally suited for the problem of hedging because simple strategies such as buy-hold strategies may not be bounded. Therefore, we propose as an alternative to use convex risk measures defined as functionals on L ² (or by simple extension L ^p , p > 1). This framework is more suitable for optimal hedging with L ²-valued financial markets. A dual representation is given for this minimum risk or market adjusted risk when the risk measure is real valued. In the general case, we introduce constrained hedging and prove that the market adjusted risk is still a L ² convex risk measure and the existence of the optimal hedge. We illustrate the practical advantage in the shortfall risk measure by showing how minimizing risk in this framework can lead to a HJB equation and we give an example of computation in a stochastic volatility model with the shortfall risk measure

In exponential semi-martingale setting for risky asset we estimate the difference of prices of options when initial physical measure P and corresponding martingale measure Q change to ̃ P and ̃Q respectively. Then, we estimate PL 1-distance of option prices for corresponding parametric models with known and estimated parameters. The results are applied to exponential Lévy models with special choice of martingale measure as Esscher measure, minimal entropy measure and Pfq-minimal martingale measure. We illustrate our results by considering GMY and CGMY models.

In the recent years especially in finance many different models either based on semimartingales, purely continuous, pure jump and a mixture of both, or fractional Brownian motion have been proposed in the literature. We provide a class of easily computable estimators which allows to infer the fine structure of the underlying process in terms of the Blumenthal-Getoor index or the Hurst exponent based on high frequency data. This method makes it possible not only to detect jumps, but also determine their activity and the regularity of continuous components, which can be used for model selection or to analyze the market microstructure by taking into account different time scales. Furthermore, our method provides a simple graphical tool for detecting jumps.