Professor V. Lakshmikantham,Editor Stochastic Analysis and Applications (1983–2012)

We use the stochastic calculus of variations for the fractional Brownian motion to derive formulas for the replicating portfolios for a class of contingent claims in a Bachelier and a Black–Scholes markets modulated by fractional Brownian motion. An example of such a model is the Black–Scholes process whose volatility solves a stochastic differential equation driven by a fractional Brownian motion that may depend on the underlying Brownian motion.     

Conditioned stochastic differential equations: theory, examples and application to finance

We generalize the notion of Brownian bridge. More precisely, we study a standard Brownian motion for which a certain functional is conditioned to follow a given law. Such processes appear as weak solutions of stochastic differential equations that we call conditioned stochastic differential equations. The link with the theory of initial enlargement of filtration is made and after a general presentation several examples are studied: the conditioning of a standard Brownian motion (and more generally of a Markov diffusion) by its value at a given date, the conditioning of a geometric Brownian motion with negative drift by its quadratic variation and finally the conditioning of a standard Brownian motion by its first hitting time of a given level. As an application, we introduce the notion of weak information on a complete market, and we give a “quantitative” value to this weak information.     

Small and Large Scale Behavior of the Poissonized Telecom Process

The stable Telecom process has infinite variance and appears as a limit of renormalized renewal reward processes. We study its Poissonized version where the infinite variance stable measure is replaced by a Poisson point measure. We show that this Poissonized version converges to the stable Telecom process at small scales and to the Gaussian fractional Brownian motion at large scales. This process is therefore locally as well as asymptotically self-similar. The value of the self-similarity parameter at large scales, namely the self-similarity parameter of the limit fractional Brownian motion, depends on the form the Poissonized Telecom process. The Poissonized Telecom process is a Poissonized mixed moving average. We investigate more general Poissonized mixed moving averages as well.     

On the most visited sites by a symmetric stable process

Summary. At time t, the most visited site of a linear Brownian motion is defined as the point which realises the supremum of the local times at time t. Let V be the time indexed process of the most visited sites by a linear Brownian motion. We show that every value is polar for V. Those results are extended from Brownian motion to symmetric stable processes, and then to the absolute value of a symmetric stable process. Received: 1 March 1996 / In revised form: 17 October 1996     

On valuing and hedging European options when volatility is estimated directly

We quantify the effects on contingent claim valuation of using an estimator for the unknown volatility σ of a geometric Brownian motion (GBM) process. The theme of the paper is to show what difficulties can arise when failing to account for estimation risk. Our narrative uses a direct estimator of volatility based on the sample standard deviation of increments of the underlying Brownian motion. After replacing the direct estimator into the GBM, we derive the resulting distribution function of the approximated GBM for any time point. This allows us to present post-estimation distributions and valuation formulae for an assortment of European contingent claims that are in accord with many of the basic properties of the underlying risk-neutral process, and yet better reflect the additional uncertainties and risks that exist in the Black-Scholes-Merton paradigm.     

Linear-quadratic optimal control under non-Markovian switching

We study a finite-dimensional continuous-time optimal control problem on finite horizon for a controlled diffusion driven by Brownian motion, in the linear-quadratic case. We admit stochastic coefficients, possibly depending on an underlying independent marked point process, so that our model is general enough to include controlled switching systems where the switching mechanism is not required to be Markovian. The problem is solved by means of a Riccati equation, which turned out to be a backward stochastic differential equation driven by the Brownian motion and by the random measure associated with the marked point process.     

Harmonic functions on Walsh’s Brownian motion

We examine a variation of two-dimensional Brownian motion introduced by Walsh that can be described as Brownian motion on the spokes of a (rimless) bicycle wheel. We construct the process by randomly assigning angles to excursions of reflecting Brownian motion. Hence, Walsh’s Brownian motion behaves like one-dimensional Brownian motion away from the origin, but differently at the origin as it is immediately sent off in random directions. Given the similarity, we characterize harmonic functions as linear functions on the rays satisfying a slope-averaging property. We also classify superharmonic functions as concave functions on the rays satisfying extra conditions.     

Modeling network traffic by a cluster Poisson input process with heavy and light-tailed file sizes

We consider a cluster Poisson model with heavy-tailed interarrival times and cluster sizes as a generalization of an infinite source Poisson model where the file sizes have a regularly varying tail distribution function or a finite second moment. One result is that this model reflects long-range dependence of teletraffic data. We show that depending on the heaviness of the file sizes, the interarrival times and the cluster sizes we have to distinguish different growths rates for the time scale of the cumulative traffic. The mean corrected cumulative input process converges to a fractional Brownian motion in the fast growth case. However, in the intermediate and the slow growth case we can have convergence to a stable Lévy motion or a fractional Brownian motion as well depending on the heaviness of the underlying distributions. These results are contrary to the idea that cumulative broadband network traffic converges in the slow growth case to a stable process. Furthermore, we derive the asymptotic behavior of the cluster Poisson point process which models the arrival times of data packets and the individual input process itself.     

Distribution of functionals of Brownian motion stopped at a time that is inverse to local time

Integral functionals of Brownian motion and of Brownian local time, as well as the supremum of Brownian motion and the supremum of Brownian local time are considered. The obtained results allow the computation of the distributions of these functionals for a Brownian motion stopped at the moment when the local time attains first a given value at one of two levels. It has been established that for this stopping time the Brownian local time is a Markov process with respect to the space variable and the generating operator of the process has been found. Examples of the computation of the distributions of certain functionals are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 184, pp. 37–61, 1990.     

Can properly discounted projects follow geometric Brownian motion?

The geometric Brownian motion is routinely used as a dynamic model of underlying project value in real option analysis, perhaps for reasons of analytic tractability. By characterizing a stochastic state variable of future cash flows, this paper considers how transformations between a state variable and cash flows are related to project volatility and drift, and specifies necessary and sufficient conditions for project volatility and drift to be time-varying, a topic that is important for real option analysis because project value and its fluctuation can only seldom be estimated from data. This study also shows how fixed costs can cause project volatility to be mean-reverting. We conclude that the conditions of geometric Brownian motion can only rarely be met, and therefore real option analysis should be based on models of cash flow factors rather than a direct model of project value.     

Self-decomposability of weak variance generalised gamma convolutions

Weak variance generalised gamma convolution processes are multivariate Brownian motions weakly subordinated by multivariate Thorin subordinators. Within this class, we extend a result from strong to weak subordination that a driftless Brownian motion gives rise to a self-decomposable process. Under moment conditions on the underlying Thorin measure, we show that this condition is also necessary. We apply our results to some prominent processes such as the weak variance alpha–gamma process, and illustrate the necessity of our moment conditions in some cases.     

A Set-indexed Fractional Brownian Motion

We define and prove the existence of a fractional Brownian motion indexed by a collection of closed subsets of a measure space. This process is a generalization of the set-indexed Brownian motion, when the condition of independance is relaxed. Relations with the Lévy fractional Brownian motion and with the fractional Brownian sheet are studied. We prove stationarity of the increments and a property of self-similarity with respect to the action of solid motions. Moreover, we show that there no “really nice” set indexed fractional Brownian motion other than set-indexed Brownian motion. Finally, behavior of the set-indexed fractional Brownian motion along increasing paths is analysed.      

Integral Representation with Adapted Continuous Integrand with Respect to Fractional Brownian Motion

We show that if a random variable is a final value of an adapted Hölder continuous process, then it can be represented as a stochastic integral with respect to fractional Brownian motion, and the integrand is an adapted process, continuous up to the final point.     

Laplace approximation of transition densities posed as Brownian expectations

We construct the Laplace approximation of the Lebesgue density for a discrete partial observation of a multi-dimensional stochastic differential equation. This approximation may be computed integrating systems of ordinary differential equations. The construction of the Laplace approximation begins with the definition of the point of minimum energy. We show how such a point can be defined in the Cameron–Martin space as a maximum a posteriori estimate of the underlying Brownian motion given the observation of a finite-dimensional functional. The definition of the MAP estimator is possible via a renormalization of the densities of piecewise linear approximations of the Brownian motion. Using the renormalized Brownian density the Laplace approximation of the integral over all Brownian paths can be defined. The developed theory provides a method for performing approximate maximum likelihood estimation.     

标的资产服从几何分数布朗运动的期权定价

本文在M ogens B ladt和T ina H av iid R ydberg无市场假设,仅利用价格过程的实际概率的期权保险精算定价模型的基础上,得出了标的资产服从几何分数布朗运动的欧式期权定价公式,并说明了几何布朗运动是本文的一种特殊情况.     

Stationary distributions for diffusions with inert drift

Consider reflecting Brownian motion in a bounded domain in ${\mathbb R^d}$ that acquires drift in proportion to the amount of local time spent on the boundary of the domain. We show that the stationary distribution for the joint law of the position of the reflecting Brownian motion and the value of the drift vector has a product form. Moreover, the first component is uniformly distributed on the domain, and the second component has a Gaussian distribution. We also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential.     

Perturbed Brownian motions

Summary. We study `perturbed Brownian motions', that can be, loosely speaking, described as follows: they behave exactly as linear Brownian motion except when they hit their past maximum or/and maximum where they get an extra `push'. We define with no restrictions on the perturbation parameters a process which has this property and show that its law is unique within a certain `natural class' of processes. In the case where both perturbations (at the maximum and at the minimum) are self-repelling, we show that in fact, more is true: Such a process can almost surely be constructed from Brownian paths by a one-to-one measurable transformation. This generalizes some results of Carmona-Petit-Yor and Davis. We also derive some fine properties of perturbed Brownian motions (Hausdorff dimension of points of monotonicity for example). Received: 17 May 1996 / In revised form: 21 January 1997     

A Further Study of the Choice Between Two Hedging Strategies–the Continuous Case

In our previous work, the choice between two popular hedging strategies was studied under the assumption that the hedge position of the underlying portfolio follows a discrete-time Markov chain with boundary conditions. This paper aims to investigate the same problem for the continuous case. We first assume that the underlying hedge position follows an arbitrary continuous-time Markov process; we give the general formulas for long-run cost per unit time under two cost structures: (1) a fixed transaction cost (2) a non-fixed transaction cost. Then we consider the case where the underlying hedge position follows a Brownian motion with drift; we show that (i) re-balancing the hedge position to the initial position is always more cost-efficient than re-balancing it to the boundary for a fixed transaction cost; (ii) when the cost function satisfies certain conditions, re-balancing the hedge position to the initial position is more cost-efficient than re-balancing it to the boundary for a non-fixed transaction cost.     

Chung's law for integrated Brownian motion

The small ball problem for the integrated process of a real-valued Brownian motion is solved. In sharp contrast to more standard methods, our approach relies on the sample path properties of Brownian motion together with facts about local times and Lévy processes.

     

Applications of the continuous-time ballot theorem to Brownian motion and related processes

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to first hit λ>0 at time t is not affected by conditioning the Brownian motion to stay below a line segment from (0,c) to (t,λ). We extend a result of Bertoin by showing that the length of the first excursion below the maximum for a negative Brownian excursion plus drift is a size-biased pick from all of the excursion lengths, and we describe the law of a negative Brownian excursion plus drift after this first excursion. We then use the same methods to prove similar results for the excursions of more general Markov processes.