Omega diffusion risk model with surplus-dependent tax and capital injections

In this paper, we propose and study an Omega risk model with a constant bankruptcy function, surplus-dependent tax payments and capital injections in a time-homogeneous diffusion setting. The surplus value process is both refracted (paying tax) at its running maximum and reflected (injecting capital) at a lower constant boundary. The new model incorporates practical features from the Omega risk model (Albrecher et al., 2011), the risk model with tax (Albrecher and Hipp, 2007), and the risk model with capital injections (Albrecher and Ivanovs, 2014). The study of this new risk model is closely related to the Azéma–Yor process, which is a process refracted by its running maximum. We explicitly characterize the Laplace transform of the occupation time of an Azéma–Yor process below a constant level until the first passage time of another Azéma–Yor process or until an independent exponential time. We also consider the case when the process has a lower reflecting boundary. This result unifies and extends recent results of Li and Zhou (2013) and Zhang (2015). We explicitly characterize the Laplace transform of the time of bankruptcy in the Omega risk model with tax and capital injections up to eigen-functions, and determine the expected present value of tax payments until default. We also discuss a further extension to occupation functionals through stochastic time-change, which handles the case of a non-constant bankruptcy function. Finally we present examples using a Brownian motion with drift, and discuss the pricing of quantile options written on the Azéma–Yor process.     

Two explicit Skorokhod embeddings for simple symmetric random walk

Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centred distribution $\mu$ on integers into a simple symmetric random walk in a uniformly integrable manner. Our first construction has a simple Markovian structure: at each step, we stop if an independent coin with a state-dependent bias returns tails. Our second construction is a discrete analogue of the celebrated Azéma–Yor solution and requires independent coin tosses only when excursions away from maximum breach predefined levels. Further, this construction maximizes the distribution of the stopped running maximum among all uniformly integrable embeddings of $\mu$.     

Explicit Martingale Representations for Brownian Functionals and Applications to Option Hedging

Abstract

Using Clark-Ocone formula, explicit martingale representations for path-dependent Brownian functionals are computed. As direct consequences, explicit martingale representations of the extrema of geometric Brownian motion and explicit hedging portfolios of path-dependent options are obtained.     

Stochastic Integral Representations of the Extrema of Time-homogeneous Diffusion Processes

The stochastic integral representations (martingale representations) of square integrable processes are well-studied problems in applied probability with broad applications in finance. Yet finding explicit expression is not easy and typically done through the Clack-Ocone formula with the advanced machinery of Malliavin calculus. To find an alternative, Shiryaev and Yor (Teor Veroyatnost i Primenen 48(2):375–385, 2003) introduced a relatively simple method using Itô’s formula to develop representations for extrema of Brownian motion. In this paper, we extend their work to provide representations of functionals of time-homogeneous diffusion processes based on the Itô’s formula.     

Backward stochastic differential equations with Azéma's martingale

The aim of this paper is to study backward stochastic differential equations (BSDE) driven by Azéma's martingale and the associated deterministic functional equations. More precisely, we introduce BSDE's vs. Azéma's martingale in a general frame, then we prove that the existence of a solution to a Markovian BSDE implies the existence of a solution to a deterministic functional equation of a new type. Uniqueness for the functional equation is proved in a particular case. Then we discuss BSDE's vs. an asymmetric martingale: half Brownian motion/half Azéma's martingale, which leads to an asymmetric deterministic functional equation.     

Martingale solutions for the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes

In this paper, the three-dimensional stochastic nonhomogeneous incompressible Navier–Stokes equations driven by Lévy processes consisting of the Brownian motion, the compensated Poisson random measure and the Poisson random measure are considered in a bounded domain. We obtain the existence of martingale solutions. The construction of the solution is based on the classical Galerkin approximation method, the stopping times, the stochastic compactness method and the Jakubowski–Skorokhod theorem.     

Exponential Functionals of Brownian Motion and Explosion Times of a System of Semilinear SPDEs

We investigate lower and upper bounds for the blowup times of a system of semilinear SPDEs. Under certain conditions on the system parameters, we obtain explicit solutions of a related system of random PDEs, which allows us to use a formula due to Yor to obtain the distribution functions of several explosion times. We also give the Laplace transforms at independent exponential times of related exponential functionals of Brownian motion.     

On functional laws of the iterated logarithm

Summary A Skorokhod embedding approach is used to give functional laws of the iterated logarithm which involve the process up to timen in the reverse martingale case and the tail of the process in the martingale case. This complements the more usual versions of the iterated logarithm laws for martingales and reverse martingales.     

Some Results on Skorokhod Embedding and Robust Hedging with Local Time

In this paper, we provide some results on Skorokhod embedding with local time and its applications to the robust hedging problem in finance. First we investigate the robust hedging of options depending on the local time by using the recently introduced stochastic control approach, in order to identify the optimal hedging strategies, as well as the market models that realize the extremal no-arbitrage prices. As a by-product, the optimality of Vallois’ Skorokhod embeddings is recovered. In addition, under appropriate conditions, we derive a new solution to the two-marginal Skorokhod embedding as a generalization of the Vallois solution. It turns out from our analysis that one needs to relax the monotonicity assumption on the embedding functions in order to embed a larger class of marginal distributions. Finally, in a full-marginal setting where the stopping times given by Vallois are well ordered, we construct a remarkable Markov martingale which provides a new example of fake Brownian motion.     

Excessive kernels and Revuz measures

We consider a proper submarkovian resolvent of kernels on a Lusin measurable space and a given excessive measure ξ. With every quasi bounded excessive function we associate an excessive kernel and the corresponding Revuz measure. Every finite measure charging no ξ–polar set is such a Revuz measure, provided the hypothesis (B) of Hunt holds. Under a weak duality hypothesis, we prove the Revuz formula and characterize the quasi boundedness and the regularity in terms of Revuz measures. We improve results of Azéma [2] and Getoor and Sharpe [20] for the natural additive functionals of a Borel right process. Received: 30 April 1997 / Revised version: 17 September 1999 /?Published online: 11 April 2000     

Tanaka Formula for Multidimensional Brownian Motions

We study the Tanaka formula for multidimensional Brownian motions in the framework of generalized Wiener functionals. More precisely, we show that the submartingale U(B _t –x) is decomposed in the sence of generalized Wiener functionals into the sum of a martingale and the Brownian local time, U being twice of the kernel of Newtonian potential and B _t the multidimensional Brownian motion. We also discuss on an aspect of the Tanaka formula for multidimensional Brownian motions as the Doob–Meyer decomposition.     

Tribute to Endre Csáki and Pál Révész

Summary This article provides a glimpse of some of the highlights of the joint work of Endre Csáki and Pál Révész since 1979. The topics of this short exploration of the rich stochastic milieu of this inspiring collaboration revolve around Brownian motion, random walks and their long excursions, local times and additive functionals, iterated processes, almost sure local and global central limit theorems, integral functionals of geometric stochastic processes, favourite sites--favourite values and jump sizes for random walk and Brownian motion, random walking in a random scenery, and large void zones and occupation times for coalescing random walks.     

A trivariate law for certain processes related to perturbed Brownian motions

D. Williams' path decomposition and Pitman's representation theorem for BES(3) are expressions of some deep relations between reflecting Brownian motion and the 3-dimensional Bessel process.In [Ph. Carmona et al., Stochastic Process. Appl. 7 (1999) 323–333], we presented an attempt to relate better reflecting Brownian motion and the 2-dimensional Bessel process, using space and time changes related to the Ray–Knight theorems on local times, in the manner of Jeulin [Lect. Notes Math., vol. 1118, Springer, Berlin, 1985] and Biane–Yor [Bull. Sci. Math. 2ème Sér. 111 (1987) 23–101].Here, we characterize the law of a triplet linked to the perturbed Brownian motion which naturally arises in [Ph. Carmona et al., Stochastic Proc. Appl. 7 (1999) 323–333], and we point out its relations with Bessel processes of several dimensions.The results provide some new understanding of the generalizations of Lévy's arc sine law for perturbed Brownian motions previously obtained by the second author.     

Approximations to permutation and exchangeable processes

We obtain weighted approximations by a Brownian bridge to permutation and exchangeable processes and to appropriately defined inverse processes. Our results provide as special cases useful weighted approximations to the uniform empirical and quantile processes and to generalized bootstrapped versions of these processes. A number of other applications are discussed. Our approach is based on the Skorokhod embedding for martingales.     

A note on convergence of option prices and their Greeks for Lévy models

We study the robustness of option prices to model variation after a change of measure where the measure depends on the model choice. We consider geometric Lévy models in which the infinite activity of the small jumps is approximated by a scaled Brownian motion. For the Esscher transform, the minimal entropy martingale measure, the minimal martingale measure and the mean variance martingale measure, we show that the option prices and their corresponding deltas converge as the scaling of the Brownian motion part tends to zero. We give some examples illustrating our results.     

Non-independence of excursions of the Brownian sheet and of additive Brownian motion

A classical and important property of Brownian motion is that given its zero set, distinct excursions away from zero are independent. In this paper, we examine the analogous question for the Brownian sheet, and also for additive Brownian motion. Our main result is that given the level set of the Brownian sheet at level zero, distinct excursions of the sheet away from zero are not independent. In fact, given the zero set of the Brownian sheet in the entire non-negative quadrant, and the sign of all but a finite number of excursions away from zero, the signs of the remaining excursions are determined. For additive Brownian motion, we prove the following definitive result: given the zero set of additive Brownian motion and the sign of a single excursion, the signs of all other excursions are determined. In an appendix by John B. Walsh, it is shown that given the absolute value of the sheet in the entire quadrant and, in addition, the sign of the sheet at a fixed, non-random time point, then the whole sheet can be recovered.

     

Stochastic Navier–Stokes Equations Driven by Lévy Noise in Unbounded 3D Domains

Martingale solutions of the stochastic Navier–Stokes equations in 2D and 3D possibly unbounded domains, driven by the Lévy noise consisting of the compensated time homogeneous Poisson random measure and the Wiener process are considered. Using the classical Faedo–Galerkin approximation and the compactness method we prove existence of a martingale solution. We prove also the compactness and tightness criteria in a certain space contained in some spaces of càdlàg functions, weakly càdlàg functions and some Fréchet spaces. Moreover, we use a version of the Skorokhod Embedding Theorem for nonmetric spaces.     

On Distribution of Functionals of a Brownian Motion Stopped at the Inverse Range Time

In this paper, we develop methods for computation of distributions of functionals of a Brownian motion stopped at the inverse range time. Some explicit formulas for distributions of functionals are obtained. Bibliography: 7 titles.     

Hypersurfaces in and the variance of exit times for Brownian motion

Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.