Skorokhod Embedding Research Articles

Pathwise inequalities for local time: Applications to Skorokhod embeddings and optimal stopping

We develop a class of pathwise inequalities of the form $H(B_t)\ge M_t+F(L_t)$, where $B_t$ is Brownian motion, $L_t$ its local time at zero and $M_t$ a local martingale. The concrete nature of the representation makes the inequality useful for a variety of applications. In this work, we use the inequalities to derive constructions and optimality results of Vallois' Skorokhod embeddings. We discuss their financial interpretation in the context of robust pricing and hedging of options written on the local time. In the final part of the paper we use the inequalities to solve a class of optimal stopping problems of the form $\sup_{\tau}\mathbb{E}[F(L_{\tau})-\int _0^{\tau}\beta(B_s) ds]$. The solution is given via a minimal solution to a system of differential equations and thus resembles the maximality principle described by Peskir. Throughout, the emphasis is placed on the novelty and simplicity of the techniques.

Binomial approximations of shortfall risk for game options

We show that the shortfall risk of binomial approximations of game (Israeli) options converges to the shortfall risk in the corresponding Black–Scholes market considering Lipschitz continuous path dependent payoffs for both discrete and continuous time cases. This results are new also for usual American style options. The paper continues and extends the study of [6] where estimates for binomial approximations of prices of game options were obtained. Our arguments rely, in particular, on strong invariance principle type approximations via the Skorokhod embedding, estimates from [6] and the existence of optimal shortfall hedging in the discrete time established in [2]. 1. Introduction. This paper deals with game (Israeli) options introduced in [5] sold in a standard securities market consisting of a nonrandom component bt representing the value of a savings account at time t with an interest rate r and of a random component St representing the stock price at time t. As usual, we view St, t > 0 as a stochastic process on a probability space (, F, P) and we assume that it generates a right continuous filtration {Ft}. The setup includes also two continuous stochastic payoff processes Xt ≥ Yt ≥ 0 adapted to the above filtration. Recall, that game contingent claim (GCC) or game option is defined as a contract between the seller and the buyer of the option such that both have the right to exercise it at any time up to a maturity date (horizon) T. If the buyer exercises the contract at time t then he receives the payment Yt, but if the seller exercises (cancels) the contract before the buyer then the latter receives Xt. The dif

A new method of normal approximation

We introduce a new version of Stein's method that reduces a large class of normal approximation problems to variance bounding exercises, thus making a connection between central limit theorems and concentration of measure. Unlike Skorokhod embeddings, the object whose variance must be bounded has an explicit formula that makes it possible to carry out the program more easily. As an application, we derive a general CLT for functions that are obtained as combinations of many local contributions, where the definition of "local" itself depends on the data. Several examples are given, including the solution to a nearest-neighbor CLT problem posed by P. Bickel.

On an Explicit Skorokhod Embedding for Spectrally Negative Lévy Processes

We present an explicit solution to the Skorokhod embedding problem for spectrally negative Lévy processes. Given a process X and a target measure μ satisfying an explicit admissibility condition we define functions φ ± such that the stopping time T=inf {t>0:X t ∈{−φ −(L t ),φ +(L t )}} induces X T ∼μ, where (L t ) is the local time in zero of X. We also treat versions of T which take into account the sign of the excursion straddling time t. We prove that our stopping times are minimal and we describe criteria under which they are integrable. We compare our solution with the one proposed by Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]). In particular, we compute explicitly the quantities introduced in Bertoin and Le Jan (Ann. Probab. 20(1):538–548, [1992]) in our setup.Our method relies on some new explicit calculations relating scale functions and the Itô excursion measure of X. More precisely, we compute the joint law of the maximum and minimum of an excursion away from 0 in terms of the scale function.

A BSDE APPROACH TO THE SKOROKHOD EMBEDDING PROBLEM FOR THE BROWNIAN MOTION WITH DRIFT

We solve Skorokhod's embedding problem for Brownian motion with linear drift (Wt + κ t)t≥0 by means of techniques of stochastic control theory. The search for a stopping time T such that the law of WT + κ T coincides with a prescribed law μ possessing the first moment is based on solutions of backward stochastic differential equations of quadratic type. This new approach generalizes an approach by Bass [3] of the classical version of Skorokhod's embedding problem using martingale representation techniques.

Symmetrization of Bernoulli

We show that an asymmetric Bernoulli random variable is symmetry resistant in the sense that any independent random variable, which when added to it produces a symmetric sum, must have a variance at least as much as itself. The main instrument is to use Skorokhod embedding to transfer the discrete problem to the realm of stochastic calculus.

Classes of measures which can be embedded in the Simple Symmetric Random Walk

We characterize the possible distributions of a stopped simple symmetric random walk $X_\tau$, where $\tau$ is a stopping time relative to the natural filtration of $(X_n)$. We prove that any probability measure on $\mathbb{Z}$ can be achieved as the law of $X_\tau$ where $\tau$ is a minimal stopping time, but the set of measures obtained under the further assumption that $(X_{n\land \tau}:n\geq 0)$ is a uniformly integrable martingale is a fractal subset of the set of all centered probability measures on $\mathbb{Z}$. This is in sharp contrast to the well-studied Brownian motion setting. We also investigate the discrete counterparts of the Chacon-Walsh (1976) and Azema-Yor (1979) embeddings and show that they lead to yet smaller sets of achievable measures. Finally, we solve explicitly the Skorokhod embedding problem constructing, for a given measure $\mu$, a minimal stopping time $\tau$ which embeds $\mu$ and which further is uniformly integrable whenever a uniformly integrable embedding of $\mu$ exists.

Optimal stopping and strong approximation theorems†

Strong approximation theorems known also as (strong) invariance principles provide uniform (in time) almost sure or in average approximations (as opposed to the convergence in distribution) in the central limit theorem type results which is done by redefining in certain ways corresponding random variables or vectors on one probability space without changing their distributions. Three methods are known at present to provide appropriate constructions and they yield also estimates of approximation errors. In this paper, we are interested in error estimates for approximations of values of Dynkin's optimal stopping games with payoff processes which are Lipschitz functionals of the Brownian motion by values of sequences of Dynkin's games with payoff processes converging to payoffs of original games. The strong approximations described above play a crucial role here but we have to face an additional substantial complication to have all stopping times defined with respect to the same filtration. The Skorokhod embedding method employed in Kifer (2006), The Annals of Applied Probability, vol. 16, pp. 984–1033, is well suited for this purpose but it does not work in the multidimensional case where another method from Berkes and Philipp (1979), The Annals of Probability, vol. 7, pp. 29–54, should be applied. The most precise quantile method of strong approximations does not seem to work for this type of problems and the question about optimality of error estimates is not clear yet. The results are new for convergence of corresponding Snell's envelopes, as well, and they are motivated, in particular, by financial mathematics applications of approximations of fair prices of American and game (Israeli) options introduced in Kifer (2000), Finance and Stochastics, vol. 4, pp. 443–463.

A unifying class of Skorokhod embeddings: connecting the Azéma–Yor and Vallois embeddings

We consider the Skorokhod embedding problem in Brownian motion. In particular, we give a solution based on the local time at zero of a variably skewed Brownian motion related to the underlying Brownian motion. Special cases of the construction include the Azéma–Yor and Vallois embeddings. In turn, the construction has an interpretation in the Chacon–Walsh framework.

An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of excursions of Markov processes. Our setting allows us to solve the Skorokhod embedding problem, in particular, for the age process of excursions of a Markov process, for diffusions and their signed age processes, for Azéma’s martingale and for Bessel processes of dimension smaller than 2. This work is a continuation and an important generalization of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110]. Our methodology is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Itô measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110] for measures with atoms.

HEAVY TRAFFIC LIMITS VIA BROWNIAN EMBEDDINGS

For theGI/GI/1 queue we show that the scaled queue size converges to reflected Brownian motion in a critical queue and converges to reflected Brownian motion with drift for a sequence of subcritical queuing models that approach a critical model. Instead of invoking the topological argument of the usual continuous-mapping approach, we give a probabilistic argument using Skorokhod embeddings in Brownian motion.

Skorokhod embeddings, minimality and non-centred target distributions

In this paper we consider the Skorokhod embedding problem for target distributions with non-zero mean. In the zero-mean case, uniform integrability provides a natural restriction on the class of embeddings, but this is no longer suitable when the target distribution is not centred. Instead we restrict our class of stopping times to those which are minimal, and we find conditions on the stopping times which are equivalent to minimality.

The law of the iterated logarithm for character ratios

Recently, Fulman developed some general connections between martingales and character ratios of a random representation of the symmetric group on transitions, and obtained a convergence rate in a central limit theorem. In this work we aim to establish the law of the iterated logarithm for character ratios. The technique is a well-known Skorokhod embedding theorem for martingales and strong approximation argument. Also, bounded martingale difference methods are used to obtain a large deviation for character ratios.

The Skorokhod embedding problem and its offspring

This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are studied in detail and their proofs are presented. A certain unification of proofs, thanks to real potential theory, is made. Some new facts which appeared in a natural way when different solutions were cross-examined, are reported. Azema and Yor's and Root's solutions are studied extensively. A possible use of the latter is suggested together with a conjecture.

An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale

A general methodology allowing to solve the Skorokhod stopping problem for positive functionals of Brownian excursions, with the help of Brownian local time, is developed. The stopping times we consider have the following form: T μ= inf{t>0 : F t⩾ϕ μ F(L t)} . As an application, the Skorokhod embedding problem for a number of functionals (F t : t⩾0) , including the age (length) and the maximum (height) of excursions, is solved. Explicit formulae for the corresponding stopping times T μ , such that F T μ ∼ μ, are given. It is shown that the function ϕ μ F is the same for the maximum and for the age, ϕ μ = ψ μ −1, where ψ μ(x)= ∫ [0, x] (y/ μ ̄ (y)) dμ(y) . The joint law of ( g T μ , T μ , L T μ ), in the case of the age functional, is characterized. Examples for specific measures μ are discussed. Finally, a randomized solution to the embedding problem for Azéma martingale is deduced. Throughout the article, two possible approaches, using excursions and martingale theories, are presented in parallel.

The rate of convergence of the binomial tree scheme

Abstract. We study the detailed convergence of the binomial tree scheme. It is known that the scheme is first order. We find the exact constants, and show it is possible to modify Richardson extrapolation to get a method of order three-halves. We see that the delta, used in hedging, converges at the same rate. We analyze this by first embedding the tree scheme in the Black-Scholes diffusion model by means of Skorokhod embedding. We remark that this technique applies to much more general cases.

The minimum maximum of a continuous martingale with given initial and terminal laws

Let $(M_t)_{0 \leq t \leq 1} be a continuous martingale with initial law $M_0 \sim \mu_0$, and terminal law $M_1 \sim \mu_1$, and let $S = \sup_{0 \leq t \leq 1} M_t$. In this paper we prove that there exists a greatest lower bound with respect to stochastic ordering of probability measures, on the law of $S$. We give an explicit construction of this bound. Furthermore a martingale is constructed which attains this minimum by solving a Skorokhod embedding problem. The form of this martingale is motivated by a simple picture. The result is applied to the robust hedging of a forward start digital option.

A solution to Skorokhod's embedding for linear Brownian motion and its local time

A solution to Skorokhod's embedding for linear Brownian motion and its local time

The Azéma–Yor embedding in non-singular diffusions

Let ( X t ) t⩾0 be a non-singular (not necessarily recurrent) diffusion on R starting at zero, and let ν be a probability measure on R. Necessary and sufficient conditions are established for ν to admit the existence of a stopping time τ ∗ of ( X t ) solving the Skorokhod embedding problem, i.e. X τ ∗ has the law ν. Furthermore, an explicit construction of τ ∗ is carried out which reduces to the Azéma–Yor construction (Séminaire de Probabilités XIII, Lecture Notes in Mathematics, Vol. 721, Springer, Berlin, p. 90) when the process is a recurrent diffusion. In addition, this τ ∗ is characterized uniquely to be a pointwise smallest possible embedding that stochastically maximizes (minimizes) the maximum (minimum) process of ( X t ) up to the time of stopping.

Tightness of localization and return time in random environment

Consider a class of diffusions with random potentials which behave asymptotically as Brownian motion. We study the tightness of localization around the bottom of some Brownian valley, and determine the limit distribution of the return time to the origin after a typical time. Via the Skorokhod embedding in random environment, we also solve the return time problem for Sinai's walk.