Skorokhod Embedding Research Articles

Some perpetual integral functionals of the three-dimensional Bessel process

We compute the Laplace transforms of some integral functionals of the three-dimensional Bessel process in terms of modified Bessel functions, Gauss’ hypergeometric functions, and confluent hypergeometric functions. Some new results are obtained, and several established results, such as Dufresne’s perpetuity and a particular case of its translated version, are recovered. In particular, we derive the Laplace transform of the two-dimensional random variable [Formula: see text] where [Formula: see text] is a three-dimensional Bessel process and [Formula: see text]. A crucial step of the derivation is to construct a martingale, as performed in the Bass solution of the Skorokhod embedding problem, with respect to an enlarged filtration.

On the continuity of the root barrier

We show that the barrier function in Root’s solution to the Skorokhod embedding problem is continuous and finite at every point where the target measure has no atom and its absolutely continuous part is locally bounded away from zero.

Approximation of martingale couplings on the line in the adapted weak topology

Our main result is to establish stability of martingale couplings: suppose that pi is a martingale coupling with marginals mu , nu . Then, given approximating marginal measures tilde{mu }approx mu , tilde{nu }approx nu in convex order, we show that there exists an approximating martingale coupling tilde{pi }approx pi with marginals tilde{mu }, tilde{nu }. In mathematical finance, prices of European call/put option yield information on the marginal measures of the arbitrage free pricing measures. The above result asserts that small variations of call/put prices lead only to small variations on the level of arbitrage free pricing measures. While these facts have been anticipated for some time, the actual proof requires somewhat intricate stability results for the adapted Wasserstein distance. Notably the result has consequences for several related problems. Specifically, it is relevant for numerical approximations, it leads to a new proof of the monotonicity principle of martingale optimal transport and it implies stability of weak martingale optimal transport as well as optimal Skorokhod embedding. On the mathematical finance side this yields continuity of the robust pricing problem for exotic options and VIX options with respect to market data. These applications will be detailed in two companion papers.

The potential of the shadow measure

It is well known that given two probability measures μ and ν on R in convex order there exists a discrete-time martingale with these marginals. Several solutions are known (for example from the literature on the Skorokhod embedding problem in Brownian motion). But, if we add a requirement that the martingale should minimise the expected value of some functional of its starting and finishing positions then the problem becomes more difficult. Beiglböck and Juillet (Ann. Probab. 44 (2016) 42–106) introduced the shadow measure which induces a family of martingale couplings, and solves the optimal martingale transport problem for a class of bivariate objective functions. In this article we extend their (existence and uniqueness) results by providing an explicit construction of the shadow measure and, as an application, give a simple proof of its associativity.

Fine properties of the optimal Skorokhod embedding problem

We study the problem of stopping a Brownian motion at a given distribution \nu while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set \mathcal{T}(\nu) of stopping times embedding \nu is weakly dense in the set \mathcal{R}(\nu) of randomized embeddings. In particular, the optimal Skorokhod embedding problem over \mathcal{T}(\nu) has the same value as the relaxed one over \mathcal{R}(\nu) when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem that exploits a novel compactness property and yields existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglböck, Cox and Huesmann [Optimal transport and Skorokhod embedding, Invent. Math. 208, 327–400 (2017)]. We show that these results can be applied to characterize the geometry of optimal embeddings through a variational condition.

A free boundary characterisation of the Root barrier for Markov processes

We study the existence, optimality, and construction of non-randomised stopping times that solve the Skorokhod embedding problem (SEP) for Markov processes which satisfy a duality assumption. These stopping times are hitting times of space-time subsets, so-called Root barriers. Our main result is, besides the existence and optimality, a potential-theoretic characterisation of this Root barrier as a free boundary. If the generator of the Markov process is sufficiently regular, this reduces to an obstacle PDE that has the Root barrier as free boundary and thereby generalises previous results from one-dimensional diffusions to Markov processes. However, our characterisation always applies and allows, at least in principle, to compute the Root barrier by dynamic programming, even when the well-posedness of the informally associated obstacle PDE is not clear. Finally, we demonstrate the flexibility of our method by replacing time by an additive functional in Root’s construction. Already for multi-dimensional Brownian motion this leads to new class of constructive solutions of (SEP).

Model-independent pricing with insider information: a skorokhod embedding approach

AbstractIn this paper we consider the pricing and hedging of financial derivatives in a model-independent setting, for a trader with additional information, or beliefs, on the evolution of asset prices. In particular, we suppose that the trader wants to act in a way which is independent of any modelling assumptions, but that she observes market information in the form of the prices of vanilla call options on the asset. We also assume that both the payoff of the derivative, and the insider’s information or beliefs, which take the form of a set of impossible paths, are time-invariant. In this way we accommodate drawdown constraints, as well as information/beliefs on quadratic variation or on the levels hit by asset prices. Our setup allows us to adapt recent work of [12] to prove duality results and a monotonicity principle. This enables us to determine geometric properties of the optimal models. Moreover, for specific types of information, we provide simple conditions for the existence of consistent models for the informed agent. Finally, we provide an example where our framework allows us to compute the impact of the information on the agent’s pricing bounds.

On the convergence of carathéodory numerical scheme for Mckean-Vlasov equations

We study the strong convergence of the Carathéodory numerical scheme for a class of nonlinear McKean-Vlasov stochastic differential equations (MVSDE). We prove, under Lipschitz assumptions, the convergence of the approximate solutions to the unique solution of the MVSDE. Moreover, we show that the result remains valid, under continuous coefficients, provided that pathwise uniqueness holds. The proof is based on weak convergence techniques and the Skorokhod embedding theorem. In particular, this general result allows us to construct the unique strong solution of a MVSDE by using the Carathéodory numerical scheme. Examples under which pathwise uniqueness holds are given.

Mean square rate of convergence for random walk approximation of forward-backward SDEs

AbstractLet (Y,Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk$B^n$from the underlying Brownian motionBby Skorokhod embedding, one can show$L_2$-convergence of the corresponding solutions$(Y^n,Z^n)$to$(Y, Z).$We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in$C^{2,\alpha}$. The proof relies on an approximative representation of$Z^n$and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

The CLT in high dimensions: Quantitative bounds via martingale embedding

We introduce a new method for obtaining quantitative convergence rates for the central limit theorem (CLT) in a high-dimensional setting. Using our method, we obtain several new bounds for convergence in transportation distance and entropy, and in particular: (a) We improve the best known bound, obtained by the third named author (Probab. Theory Related Fields 170 (2018) 821–845), for convergence in quadratic Wasserstein transportation distance for bounded random vectors; (b) we derive the first nonasymptotic convergence rate for the entropic CLT in arbitrary dimension, for general log-concave random vectors (this adds to (Ann. Inst. Henri Poincaré Probab. Stat. 55 (2019) 777–790), where a finite Fisher information is assumed); (c) we give an improved bound for convergence in transportation distance under a log-concavity assumption and improvements for both metrics under the assumption of strong log-concavity. Our method is based on martingale embeddings and specifically on the Skorokhod embedding constructed in (Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016) 1259–1280).

A new solution to the conformal Skorokhod embedding problem and applications to the Dirichlet eigenvalue problem

In a recent work by Gross, the following problem was posed and solved: given a measure μ on R with finite second moment, find a simply connected domain U in C such that the real part of the random variable ZτU has the distribution μ, where Z is a planar Brownian motion and τU is the exit time from U. The construction developed by Gross yields a domain which is symmetric with respect to the real axis, but it has been noted by other authors that other domains are also possible, in particular there are a number of examples which have the property that a vertical ray starting at a point in the domain lies entirely within the domain. In this paper we give a new solution to the problem posed by Gross, and show that these other cases noted before are special cases of this method. We further show, following a method due to Mariano and Panzo, that the domain generated by this method has the property that it always has the minimal rate (as defined in terms of the spectrum of the Laplacian operator) among all possible domains corresponding to a fixed distribution μ. This gives a partial solution to a question posed by Mariano and Panzo. We show that the domain is unique, provided certain conditions are imposed, and use this to give several examples. We also describe a method for identifying the boundary curve of the domain, and discuss several other related topics.

An explicit solution to the Skorokhod embedding problem for double exponential increments

Strong approximations of uniform transport processes to the standard Brownian motion rely on the Skorokhod embedding of random walk with centered double exponential increments. In this note we make such an embedding explicit by means of a Poissonian scheme, which both simplifies classic constructions of strong approximations of uniform transport processes (Griego, 1971) and improves their rate of strong convergence (Gorostiza and Griego, 1980). We finalize by providing an extension regarding the embedding of a random walk with asymmetric double exponential increments.

The Skorokhod embedding problem for inhomogeneous diffusions

Nous resolvons le probleme de plongement de Skorokhod pour une classe de processus stochastiques satisfaisant une equation differentielle stochastique (EDS) non homogene de la forme $\mathrm{d}A_{t}=\mu(t,A_{t})\,\mathrm{d}t+\sigma(t,A_{t})\,\mathrm{d}W_{t}$. Nous fournissons des conditions suffisantes garantissant que, pour une mesure de probabilite $\nu$ sur $\mathbb{R}$, il existe un temps d’arret borne $\tau$ et un reel $a$ tels que la solution $(A_{t})$ de l’EDS avec condition initiale $a$ satisfait $A_{\tau}\sim\nu$. Nous distinguons ici les cas ou $(A_{t})$ est une solution de l’EDS dans un sens faible ou fort. Notre construction des temps d’arret de plongement est basee sur une solution d’une EDS progressive retrograde totalement couplee. Nous utilisons la methode des decoupling fields pour verifier que l’EDSPR a une solution unique. Enfin, nous esquissons un algorithme pour mettre en pratique notre construction theorique et l’illustrons par une simulation numerique.

Transport Plans with Domain Constraints

This paper focuses on martingale optimal transport problems when the martingales are assumed to have bounded quadratic variation. First, we give a result that characterizes the existence of a probability measure satisfying some convex transport constraints in addition to having given initial and terminal marginals. Several applications are provided: martingale measures with volatility uncertainty, optimal transport with capacity constraints, and Skorokhod embedding with bounded times. Next, we extend this result to multi-marginal constraints. Finally, we consider an optimal transport problem with constraints and obtain its Kantorovich duality. A corollary of this result is a monotonicity principle which gives a geometric way of identifying the optimizer.

On the Root Solution to the Skorokhod Embedding Problem Given Full Marginals

This paper examines the Root solution of the Skorokhod embedding problem given full marginals on some compact time interval. Our results are obtained by limiting arguments based on the finitely man...

Conformal Skorokhod embeddings and related extremal problems

The conformal Skorokhod embedding problem (CSEP) is a planar variant of the classical problem where the solution is now a simply connected domain $D\subset \mathbb {C}$ whose exit time embeds a given probability distribution $\mu $ by projecting the stopped Brownian motion onto the real axis. In this paper we explore two new research directions for the CSEP by proving general bounds on the principal Dirichlet eigenvalue of a solution domain in terms of the corresponding $\mu $ and by proposing related extremal problems. Moreover, we give a new and nontrivial example of an extremal domain $\mathbb {U}$ that attains the lowest possible principal Dirichlet eigenvalue over all domains solving the CSEP for the uniform distribution on $[-1,1]$. Remarkably, the boundary of $\mathbb {U}$ is related to the Grim Reaper translating solution to the curve shortening flow in the plane. The novel tool used in the proof of the sharp lower bound is a precise relationship between the widths of the orthogonal projections of a simply connected planar domain and the support of its harmonic measure that we develop in the paper. The upper bound relies on spectral bounds for the torsion function which have recently appeared in the literature.

The geometry of multi-marginal Skorokhod Embedding

The Skorokhod Embedding Problem is one of the classical problems in the theory of stochastic processes, with applications in many different fields [cf. the surveys (Hobson in: Paris-Princeton lectures on mathematical finance 2010, Volume 2003 of Lecture Notes in Mathematics, Springer, Berlin, 2011; Obłój in: Probab Surv 1:321–390, 2004)]. Many of these applications have natural multi-marginal extensions leading to the (optimal) multi-marginal Skorokhod problem. Some of the first papers to consider this problem are Brown et al. (Probab Theory Relat Fields 119(4):558–578, 2001), Hobson (Séminaire de Probabilités, XXXII, Volume 1686 of Lecture Notes in Mathematics, Springer, Berlin, 1998), Madan and Yor (Bernoulli 8(4):509–536, 2002). However, this turns out to be difficult using existing techniques: only recently a complete solution was be obtained in Cox et al. (Probab Theory Relat Fields 173:211–259, 2018) establishing an extension of the Root construction, while other instances are only partially answered or remain wide open. In this paper, we extend the theory developed in Beiglböck et al. (Invent Math 208(2):327–400, 2017) to the multi-marginal setup which is comparable to the extension of the optimal transport problem to the multi-marginal optimal transport problem. As for the one-marginal case, this viewpoint turns out to be very powerful. In particular, we are able to show that all classical optimal embeddings have natural multi-marginal counterparts. Notably these different constructions are linked through a joint geometric structure and the classical solutions are recovered as particular cases. Moreover, our results also have consequences for the study of the martingale transport problem as well as the peacock problem.

On Skorokhod embeddings and Poisson equations

The classical Skorokhod embedding problem for a Brownian motion $W$ asks to find a stopping time $\tau $ so that $W_{\tau }$ is distributed according to a prescribed probability distribution $\mu $. Many solutions have been proposed during the past 50 years and applications in different fields emerged. This article deals with a generalized Skorokhod embedding problem (SEP): Let $X$ be a Markov process with initial marginal distribution $\mu_{0}$ and let $\mu_{1}$ be a probability measure. The task is to find a stopping time $\tau $ such that $X_{\tau }$ is distributed according to $\mu_{1}$. More precisely, we study the question of deciding if a finite mean solution to the SEP can exist for given $\mu_{0}$, $\mu_{1}$ and the task of giving a solution which is as explicit as possible. If $\mu_{0}$ and $\mu_{1}$ have positive densities $h_{0}$ and $h_{1}$ and the generator $\mathcal{A}$ of $X$ has a formal adjoint operator $\mathcal{A}^{*}$, then we propose necessary and sufficient conditions for the existence of an embedding in terms of the Poisson equation $\mathcal{A}^{*}H=h_{1}-h_{0}$ and give a fairly explicit construction of the stopping time using the solution of the Poisson equation. For the class of Lévy processes, we carry out the procedure and extend a result of Bertoin and Le Jan to Lévy processes without local times.

Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type

Abstract In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic nonlocal Cahn–Hilliard–Navier–Stokes system driven by a pure jump noise in both 2D and 3D bounded domains. Our goal is achieved by using the classical Faedo–Galerkin approximation, a compactness method and a version of the Skorokhod embedding theorem for nonmetric spaces. In the 2D case, we prove the pathwise uniqueness of the solution and use the Yamada–Watanabe classical result to derive the existence of a strong solution.

PDE methods for optimal Skorokhod embeddings

We consider cost minimizing stopping time solutions to Skorokhod embedding problems, which deal with transporting a source probability measure to a given target measure through a stopped Brownian process. PDEs and a free boundary problem approach are used to address the problem in general dimensions with space–time inhomogeneous costs given by Lagrangian integrals along the paths. We introduce an Eulerian—mass flow—formulation of the problem, whose dual is given by Hamilton–Jacobi–Bellman type variational inequalities. Our key result is the existence (in a Sobolev class) of optimizers for this new dual problem, which in turn determines a free boundary, where the optimal Skorokhod transport drops the mass in space–time. This complements and provides a constructive PDE alternative to recent results of Beiglbock, Cox, and Huesmann, and is a first step towards developing a general optimal mass transport theory involving mean field interactions and noise.