Skorokhod Embedding Research Articles

Martingale solutions of stochastic fractional nonlinear Schrödinger equation on a bounded interval*

This paper is concerned with stochastic fractional nonlinear Schrödinger equation, which plays a very important role in fractional nonrelativistic quantum mechanics. Due to disturbing and interacting of the fractional Laplacian operator on a bounded interval with white noise, the stochastic fractional nonlinear Schrödinger equation is too complicated to be understood. This paper would explore and analyze this stochastic fractional system. Using a suitable weighted space with some fractional operator skills, it overcame the difficulties coming from the fractional Laplacian operator on a bounded interval. Applying the tightness instead of the common compactness, and combining Prokhorov theorem with Skorokhod embedding theorem, it solved the convergence problem in the case of white noise. It finally established the existence of martingale solutions for the stochastic fractional nonlinear Schrödinger equation on a bounded interval.

Skorokhod embeddings via stochastic flows on the space of Gaussian measures

Nous proposons une nouvelle construction d’un plongement de Skorokhod: étant donnée une mesure de probabilité $\mu$ avec espérance nulle et variance finie, nous construisons un temps d’arrêt intégrable $T$ adapté à la filtration $\mathcal{F}_{t}$, tel que $W_{T}$ possède la loi $\mu$ et $W$ est un processus de Wiener standard adapté à la même filtration. Nous trouvons plusieurs conditions suffisantes pour que le temps d’arrêt $T$ soit borné ou ait des queues sous-exponentielles. En particulier, notre plongement semble assez naturel dans le cas où $\mu$ est log-concave et $T$ satisfait plusieurs estimations fortes. Notre plongement a la propriété suivante : le processus stochastique à valeur dans les mesures $\{\mu_{t}\}_{t\geq0}$, où $\mu_{t}$ est la loi de $W_{T}$ conditionnée par $\mathcal{F}_{t}$, est un processus de Markov. Compte tenu de cette propriété, nous allons considérer une famille plus générale de plongements de Skorokhod qui peuvent être construits à l’aide d’un noyau générant un flot stochastique sur l’espace des mesures. Cette famille inclut des constructions déjà existantes comme celle d’Azéma–Yor (In Séminaire de Probabilités XIII (1979) 90–115 Springer) et celle de Bass (In Séminaire de Probabilités XVII (1983) 221–224 Springer), suggérant ainsi un point de vue nouveau sur ces constructions.

Numerical approximation of irregular SDEs via Skorokhod embeddings

We provide a new algorithm for approximating the law of a one-dimensional diffusion M solving a stochastic differential equation with possibly irregular coefficients. The algorithm is based on the construction of Markov chains whose laws can be embedded into the diffusion M with a sequence of stopping times. The algorithm does not require any regularity or growth assumption; in particular it applies to SDEs with coefficients that are nowhere continuous and that grow superlinearly. We show that if the diffusion coefficient is bounded and bounded away from 0, then our algorithm has a weak convergence rate of order 1/4. Finally, we illustrate the algorithm's performance with several examples.

The maximum maximum of a martingale with given $\mathbf{n}$ marginals

We obtain bounds on the distribution of the maximum of a martingale with fixed marginals at finitely many intermediate times. The bounds are sharp and attained by a solution to $n$-marginal Skorokhod embedding problem in Ob{\l}\'oj and Spoida [An iterated Az\'ema-Yor type embedding for finitely many marginals (2013) Preprint]. It follows that their embedding maximizes the maximum among all other embeddings. Our motivating problem is superhedging lookback options under volatility uncertainty for an investor allowed to dynamically trade the underlying asset and statically trade European call options for all possible strikes and finitely-many maturities. We derive a pathwise inequality which induces the cheapest superhedging value, which extends the two-marginals pathwise inequality of Brown, Hobson and Rogers [Probab. Theory Related Fields 119 (2001) 558-578]. This inequality, proved by elementary arguments, is derived by following the stochastic control approach of Galichon, Henry-Labord\`ere and Touzi [Ann. Appl. Probab. 24 (2014) 312-336].

On the Monotonicity Principle of Optimal Skorokhod Embedding Problem

This is a continuation of our accompanying paper [SIAM J. Control Optim., 54 (2016), pp. 2174--2201] We provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established in [Beiglböck, Cox, and Huesmann, Optimal Transport and Skorokhod Embedding, preprint, 2013]. Our proof is based on the adaptation of the Monge--Kantorovich duality in our context, a delicate application of the optional cross-section theorem, and a clever conditioning argument introduced in [Beiglböck, Cox, and Huesmann, Optimal Transport and Skorokhod Embedding, preprint, 2013].

Optimal Skorokhod Embedding Under Finitely Many Marginal Constraints

The Skorokhod embedding problem aims to represent a given probability measure on the real line as the distribution of Brownian motion stopped at a chosen stopping time. In this paper, we consider an extension of the weak formulation of the optimal Skorokhod embedding problem in Beiglböck, Cox, and Huesmann [Optimal Transport and Skorokhod Embedding, Preprint, 2013] to the case of finitely many marginal constraints. Using the classical convex duality approach together with the optimal stopping theory, we establish some duality results under more general conditions than Beiglböck, Cox, and Huesmann. We also relate these results to the problem of martingale optimal transport under multiple marginal constraints.

Robust Hedging of Options on Local Time

In this paper, we focus on model-free pricing and robust hedging of options depending on the local time when one or more marginals of the underlying price process are known. By using the stochastic control approach initiated in Galichon, Henry-Labord\`ere and Touzi, we identify the optimal hedging strategies and the corresponding prices in the one-marginal case. As a by-product, we recover the property that the Vallois solutions to the Skorokhod embedding problem (SEP) maximize and minimize the expectation of any convex function of the local time. Furthermore, we extend the analysis to the two-marginal case, where we provide candidates for the optimal hedging strategies, and, we construct a new solution to the two-marginal SEP as a generalization of the Vallois embedding. Finally, a special multi-marginal case is studied, where the stopping times given by Vallois are well-ordered. In the $n$-marginal case, we solve the robust hedging problem as it essentially reduces to the one-marginal case. In the full marginal setting, we construct a remarkable Markov martingale and compute its generator explicitly. In particular, this provides a new example of fake Brownian motion.

On the Study of Processes of $$\sum (H)$$ ∑ ( H ) and $$\sum _\mathrm{s}(H)$$ ∑ s ( H ) Classes

In papers by Yor, a remarkable class $$({\varSigma })$$ of submartingales is introduced, which, up to technicalities, are submartingales $$(X_{t})_{t\ge 0}$$ whose increasing process is carried by the times t such that $$X_{t}=0$$ . These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class $$({\varSigma })$$ have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Cheridito and Platen). On the other hand, stochastic calculus has been extended to signed measures by Ruiz de Chavez (Le theoreme de Paul Levy pour des mesures signees. Seminaire de probabilites (Strasbourg). Springer, Berlin, 1984) and Beghdadi-Sakrani (Calcul stochastique pour les mesures signees. Seminaire de probabilites (Strasbourg). Springer, Berlin, 2003). In Eyi Obiang et al. (Stochastics 86(1):70–86, 2014), the authors of the present paper have extended the notion of submartingales of class $$({\varSigma })$$ to the setting of Ruiz de Chavez (1984) and Beghdadi-Sakrani (2003), giving two different classes of stochastic processes named classes $$\sum (H)$$ and $$\sum _\mathrm{s}(H)$$ where from tools of the theory of stochastic calculus for signed measures, the authors provide general frameworks and methods for dealing with processes of these classes. In this work, we first give some formulas of multiplicative decomposition for processes of these classes. Afterward, we shall establish some representation results allowing to recover any process of one of these classes from its final value and the last time it visited the origin.

Root’s barrier, viscosity solutions of obstacle problems and reflected FBSDEs

We revisit work of Rost (1976), Dupire (2005) and Cox–Wang (2013) on connections between Root’s solution of the Skorokhod embedding problem and obstacle problems. We develop an approach based on viscosity sub- and supersolutions and an accompanying comparison principle. This gives new, constructive and simple proofs of the existence and minimality properties of Root type solutions as well as their complete characterization. The approach is self-contained and covers martingale diffusions with degenerate elliptic or time-dependent volatility as well as Rost’s reversed Root barriers; it also provides insights about the dynamics of general Skorokhod embeddings by identifying them as supersolutions of certain nonlinear PDEs.

An integral equation for Root’s barrier and the generation of Brownian increments

We derive a nonlinear integral equation to calculate Root's solution of the Skorokhod embedding problem for atom-free target measures. We then use this to efficiently generate bounded time-space increments of Brownian motion and give a parabolic version of Muller's classic "Random walk over spheres" algorithm.

Skorokhod embeddings for two-sided Markov chains

Let $$(X_n :n\in \mathbb {Z})$$ be a two-sided recurrent Markov chain with fixed initial state $$X_0$$ and let $$\nu $$ be a probability measure on its state space. We give a necessary and sufficient criterion for the existence of a non-randomized time T such that $$(X_{T+n} :n\in \mathbb {Z})$$ has the law of the same Markov chain with initial distribution $$\nu $$ . In the case when our criterion is satisfied we give an explicit solution, which is also a stopping time, and study its moment properties. We show that this solution minimizes the expectation of $$\psi (T)$$ in the class of all non-negative solutions, simultaneously for all non-negative concave functions $$\psi $$ .

Finite, integrable and bounded time embeddings for diffusions

We solve the Skorokhod embedding problem (SEP) for a general time-homogeneous diffusion $X$: given a distribution $\rho$, we construct a stopping time $\tau$ such that the stopped process $X_{\tau}$ has the distribution $\rho$. Our solution method makes use of martingale representations (in a similar way to Bass (In Seminar on Probability XVII. Lecture Notes in Math. 784 (1983) 221-224 Springer) who solves the SEP for Brownian motion) and draws on law uniqueness of weak solutions of SDEs. Then we ask if there exist solutions of the SEP which are respectively finite almost surely, integrable or bounded, and when does our proposed construction have these properties. We provide conditions that guarantee existence of finite time solutions. Then, we fully characterize the distributions that can be embedded with integrable stopping times. Finally, we derive necessary, respectively sufficient, conditions under which there exists a bounded embedding.

On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale

We study the joint laws of the maximum and minimum of a continuous, uniformly integrable martingale. In particular, we give explicit martingale inequalities which provide upper and lower bounds on the joint exit probabilities of a martingale, given its terminal law. Moreover, by constructing explicit and novel solutions to the Skorokhod embedding problem, we show that these bounds are tight. Together with previous results of Azéma & Yor, Perkins, Jacka and Cox & Obłój, this allows us to completely characterise the upper and lower bounds on all possible exit/no-exit probabilities, subject to a given terminal law of the martingale. In addition, we determine some further properties of these bounds, considered as functions of the maximum and minimum.

An FBSDE approach to the Skorokhod embedding problem for Gaussian processes with non-linear drift

We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.

Integrability of solutions of the Skorokhod embedding problem for diffusions

Suppose $X$ is a time-homogeneous diffusion on an interval $I^X \subseteq {\mathbb R}$ and let $\mu$ be a probability measure on $I^X$. Then $\tau$ is a solution of the Skorokhod embedding problem (SEP) for $\mu$ in $X$ if $\tau$ is a stopping time and $X_\tau \sim \mu$. There are well-known conditions which determine whether there exists a solution of the SEP for $\mu$ in $X$. We give necessary and sufficient conditions for there to exist an integrable solution. Further, if there exists a solution ofthe SEP then there exists a minimal solution. We show that every minimal solution of the SEP has the same first moment. When $X$ is Brownian motion, there exists an integrable embedding of $\mu$ if and only if $\mu$ is centred and in $L^2$. Further,every integrable embedding is minimal. When $X$ is a general time-homogeneous diffusion the situation is more subtle. The case with drift can be reduced to the local martingale case by a change of scale. If $Y$ is a diffusion in natural scale, and if the target law is centred, then as in the Brownian case, there is an integrable embedding if the target law satisfies an integral condition. However, unlike in the Brownian case, there exist integrable embeddings of target laws which are not centred. Further, there exist integrable embeddings which are not minimal. Instead, if there exists an integrable embedding, then the set of minimal embeddings is the set of embeddings such that the mean equals a certain quantity, which we identify.

MRL order, log-concavity and an application to peacocks

We provide an equivalent log-concavity condition to the mean residual life (MRL) ordering for real-valued processes. This result, combined with classical properties of total positivity of order 2, allows to exhibit new families of integrable processes which increase in the MRL order (MRL processes). Note that MRL processes with constant mean are peacocks to which the Azéma–Yor (Skorokhod embedding) algorithm yields an explicit associated martingale.

A connection of the Brascamp-Lieb inequality with Skorokhod embedding

We reveal a connection of the Brascamp-Lieb inequality with Skorokhod embedding. Error bounds for the inequality in terms of variance are also provided.

The Generalized Shiryaev Problem and Skorokhod Embedding

In this paper we consider a connection between the famous Skorokhod embedding problem and the Shiryaev inverse problem for the first hitting time distribution of a Brownian motion: given a probability distribution, $F$, find a boundary such that the first hitting time distribution is $F$. By randomizing the initial state of the process we show that the inverse problem becomes analytically tractable. The randomization of the initial state allows us to significantly extend the class of target distributions in the case of a linear boundary and, moreover, allows us to establish a connection with the Skorokhod embedding problem.

From minimal embeddings to minimal diffusions

We show that there is a one-to-one correspondence between diffusions and the solutions of the Skorokhod Embedding Problem due to Bertoin and Le-Jan. In particular, the embedding corresponds to a minimal local martingale diffusion, which is a notion we introduce in this article. Minimality is closely related to the martingale property. A diffusion is if it minimises the expected local time at every point among all diffusions with a given distribution at an exponential time. Our approach makes explicit the connection between the boundary behaviour, the martingale property and the local time characteristics of time-homogeneous diffusions.

Maximizing functionals of the maximum in the Skorokhod embedding problem and an application to variance swaps

The Az\'{e}ma-Yor solution (resp., the Perkins solution) of the Skorokhod embedding problem has the property that it maximizes (resp., minimizes) the law of the maximum of the stopped process. We show that these constructions have a wider property in that they also maximize (and minimize) expected values for a more general class of bivariate functions $F(W_{\tau},S_{\tau})$ depending on the joint law of the stopped process and the maximum. Moreover, for monotonic functions $g$, they also maximize and minimize $\mathbb {E}[\int_0^{\tau}g(S_t)\,dt]$ amongst embeddings of $\mu$, although, perhaps surprisingly, we show that for increasing $g$ the Az\'{e}ma-Yor embedding minimizes this quantity, and the Perkins embedding maximizes it. For $g(s)=s^{-2}$ we show how these results are useful in calculating model independent bounds on the prices of variance swaps. Along the way we also consider whether $\mu_n$ converges weakly to $\mu$ is a sufficient condition for the associated Az\'{e}ma-Yor and Perkins stopping times to converge. In the case of the Az\'{e}ma-Yor embedding, if the potentials at zero also converge, then the stopping times converge almost surely, but for the Perkins embedding this need not be the case. However, under a further condition on the convergence of atoms at zero, the Perkins stopping times converge in probability (and hence converge almost surely down a subsequence).