In order to approximate a continuous time stochastic process by discrete time Markov chains one has several options to embed the Markov chains into continuous time processes. On the one hand there is the Markov embedding, which uses exponential waiting times. On the other hand each Skorokhod topology naturally suggests a certain embedding. These are the step function embedding for $J_1$, the linear interpolation embedding for $M_1$, the multi step embedding for $J_2$ and a more general embedding for $M_2$. We show that the convergence of the step function embedding in $J_1$ implies the convergence of the other embeddings in the corresponding topologies, respectively. For the converse statement a $J_1$-tightness condition for embedded Markov chains is given.
The result relies on various representations of the Skorokhod topologies. Additionally it is shown that $J_1$ convergence is equivalent to the joint convergence in $M_1$ and $J_2$.
We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of Markovian excursions. Our setting allows to solve the Skorokhod embedding problem, in particular, for diffusions and their (signed, scaled) age processes, for Azema's martingale, for spectrally one-sided Levy processes and their reflected versions, for Bessel processes of dimension smaller than 2, and for their age processes, as well as for the age process of excursions of Cox-Ingersoll-Ross processes. This work is a continuation and an important generalization of Obl…
Let $(X_n \colon n\in\Z)$ be a two-sided recurrent Markov chain with fixed initial state $X_0$ and let $ν$ 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 $(X_{T+n} \colon n\in\Z)$ has the law of the same Markov chain with initial distribution $ν$. 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 $ψ(T)$ in the class of all non-negative solutions, simulta…
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.
Motivated by problems in behavioural finance, we provide two explicit constructions of a randomized stopping time which embeds a given centered distribution $μ$ 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…
Necessary and sufficient conditions for convergence in distribution of first-rare-event times and convergence in Skorokhod J-topology of first-rare-event-time processes for perturbed semi-Markov processes with finite phase space are obtained.
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…
The classical Skorokhod embedding problem for a Brownian motion $W$ asks to find a stopping time $τ$ so that $W_τ$ is distributed according to a prescribed probability distribution $μ$. 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 $μ_0$ and let $μ_1$ be a probability measure. The task is to find a stopping time $τ$ such that $X_τ$ is distributed according to $μ_1$. More precisely, we study the questio…
The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optim…
We prove continuity of a controlled SDE solution in Skorokhod's $M_1$ and $J_1$ topologies and also uniformly, in probability, as a non-linear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that $M_1$-continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic co…
We present a new construction of a Skorohod embedding, namely, given a probability measure mu with zero expectation and finite variance, we construct an integrable stopping time T adapted to a filtration F_t, such that W_t has the law mu, where W_t is a standard Wiener process adapted to the same filtration. We find several sufficient conditions for the stopping time T to be bounded or to have a sub-exponential tail. In particular, our embedding seems rather natural for the case that mu is a log-concave measure and the tail behaviour of $T$ admits some tight bounds in that case. Our embedding …
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 fro…
Regime-switching processes contain two components: continuous component and discrete component, which can be used to describe a continuous dynamical system in a random environment. Such processes have many different properties than general diffusion processes, and much more difficulties are needed to be overcome due to the intensive interaction between continuous and discrete component. We give conditions for the existence and uniqueness of invariant measures for state-dependent regime-switching diffusion processes by constructing a new Markov chain to control the evolution of the state-depend…
When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $ξ=\sum_{j \in Λ} δ_{x_j}$, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function $Φ$ having zeros in $\supp ξ$. Using such entire functions $Φ$, we define new topologies called the $Φ$-moderate topologies. Then we construct three infinite-dimensional determinantal processes, …
We consider the problem of balancing load items (tokens) in networks. Starting with an arbitrary load distribution, we allow nodes to exchange tokens with their neighbors in each round. The goal is to achieve a distribution where all nodes have nearly the same number of tokens. For the continuous case where tokens are arbitrarily divisible, most load balancing schemes correspond to Markov chains, whose convergence is fairly well-understood in terms of their spectral gap. However, in many applications, load items cannot be divided arbitrarily, and we need to deal with the discrete case where th…
In this paper we investigate three discrete or semi-discrete approximation schemes for reflected Brownian motion on bounded Euclidean domains. For a class of bounded domains $D$ in $\mathbb{R}^n$ that includes all bounded Lipschitz domains and the von Koch snowflake domain, we show that the laws of both discrete and continuous time simple random walks on $D\cap2^{-k}\mathbb{Z}^n$ moving at the rate $2^{-2k}$ with stationary initial distribution converge weakly in the space $\mathbf{D}([0,1],\mathbb{R}^n)$, equipped with the Skorokhod topology, to the law of the stationary reflected Brownian mo…
We investigate the scaling limit of the seed bank diffusion when reproduction and migration (to and from the seed bank) happen on different time-scales. More precisely, we consider the case when migration is `slow' and reproduction is `standard' (in the original time-scale) and then switch to a new, accelerated time-scale, where migration is `standard' and reproduction is `fast'. This is motivated by models for bacterial dormancy, where periods of quiescence can be orders of magnitude larger than reproductive times, and where it is expected to find non-trivial degenerate genealogies on the evo…
