Càdlàg Functions

This chapter considers the space D of functions on the unit interval that are continuous on the right and with left limits, known as càdlàg functions. D contains and extends the space C, but is nonseparable under the uniform metric so to work with it calls for new techniques. By defining a new topology for D (the Skorokhod topology), families of measures on D can be constructed and sufficient conditions for weak convergence of partial sum processes specified.

A GENERAL ORNSTEIN–UHLENBECK STOCHASTIC VOLATILITY MODEL WITH LÉVY JUMPS

We present a general class of stochastic volatility models with jumps where the stochastic variance process follows a Lévy-driven Ornstein–Uhlenbeck (OU) process and the jumps in the log-price process follow a Lévy process. This financial market model is a true extension of the Barndorff-Nielsen–Shephard (BNS) model class and can establish a weak link between log-price jumps and volatility jumps. Furthermore, we investigate the weak-link [Formula: see text]-OU-BNS model as a special case, where we calculate the characteristic function of the logarithmic price in closed form. The classical [Formula: see text]-OU-BNS model can be obtained as a limit of weak-link [Formula: see text]-OU-BNS models in the Skorokhod topology. We highlight that the weak-link property may be a useful model extension in the case of pricing barrier options.

Bounded-Size Rules: The Barely Subcritical Regime

Bounded-size rules (BSRs) are dynamic random graph processes which incorporate limited choice along with randomness in the evolution of the system. Typically one starts with the empty graph and at each stage two edges are chosen uniformly at random. One of the two edges is then placed into the system according to a decision rule based on the sizes of the components containing the four vertices. For bounded-size rules, all components of size greater than some fixed K ≥ 1 are accorded the same treatment. Writing BSR(t) for the state of the system with ⌊nt/2⌋ edges, Spencer and Wormald [26] proved that for such rules, there exists a (rule-dependent) critical time tc such that when t < tc the size of the largest component is of order log n, while for t > tc, the size of the largest component is of order n. In this work we obtain upper bounds (that hold with high probability) of order n2γ log4n, on the size of the largest component, at time instants tn = tc−n−γ, where γ ∈ (0,1/4). This result for the barely subcritical regime forms a key ingredient in the study undertaken in [4], of the asymptotic dynamic behaviour of the process describing the vector of component sizes and associated complexity of the components for such random graph models in the critical scaling window. The proof uses a coupling of BSR processes with a certain family of inhomogeneous random graphs with vertices in the type space $\mathbb{R}_+\times \mathcal{D}([0,\infty):\mathbb{N}_0)$, where $\mathcal{D}([0,\infty):\mathbb{N}_0)$ is the Skorokhod D-space of functions that are right continuous and have left limits, with values in the space of non-negative integers $\mathbb{N}_0$, equipped with the usual Skorokhod topology. The coupling construction also gives an alternative characterization (from the usual explosion time of the susceptibility function) of the critical time tc for the emergence of the giant component in terms of the operator norm of integral operators on certain L2 spaces.

Approximating Multivariate Tempered Stable Processes

We give a simple method to approximate multidimensional exponentially tempered stable processes and show that the approximating process converges in the Skorokhod topology to the tempered process. The approximation is based on the generation of a random angle and a random variable with a lower-dimensional Lévy measure. We then show that if an arbitrarily small normal random variable is added, the marginal densities converge uniformly at an almost linear rate, providing a critical tool to assess the performance of existing particle tracking codes.

Approximating Multivariate Tempered Stable Processes

We give a simple method to approximate multidimensional exponentially tempered stable processes and show that the approximating process converges in the Skorokhod topology to the tempered process. The approximation is based on the generation of a random angle and a random variable with a lower-dimensional Lévy measure. We then show that if an arbitrarily small normal random variable is added, the marginal densities converge uniformly at an almost linear rate, providing a critical tool to assess the performance of existing particle tracking codes.

Coalescence theory for a general class of structured populations with fast migration

In this paper we study a general class of population genetic models where the total population is divided into a number of subpopulations or types. Migration between subpopulations is fast. Extending the results of Nordborg and Krone (2002) and Sagitov and Jagers (2005), we prove, as the total population sizeNtends to ∞, weak convergence of the joint ancestry of a given sample of haploid individuals in the Skorokhod topology towards Kingman's coalescent with a constant change of time scalec. Our framework includes age-structured models, geographically structured models, and combinations thereof. We also allow each individual to have offspring in several subpopulations, with general dependency structures between the number of offspring of various types. As a byproduct, explicit expressions for the coalescent effective population sizeN/care obtained.

Coalescence theory for a general class of structured populations with fast migration

In this paper we study a general class of population genetic models where the total population is divided into a number of subpopulations or types. Migration between subpopulations is fast. Extending the results of Nordborg and Krone (2002) and Sagitov and Jagers (2005), we prove, as the total population size N tends to ∞, weak convergence of the joint ancestry of a given sample of haploid individuals in the Skorokhod topology towards Kingman's coalescent with a constant change of time scale c. Our framework includes age-structured models, geographically structured models, and combinations thereof. We also allow each individual to have offspring in several subpopulations, with general dependency structures between the number of offspring of various types. As a byproduct, explicit expressions for the coalescent effective population size N/c are obtained.

The coalescent process in a population with stochastically varying size

We study the genealogical structure of a population with stochastically fluctuating size. If such fluctuations, after suitable rescaling, can be approximated by a nice continuous-time process, we prove weak convergence in the Skorokhod topology of the scaled ancestral process to a stochastic time change of Kingman's coalescent, the time change being given by an additive functional of the limiting backward size process.