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We propose a class of stochastic processes that we call captive diffusions, which evolve within measurable pairs of càdlàg bounded functions that admit bounded right-derivatives at points where they are continuous. In full generality, such processes allow reflection and absorption dynamics at their boundaries—possibly in a hybrid manner over non-overlapping time periods—and if they are martingales, continuous boundaries are necessarily monotonic. We employ multi-dimensional captive diffusions equipped with a totally ordered set of boundaries to model random processes that preserve an initially determined rank. We run numerical simulations on several examples governed by different drift and diffusion coefficients. Applications include interacting particle systems, random matrix theory, epidemic modelling and stochastic control.

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We model continuous-time information flows generated by a number of information sources that switch on and off at random times. By modulating a multi-dimensional Lévy random bridge over a random point field, our framework relates the discovery of relevant new information sources to jumps in conditional expectation martingales. In the canonical Brownian random bridge case, we show that the underlying measure-valued process follows jump-diffusion dynamics, where the jumps are governed by information switches. The dynamic representation gives rise to a set of stochastically-linked Brownian motions on random time intervals that capture evolving information states, as well as to a state-dependent stochastic volatility evolution with jumps. The nature of information flows usually exhibits complex behavior, however, we maintain analytic tractability by introducing what we term the effective and complementary information processes, which dynamically incorporate active and inactive information, respectively. As an application, we price a financial vanilla option, which we prove is expressed by a weighted sum of option values based on the possible state configurations at expiry. This result may be viewed as an information-based analogue of Merton’s option price, but where jump-diffusion arises endogenously. The proposed information flows also lend themselves to the quantification of asymmetric informational advantage among competitive agents, a feature we analyze by notions of information geometry.

Archimedean copulas are popular in the world of multivariate modelling as a result of their breadth, tractability, and flexibility. McNeil and Nešlehová (2009) showed that the class of Archimedean copulas coincides with the class of positive multivariate ℓ 1norm symmetric distributions. Building upon their results, we introduce a class of multivariate Markov processes that we call 'Archimedean survival processes' (ASPs). An ASP is defined over a finite time interval, is equivalent in law to a vector of independent gamma processes, and its terminal value has an Archimedean survival copula. There exists a bijection from the class of ASPs to the class of Archimedean copulas. We provide various characterisations of ASPs, and a generalisation.