We set up a model for electronic foreign-exchange markets, suggesting subordinators to represent sellers’ and buyers’ offers. Its analysis naturally leads to the study of level passage events. The classical level passage event concerns the joint law of the time, height, and jump size observed when a real-valued stochastic process first exceeds a given level h. We provide an up-to-date treatment...

We consider investment problems where an investor can invest in a savings account, stocks and bonds and tries to maximize her utility from terminal wealth. In contrast to the classical Merton problem we assume a stochastic interest rate. To solve the corresponding control problems it is necessary to prove a veri cation theorem without the usual Lipschitz assumptions.

Fractional-order SIR models have become increasingly popular in the literature in recent years, however unlike the standard SIR model, they often lack a derivation from an underlying stochastic process. Here we derive a fractional-order infectivity SIR model from a stochastic process that incorporates a time-since-infection dependence on the infectivity of individuals. The fractional derivative...

Let (vu) i, j = 1, 2 ,..., be i.i.d. standardized random variables. For each n, let V, = (v,,) i = 1, 2 ,..., n; j = 1, 2 ,..., s = s(n), where (n/s) -+ y > 0 as n + 03, and let M, = (l/s) V, Vi. Previous results [7,8] have shown the eigenvectors of M, to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process X, on [0, 11, constructed f...

A stochastic differential equation is conjectured for approximately modelling the fluctuating size changes of an individual droplet in a fluid that is metastable with respect to nucleation of a new phase, in the limit when the critical droplet size is very large. The Freidlin-Wentzell formula for this S.D.E. is used to make estimates of large-deviation type for probabilities of such events as t...

Under very general conditions the hitting time of a set by a stochastic process is a stopping time. We give a new simple proof of this fact. The section theorems for optional and predictable sets are easy corollaries of the proof.

Multi-dimensional backward stochastic Riccati di erential equations (BSRDEs in short) are studied. A closed property for solutions of BSRDEs with respect to their coeÆcients is stated and is proved for general BSRDEs, which is used to obtain the existence of a global adapted solution to some BSRDEs. The global existence and uniqueness results are obtained for two classes of BSRDEs, whose genera...

The paper considers a particular family of set–valued stochastic processes modeling birth–and–growth processes. The proposed setting allows us to investigate the nucleation and the growth processes. A decomposition theorem is established to characterize the nucleation and the growth. As a consequence, different consistent set–valued estimators are studied for growth process. Moreover, the nucle...