In this paper, we derive fully implementable first order time-stepping schemes for point delay McKean stochastic differential equations (McKean SDEs), possibly with a drift term exhibiting super-linear growth in the state component. Specifically, we propose different tamed Milstein schemes for a time-discretised interacting particle system associated with the McKean equation and prove strong convergence of order 1 and moment stability, making use of techniques from calculus on the space of probability measures with finite second order moments. In addition, we introduce a truncated tamed Milstein scheme based on an antithetic multi-level Monte Carlo approach, which leads to optimal complexity estimators for expected functionals without the need to simulate Lévy areas.

The purpose of this paper is to study on-line portfolio selection strategies for currency exchange markets and our focus is on the markets with presence of decrements. To this end, we first analyze the main factors arising in the decrements. Then we develop a cross rate scheme which enables us to establish an on-line portfolio selection strategy for the currency exchange markets with presence of decrements. Finally, we prove the universality of our on-line portfolio selections.

In this paper, motivated by modelling currency exchange markets with matrix-valued stochastic processes, matrix-valued stochastic differential equations (SDEs) are formulated. This is done based on the matrix trace, as for the purpose of modelling currency exchange markets. To be more precise, we set up a Hilbert space structure for n × n square matrices via the trace of the Hadamard product of two matrices. With the help of this framework, one can then define stochastic integral of Itô type and Itô SDEs. Two types of sufficient conditions are discussed for the existence and uniqueness of solutions to the matrixvalued SDEs.

This work focuses on a class of retarded stochastic differential equations that need not satisfy dissipative conditions. The principle technique of our investigation is to use variation-of-constants formula to overcome the difficulties due to the lack of the information at the current time. By using variation-of-constants formula and estimating the diffusion coefficients we give sufficient conditions for p-th moment exponential stability, almost sure exponential stability and convergence of solutions from different initial value. Finally, we provide two examples to illustrate the effectiveness of the theoretical results.