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In this paper we discuss stochastic di erential delay equations with Markovian switching. Such an equation can be regarded as the result of several stochastic di erential delay equations switching from one to the others according to the movement of a Markov chain. One of the main aims of this paper is to investigate the exponential stability of the equations.

The mean-reverting square root process is a stochastic differential equation (SDE) that has found considerable use as a model for volatility, interest rate, and other financial quantities. The equation has no general, explicit, solution, although its transition density can be characterized. For valuing path-dependent options under this model, it is typically quicker and simpler to simulate the SDE directly than to compute with the exact transition density. Because the diffusion coefficient does not satisfy a global Lipschitz condition, there is currently a lack of theory to justify such simulations. We begin by showing that a natural Euler-Maruyama discretization provides qualitatively correct approximations to the first and second moments. We then derive explicitly computable bounds on the strong (pathwise) error over finite time intervals. These bounds imply strong convergence in the limit of the timestep tending to zero. The strong convergence result can be used to justify the method within Monte Carlo simulations that compute the expected payoff of financial products. We spell this out for a bond with interest rate given by the mean-reverting square root process, and for an up-and-out barrier option with asset price governed by the mean-reverting square root process. We also prove convergence for European and up-andout barrier options under Heston's stochastic volatility model-here the mean-reverting square root process feeds into the asset price dynamics as the squared volatility.