A transformed jump-adapted backward Euler method for jump-extended CIR and CEV models
A jump-adapted backward Euler method is devised for approximating the solution of a jump-diffusion Itô stochastic differential equation of the form \[ dX_t= \kappa(\theta- X_{t-})\,dt+ \sigma X^\alpha_{t-}dW_t+ g(X_{t-})\,dN_t,\quad t\in(0,T],\quad X(0)= X_0, \] where \(W_t\) is a scalar Wiener process and \(N_t\) is a scalar Poisson process. Such equations arise in mathematical finance. Under certain assumptions, it is proved that positivity will be retained and that the method is strongly convergent with order one in the \(p\)th mean. Numerical results for examples are presented in which a strong convergence rate of order one is attained and positivity is preserved.
jump-diffusion Itô stochastic differential equation
scientific article; zbMATH DE number 6674009
scientific article