A STRONG AND WEAK APPROXIMATION SCHEME FOR STOCHASTIC
DIFFERENTIAL EQUATIONS DRIVEN BY A TIME-CHANGED
BROWNIAN MOTION
Abstract: This paper establishes a discretization scheme for a large class of stochastic differential
equations driven by a time-changed Brownian motion with drift, where the time change is
given by a general inverse subordinator. The scheme involves two types of errors: one
generated by application of the Euler–Maruyama scheme and the other ascribed to simulation
of the inverse subordinator. With the two errors carefully examined, the orders of strong and
weak convergence are established. In particular, an improved error estimate for
the Euler–Maruyama scheme is derived, which is required to guarantee the strong
convergence. Numerical examples are attached to support the convergence results.
2010 AMS Mathematics Subject Classification: Primary: 60H35, 65C30; Secondary:
60H10.
Keywords and phrases: Stochastic differential equation, numerical approximation, order
of convergence, time-changed Brownian motion, inverse subordinator.