Abstract
To solve the stiff stochastic differential equations, we propose an improved Milstein method, which is constructed by adding an error correction term to the Milstein scheme. The correction term is derived from an approximation of the difference between the exact solution of stochastic differential equations and the Milstein continuous-time extension. The scheme is proved to be strongly convergent with order one and is as easy to implement as standard explicit schemes but much more efficient for solving stiff stochastic problems. The efficiency and the advantage of the method lie in its very large stability region. For a linear scalar test equation, it is shown that the mean-square stability domain of the method is much bigger than that of the Milstein method. Finally, numerical examples are reported to highlight the accuracy and effectiveness of the method.
Highlights
Stochastic differential equations (SDEs) play a prominent role in a range of scientific areas like biology, chemistry, epidemiology, mechanics, microelectronics, and finance [ – ]
We look at the following d-dimensional SDEs driven by multiplicative noise: dX(t) = f (X(t)) dt +
Here we restrict ourselves to the special case d = m = in ( . ) and refer to Section for the general case. In this setting we introduce the improved Milstein (IM) method for equation ( . ) as follows: Yn+ = Yn+ + ( – hf (Yn+ ))– h(f (Yn+ ) – f (Yn)), Yn+ = Yn + hf (Yn) + g(Yn) where Yn is the approximation of the exact solution X(t) of ( . ) at time tn = nh with h being the time step size
Summary
Stochastic differential equations (SDEs) play a prominent role in a range of scientific areas like biology, chemistry, epidemiology, mechanics, microelectronics, and finance [ – ]. Note that Y can be viewed as a one-step approximation generated by the classical Milstein method [ , ] starting with the initial value Y = X . ) have continuously bounded derivatives up to the required order for the following analysis, and the coefficient functions in Itô-Taylor expansions (up to a sufficient order) are globally Lipschitz and satisfy the linear growth conditions.
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