Abstract
In this paper, based on the iterative technique, a new explicit Magnus expansion is proposed for the nonlinear stochastic equation d y = A ( t , y ) y d t + B ( t , y ) y ∘ d W . One of the most important features of the explicit Magnus method is that it can preserve the positivity of the solution for the above stochastic differential equation. We study the explicit Magnus method in which the drift term only satisfies the one-sided Lipschitz condition, and discuss the numerical truncated algorithms. Numerical simulation results are also given to support the theoretical predictions.
Highlights
Stochastic differential equations (SDE) have been widely applied in describing and understanding the random phenomena in different areas, such as biology, chemical reaction engineering, physics, finance and so on
We prove that the explicit Magnus methods succeed in reproducing the almost sure exponential stability for the stochastic differential Equation (4) with a one-sided Lipschitz condition
We show that the nonlinear Magnus methods preserve the positivity independent of the stepsize
Summary
Stochastic differential equations (SDE) have been widely applied in describing and understanding the random phenomena in different areas, such as biology, chemical reaction engineering, physics, finance and so on. We design explicit stochastic Magnus expansions [9] for the nonlinear equation dy = A(t, y)ydt + B(t, y)y ◦ dW, y(t0 ) = y0 ∈ G,. When Equation (4) is used to describe the asset pricing model in finance, the positivity of the solution is a very important factor to be maintained in the numerical simulation This task can be accomplished by the nonlinear Magnus methods. We prove that the explicit Magnus methods succeed in reproducing the almost sure exponential stability for the stochastic differential Equation (4) with a one-sided Lipschitz condition.
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