First order strong approximations of scalar SDEs defined in a domain
In order to generate strong approximate solutions of stochastic differential equations (SDEs) of the form \[ dy(t)= a(y(t))\,dt+ b(y(t))\,dw(t) \] with coefficients that are not Lipschitz, a Lamperti-type transformation is applied to the SDE to obtain an SDE of the form \[ dx(t)= f(x(t))\,dt+\lambda dw(t), \] whose solution is approximated using a backward Euler-Maruyama method (BEM). The inverse transform of this approximate solution yields an approximate solution of the original SDE. Under certain conditions, strong convergence of order 1 of the BEM approximate solution of the transformed SDE is proved. Strong convergence of order 1 of the approximate solution of the original SDE is established for five well-known SDEs that arise in mathematical finance and biomathematics.
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