For stochastic differential equations (SDEs) of the form dX(t) = b(X)(t)) dt + sigma(X(t))dW(t) where b and sigma are Lipschitz continuous, it is shown that if we consider a fixed sigma is an element of C-5, bounded and with bounded derivatives, the random field of solutions is pathwise locally Lipschitz continuous with respect to b when the space of drift coefficients is the set of Lipschitz continuous functions of sublinear growth endowed with the sup-norm. Furthermore, it is shown that this result does not hold if we interchange the role of b and c. However for SDEs where the coefficient vector fields commute suitably we show continuity with respect to the sup-norm on the coefficients and a number of their derivatives.
- stochastic differential equations
- random field of solutions
- pathwise continuity wrt coefficients
- p-step nilpotent Lie algebras
- shuffle product