Stochastic Differential equations with $\sin(x^2)$ as drift.

Question

Can somebody help me how to solve the following SDE analytically or suggest me to go through some literature to understand this or can give me a little bit hint to work by myself. Thanks in advance.

$dX(t) =\sin (X(t)^2) dt + \cos (X(t)-2) dB(t),\quad X(0) = 1.$

Answer 1

As mentioned in the comments an analytic solution seems out of reach.

• one cannot simplify by removing the drift as done in Shreve-Karatzas because the volatility coefficient has zeroes.
• There is no clear trigonometric identity to use here for $\sin(x^{2})$ as one could in a related one to simplify Stochastic differential equation with trigonometric functions.

In terms of numerics, there are many methods out there eg. see "Lecture 9: Numerically solving SDEs".