Prove that X(t) > 0 for all t 0

Let oz 6 1R, 5 > 0 be given constants and suppose that W is a real-valued Wiener martingale. Consider axe) = aX(t)dt + exam/m), X(0) = :1: > 0. Prove that X(t) > 0 for all t 2 0. Hint: Try to guess the solution of the equation, prove that your guess is indeed the solution and conclude. We will need the multi-dimensional version of the Ito's formula. Let W be an m- dimensional Wiener martingale with respect to (F)tzo. Let o E Saxm and let b E Ad. We say that the d-dimensional process X has the stochastic differential dX (t) = b(t)dt + o(t) dW(t) (1) for t E [0, T], if X(t) = X(0)+ b(s)ds+ o(s) dW (s). Such a process is also called an Ito process