I'll try to provide some pointers:1) Your process x(t) has continuous paths2) it is an Ito process which coefficients you can write down3) its expectation is constant in time.4) it's well defined since Bt and exp are well defined5) don't worry about the diffusion exploding, forget novikov, in your case you have strong solution6) there are arbitrage reasons why the process must tend to 0. In all rigor this a well documented argument, confirmed by empirical evidence and stylized facts. For memory, think of dividends. 7)actually, absolute value of a price process can increase in expectation, cf the technical computations of Dupire formulaIt's amazing that you use such sophisticated mathematics in your work. May I ask what kind of finance you are involved in? If you want to go deeper, I can solve this kind of problem all day for a very reasonable wage.