I'm not sure if I understand the model correctly as it seems to me that the answer here is very natural if you think of your model in terms of expected "price increments" ("price changes"). The price level - not change - is given by N(0,1). Thus every time the price is positive the expected increment is negative. If the price level is negative the expected increment is positive. Therefore, the optimal strategy can be stated as: buy if the expected increment is positive and sell if it is negative. Of course it is exactly the conjectured optimal strategy.

I'm not sure if I understand the model correctly as it seems to me that the answer here is very natural if you reinterpret your model in terms of expected "price increments" ("price changes"). The model assumes that the price level - not change - is given by N(0,1). Thus every time the price is positive the expected increment is negative. If the price level is negative the expected increment is positive. Therefore, the optimal strategy can be stated as: buy if the expected increment is positive and sell if it is negative. Of course it is exactly the conjectured optimal strategy.