Consider the following stochastic integral $$ dY_t = -Y_t \hspace{1mm} dW_t + Y_t \hspace{1mm} dt $$ with $Y_0 = 1$.
I have managed to evaluate this to deduce that $$ Y_t = \exp \left( t - \int_0^t dW_t \right) $$ Thus, assuming that this is correct, I have attempted to find $Y_1$ as follows: $$ Y_1 = \exp \left( 1 - \int_0^1 dW_t \right) = \exp \left( 1 - [W_1 - W_0] \right) = \exp \left( 1 - W_1 \right) $$
However, I have written an R script which evaluates $Y_1$ 1000 times using 1000 independently simulated brownian motions. The values returned by this function have mean $e$ and variance $e^3 - e$, which would suggest that the correct solution would be $$ Y_1 = \exp \left( \frac{1}{2} - W_1 \right) $$
Which solution is correct?
This is a special case of the Geometric Brownian motion with $\mu=1$ and $\sigma=-1$.
The derivation, that has your missing factor $\frac{1}{2}$ from Itô's lemma, is concisely stated on the Wikipedia page: