# Solutions of Stochastic Differential Equations

I'm very new to Stochastic processes and SDEs, I have a question related to solutions to a SDE, The famous SDE for financial assets is:

$$dS_{t} = \mu S_{t}dt + \sigma S_{t}dW_{t}$$

the solution for it is: $$S_{t} = S_{0} \exp^{(\mu - \frac{1}{2}\sigma^2)t + \sigma W_{t}}$$

if I want to get the return for 1 period then $$\int_{0}^{t}\frac{dS_{u}}{S_{u}} = \int_{0}^{t}\mu dt + \int_{0}^{t}\sigma dW_{u} = \log\left(\frac{S_{t}}{S_{0}}\right) = \mu t + \sigma W_{t} = \log\left(\frac{S_{1}}{S_{0}}\right) = \mu + \sigma W_{1}$$ Which is correct according to the assumptions:

Returns are i.i.d normal with mean = $$\mu$$ and var = $$\sigma^2$$

So the asset price follows a random walk

But if I used the solution above and I plug t = 1 to calculate the position of the asset at time t = 1 and calculate the return from there, I can't get exactly the return from 0 -> 1 due to the random component but my expected return should be the mean $$\mu$$ $$S_{1} = S_{0} \exp^{(\mu - \frac{1}{2}\sigma^2) + \sigma W_{1}} = \log\left(\frac{S_{1}}{S_{0}}\right) = (\mu - \frac{1}{2}\sigma^2) + \sigma W_{1}$$ so how can I get rid of the $$-(\frac{1}{2}\sigma)$$ or I just can't use the solution this way and I have to go through the whole process of $$dS_{t}$$ using Ito lemma things Thank you guys so much!

You need to understand that $$dW_t$$ is much larger, in general, than $$dt$$. The random increments are so large (in the infinitesimal scale) that the accumulation of their squares is still proportional to $$dt$$ and will thus contribute to the "observable" behavior.
In more detail, you can calculate that $$1+σ\,dW_t+μ\,dt=e^{\ln(1+σ\,dW_t+μ\,dt)}=e^{σ\,dW_t+μ\,dt-\frac12(σ\,dW_t+μ\,dt)^2+\frac13(σ\,dW_t+μ\,dt)^3\mp...}$$ and the terms $$σ\,dW_t$$, $$μ\,dt$$ and $$-\frac12σ^2\,(dW_t)^2$$ all accumulate to appreciable sizes in $$σ\,ΔW_t$$, $$μ\,Δt$$ and $$-\frac12σ^2\,Δt$$ over some larger time span $$Δt$$. Only the terms following them are of a size $$\sim (dt)^{3/2}$$ that under accumulation still remain infinitesimal like $$\sqrt{dt}$$ or smaller.