Simple stochastic differential equation

Question

Solve the following stochastic differential equation: $$ dX_t=X_t\,dt+dW_t. $$ Thank you very much for help! I even don't know where to start...

Answer 1

The usual tool used here is Ito's Formula. To use it, you'll want to propose a possible solution, say $$X_t = e^{t}\left(X_0 + \int_0^t e^{-s} \, dW_s\right),$$ and use Ito, by defining a function, say, $g(t,y)$ that is a function of deterministic $t$ and stochastic $y$. For a nice introduction to SDEs I half-heartedly recommend Oksendal.

Indeed, I do not respond directly to your question. Readily available is not my intuition about generating this solution (I happened to have it "in my back pocket"), but I think it derives from intuition you might have developed in ODE (notice the exponential term, a common term in the solution to ODEs).

Answer 2

(Shreve: Stochastic Calculus for Finance Volume II, Exercise 4.8). This is a specific case of the Vasicek SDE, $$ dR_t = (\alpha - \beta R_t)dt + \sigma dW_t. $$

Use Ito's formula to compute $$d(e^{\beta t} R_t)= \beta e^{\beta t} R_tdt + e^{\beta t} dR_t = \cdots $$ the right side will not involve $R_t$. Integrate this answer to obtain the solution.