Unique strong solution of $X_t=t+\int_0^tX_sdW_s$

Question

Let $(W_t)_{t\ge0}$ be a standard brownian motion. Using Ito's lemma, I was able to show that

$$X_t:=\exp(W_t-\frac{1}{2}t)\int_0^t\exp(-(W_s-\frac{1}{2}s))ds$$

solves

$$X_t=t+\int_0^tX_sdW_s$$

Therefore $X$ is a strong solution of this equation.

Now I wonder if this solution is unique and if it relates to the function I was using in Ito's lemma, which is

$$f:\mathbb{R}^2\rightarrow\mathbb{R},\quad f(x_1,x_2)=\exp(x_1)\cdot x_2,$$

such that

$$X_t=f\big(W_t-\frac{1}{2}t,\int_0^t\exp(-(W_s-\frac{1}{2}s))ds\big)$$

Thanks for your attention.

Answer 1

in general, a stochastic differential equation has a unique solution, subject to some relatively mild hypotheses on the coefficients (which are satisfied here). A reference to the precise theorem can be found in the book by Oksendal, or here: https://en.wikipedia.org/wiki/Stochastic_differential_equation#Existence_and_uniqueness_of_solutions