# What does it mean that $X_T$ is a solution to a Brownian Motion driven SDE ($dX_t = ....$)?

Please consider the $$X_t = X_0 + \int^t_of(X_s,s)ds+\int^t_og(X_s,s)dB_s$$ where $$B_t$$ is Brownian Motion. This can also be expressed as: $$dX_t=f()dt+g()dB_t$$ What does it mean that $$X_T$$ is a solution to the above stochastic differential equation?

I can't seem to intuitive explain this. The way I will put it is:

For a given Brownian motion realization it $$X_T$$ is a solution to the SDE above.. Is this correct or is there more it?

There's actually two definitions of solution to SDE. Strong and weak.

Strong solution. Given a probability space $$(\Omega, \mathcal F, \mathcal F_t,P)$$ and a Brownian motion $$B(t)$$ on that space adapted to $$\mathcal F_t$$, a solution to $$dX=fdt+g~dB(t)$$ with initial condition $$x\in \mathbb R$$ is a continuous process adapted to $$\mathcal F_t$$ such that $$X(t)=x+\int_0^t fdt+\int_0^t g~dB(t)$$.

There is also

Weak solution. A process $$X(t)$$ is called a weak solution to $$dX=fdt+g~dB(t)$$ if there exists a probability space $$(\Omega, \mathcal F, \mathcal F_t,P)$$ and a Brownian motion $$B(t)$$ on that space adapted to $$\mathcal F_t$$ such that $$X(t)=x+\int_0^t fdt+\int_0^t g~dB(t)$$.

Note that strong solution implies weak solution.

As an example of something that has weak but not strong solution, consider Tanaka's equation

$$dX(t)=\text{sign}(X(t))dB(t)$$

It can be shown that this has no strong solution by considering local times. However let $$X(t)$$ be a Brownian motion and then $$X(t)$$ is a weak solution. Note that $$\int_0^t\text{sign}^2(s)ds=t<\infty$$ so the Ito integral $$\int_0^t\frac{1}{\text{sign}(s)}dX(s)=\int_0^t \text{sign}(s) dX(s)$$ exists. Noting that the quadratic variation of this integral is $$t$$, and Ito integrals are continuous martingales, so we know $$\int_0^t\frac{1}{\text{sign}(s)}dX(s)=\tilde{B}(t)$$ is a Brownian motion.

Thus, $$dX(t)=\text{sign}(t)d\tilde{B}(t)$$.

The difference here is that the Brownian motion is constructed in terms of our solution not the other way around.