# Condition for existence of a stochastic differential equation

With $B$ a standard Brownian motion, write $$dX_t=f_tdt+g_tdB_t.$$ What are the conditions on $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ for $X_t$ to exists?

I think $\left(f\right)_{t\ge 0}$ and $\left(g\right)_{t\ge 0}$ should be adapted to the filtration of $B$.
Do they have to be càdlag?
What about the integrability condition \begin{align} &E\left( \int_0^t f_sds\right) < \infty, \text{ and },\\ &E\left( \int_0^t g_s^2d[B]_s\right) < \infty? \end{align}

We are using several textbooks in our class, and I can't pindown the theorem that treats this.
(Kurtz, Stochastic Analysis; Oksendal; Protter)

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## 1 Answer

Well usually these functions depend also on the (yet to be found) solution $X$ and must be measurable. The most used condition that guarantees a unique continuous solution is a growth condition of the form $$|f(t,x)|+|g(t,x)|\leq C(1+|x|)\qquad x\in\mathbb{R}^n,0\leq t\leq T$$ and a Lipschitz condition $$|f(t,x)-f(t,y)|+|g(t,x)-g(t,y)|\leq L|x-y|\qquad x,y\in\mathbb{R}^n,0\leq t\leq T.$$ The proofs often use the same idea as the in ODE case (Picard-Lindelöf).

Check out Theorem 5.2.1 in Oksendal!

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