Understanding existence theorem of (strong) solution of SDE

[Paalfaal]

I'm currently working my way through the existence theorem of strong solutions for the stochastic differential equation
$X_t = X_0 + \int_0^t b(s,X_s)ds + \int_0^t \sigma(s,X_s)Bs$,
Where $\int_0^t \sigma(s,X_s)Bs$ is the Ito integral. The assumptions are:
1: $b,\sigma$ are jointly measurable and adapted to the filtration $\{ \mathcal{F}_t\}_{t\geq0}$ .
2: $b,\sigma$ satisfy the Lipschitz- and linear growth bound conditions.
3: $\left| | X_0 | \right|_{L^2(\Omega)} < \infty$ and $X_0$ is $\mathcal{F}_0$-measurable.
The iterative scheme
$\begin{align} \begin{cases} X_t^0 &= X_0, \\ X_t^{n+1} &= X_0 + \int_0^t b(s,X_s^{n})ds + \int_0^t \sigma(s,X_s^{n}) dB_s, \end{cases} \end{align}$
is introduced. A bit into the proof we need to use Doobs martingale inequality on $X_t^{n+1} - X_t^n$. My problem is that I fail to see how the previous expression is a martingale (wrt the filtration $\{ \mathcal{F}_t\}_{t\geq0}$). I know it suffices to show that $X_t^n$ is a martingale. It is true that the Ito integral is a martingale, but what happens with the Lebesgue (or Riemann) integral? I have read the proof in several textbooks (Øksendal, Klöden-Platen, Kuo), none of the argued explicitly why $X_t^n$ is a martingale, but all og them use Doobs martingale inequality.

By the assumptions it seems clear that $X_0$ is a martingale, but the Lebesgue integral seems to get in the way of proving it for general $n$.

 

[Paalfaal]

Sorry, I was wrong in two spots; Klöden-Platen does not use Doobs martingale inequality, but rather the 'Markov inequality':
$P(|X| > a) \leq \frac{1}{a^r}E[|X|^r]$
for $a,r > 0$. And thus I don't need to show that $X_t^n$ is a martingale. Moreover, Kuo uses 'Doobs submartingale inequality'. Still, I don't understand how $X_t^n$ can be a martingale..

 

[Paalfaal]

I would appreciate it if someone could clear up the martingale problem: The SDE

$X_t = X_0 + \int_0^t b(s,X_s)ds + \int_0^t \sigma(s,X_s)dB_s$

as I understand it, is NOT a martingale wrt $\{ \mathcal{F}_t:t\in [0,T] \}$ unless the drift term $b(t,x)$ is zero. In Theorem 10.2.2 in the textbook "Klöden-Platen:Numerical solution to stochastic differential equations(1990)" the Doob inequality is used in eqns (10.2.14),(10,2,15). The Doob inequality reads

$E\big[ \sup\limits_{0\leq s\leq T} |X_s|^p \big] \leq \Big(\frac{p}{p-1}\Big)^p E\big[ |X_t|^p \big].$

But this inequality is only true for martingales, and $X_t$ is not a martingale wrt $\{ \mathcal{F}_t:t\in [0,T] \}$. There are two possibilities:

1. The book is wrong (which I highly doubt)
2. I am wrong, and $X_t$ is a martingale.

Please help!