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Understanding existence theorem of (strong) solution of SDE

  1. Nov 13, 2014 #1
    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 ##.
     
    Last edited: Nov 13, 2014
  2. jcsd
  3. Nov 13, 2014 #2
    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..
     
    Last edited: Nov 13, 2014
  4. Dec 21, 2014 #3
    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!
     
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