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Let $X_{s}^{t,x}$ denote the solution at time $s$ of an Ito SDE whose coefficients are Lipschitz continuous with initial condition $X_t=x$. Let $t\leq s\leq T<\infty$. The inequality $$ \mathbb{E}\left[\left|X_{s}^{t,x}-X_{s}^{t,\hat{x}}\right|^{2}\right]\leq C\left|x-\hat{x}\right|^2\text{ for }s\in[t,T] $$ is well-known and established by Grönwall's lemma. Are there are any nontrivial cases where such an inequality, or anything implying some kind of uniform continuity (e.g. Hölder, Lipschitz, etc.) holds for a Lévy SDE?

parsiad
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  • Could you specfiy what you mean by "such an in equality"? What kind of inequality are you looking for? Since Lévy processes are homogeneous in space, we have $$X_s^x-X_s^{\hat{x}} = x-\hat{x}$$ and so the inequality $$\mathbb{E}(|X_s^x-X_s^{\hat{x}}|^2) = |x-\hat{x}|^2$$ is trivially satisfied. (Here $X_t^x$ denotes the Lévy process started at $x$.) – saz Jun 17 '16 at 17:48
  • But even the simple ODE $dX_{t}=X_{t}^{2}dt$ has solutions $X_{0}/(1-X_{0}t)$ (not homogeneous in space). Perhaps I am getting my terminology mixed up. For example, in the context of Ito processes, I am thinking of those in which the drift and scaling can depend in a Markovian way on the state (i.e., $dX_{t}=\mu(t,X_{t})dt+\sigma(t,X_{t})dW_{t}$). – parsiad Jun 18 '16 at 14:35
  • The title of your question suggests that your asking about Lévy processes ... and the solution of the ODE $dX_t = X_t^2 , dt$ is obviously not a Lévy process. I simply don't understand how the Lévy processes are coming into play... – saz Jun 18 '16 at 16:23
  • The ODE is a special case of the SDE that I wrote in the previous comment with $\mu(t,x)=x^2$ and $\sigma(t,x)=0$. – parsiad Jun 18 '16 at 16:57
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    Yes, this is possible. See e.g. Proposition 6.6.2 in Applebaum Levy processes and Stochastic Calculus. – zhoraster Jun 19 '16 at 08:20

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