When is a stochastic process defined via a SDE Markovian?

Question

I was wondering when a stochastic process defined via a SDE is Markovian? The SDE may involved Ito integral, Lebesgue integral, jump component, and any other things. The reason I ask this question is that I don't understand when we can and cannot discuss properties of a Markov process, such as Kolmogorov equations, for a process defined by a SDE. Thanks and regards!

Answer 1

If SDEs are of the form $\mathrm dX_t = \sum_{i=1}^n f(X_t)\mathrm dY^i_t$ where $Y^i$ are Markov processes and $f$ are some functions, than unless $Y^i$ and $f$ are not "nice enough", $X$ will be a Markov process. Intuitively, this is the case since the future dynamics of $X_t$ is independent from its past: e.g. $$ \mathrm dX_t = X_t\mathrm dW_t $$ is a Markov process whereas $$ \mathrm dX_t = \left(\int_0^t X_s\mathrm ds\right)\mathrm dW_t $$ is not. I am not sure that there is a complete answer, which "nice" conditions $Y^i$ and $f$ have to verify in order for $X$ to be a (strong) Markov process - but I guess you are aware of many example of such conditions for jump diffusion.

Answer 2

Consider the following SDE

$$ X_t ^{s,x} = x +\int _s^t \sigma(u,X_t ^{u,x}) ~ dW_u+\int _s^t b(u,X_t ^{u,x}) ~ du $$

satisfing the following hypothesis:

there is $C>0$ such that, for all $(x,y) \in \mathbb R ^p \times \mathbb R ^p$ and $u \in \mathbb R _+$

$$\left|\sigma(u,x) -\sigma(u,y) \right|+\left|b(u,x) -b(u,y) \right|\leq C\left|x-y \right|$$

for all $t\in \mathbb R _+$ and $x\in \mathbb R ^p$

$$ \int_s^t \left(\left| \sigma \right|^2(u,x)+\left|b \right|^2(u,x)\right)~du <+ \infty$$

If $b$ and $\sigma$ are time-homogeneous, ie, $\sigma(u,x) =\sigma(x)$ and $ b(u,x)= b(x)$, the simple Markov property applies to the solution of this SDE.

Answer 3

A good example of this is the time-integral of the Ornstein-Uhlenbeck process. On its own, this is not a Markov process. However, a two dimensional stochastic process, with one co-ordinate being the Ornstein-Uhlenbeck process and the other being its time-integral, will be a two dimensional Markov process. Similarly, a one-dimensional Markov process and its supremum process, comprise a two-dimensional Markov process.