SDE with respect to an Ornstein-Uhlenbeck process

stochastic-analysisstochastic-calculusstochastic-differential-equationsstochastic-integralsstochastic-processes

I have come across the following equation
\begin{equation}dX_t=-\lambda X_t dt +dU_t\quad (1)\end{equation}
where $U_t$ is an Ornstein Uhlenbeck process:
$$dU_t=-\theta U_tdt + \sigma dW_t$$
The context in which this was introduced is an applied modeling setting, so their only interest was to integrate this equation in a time discrete way. This is no problem: First you create a sample path of $U_t$ using the Euler-Maruyama method and then similarly integrate $X_t$:
$$X_{t+\Delta t}-X_t=-\lambda X_t *\Delta t +\Delta U_t=-\lambda X_t *\Delta t+(U_{t+\Delta t}-U_t)$$
I want to know more about the analytical solution of (1) though.

  1. Am I correct, that this is technically not an SDE? The most general equation allowed for that seems to be
    $$dX_t=a(t,X_t)dt+b(t,X_t)dW_t$$
  2. Is equation (1) well-defined? I only know that the Ito integral is defined with respect to semi martingales. Is $U_t$ a semi martingale? Can you even meaningfully integrate this equation if it is not?
  3. You could rewrite (1) as \begin{equation}dX_t=-(\lambda X_t +\theta U_t)dt +\sigma dW_t\end{equation}Note that this is not an answer to question 1. since $-(\lambda X_t +\theta U_t)\neq a(t,X_t)$. We could then discretize a second way: $$X_{t+\Delta t}-X_t=-(\lambda X_t +\theta U_t)*\Delta t +\sigma\Delta W_t$$ I assume the two different Euler Maruyama discretizations converge to the same process, right? (At let distributionally speaking)

Best Answer

You can interpret the given SDE $dX_t=-\lambda X_t dt +dU_t$ simply by integrating: $$X_t - X_0 = -\lambda \int_0^t X_s ds + U_t - U_0$$

You may even solve the SDE for $U_t$ by using an integration factor, to arrive at: $$U_t = U_0e^{-\theta t} + \sigma \int_0^t e^{-\theta (t-s)} dW_s$$ So that ultimately, $X_t$ is given by: $$X_t - X_0 = -\lambda \int_0^t X_s ds + U_0e^{-\theta t} - U_0 + \sigma \int_0^t e^{-\theta (t-s)} dW_s$$ There is no ambiguity in this expression, as the first integral is a Riemann integral, while the second is a usual Itô integral.

Am I correct, that this is technically not an SDE? The most general equation allowed for that seems to be $dX_t=a(t,X_t)dt+b(t,X_t)dW_t$

As a consequence of the Bichteler-Dellacherie theorem, the most general Itô integrators are semi-martingales; thus the most general SDEs that can be interpreted in the Itô sense are those driven by semi-martingales. The above is one example.

Is $U_t$ a semi-martingale?

Yes; this can be seen by noting it is an Itô process. Thus, $U_t$ is an admissible stochastic integrator.

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