# Time-dependent drift in an SDE

Given an SDE $$dX_t = b(t,X_t)dt + dZ_t$$ where $$Z_t$$ is a Levy process. I am curious about the infinitesimal generator of this process.

If the SDE was say $$dX_t = b(X_t)dt + dW_t$$ where $$W_t$$ was a Wiener process then we would have the generator being

$$Af(x) = \sum_i b_i(x)\frac{\partial f}{\partial x_i} + \frac{1}{2}\sum_{i,j}(\sigma\sigma^T)_{i,j}(x)\frac{\partial^2 f}{\partial x_i\partial x_j}$$

where $$f \in C_{0}^{2}(\mathbb{R}^n)$$.

My question is does time-dependent drift make a change to what the generator would be? I know there is a difference for the Lévy process vs just Wiener process but for the example only wanted to put the Wiener process version. I am only considered in the case where the drift is the only coefficient containing time-dependency.

I looked for a reference covering this case but could not find one.

• Try to figure out what the generator of the process $(t,X_t)$ should be. Feb 1, 2023 at 6:14
• Extending the example above should be $$Af(t,x) = \sum_i b_i(t,x)\frac{\partial f}{\partial t} + \sum_i b_i(t,x)\frac{\partial f}{\partial x_i} + \frac{1}{2}\sum_{i,j}(\sigma\sigma^T)_{i,j}(t,x)\frac{\partial^2 f}{\partial x_i\partial x_j}$$ However, that include time-dependency in the diffusive portion. Feb 1, 2023 at 16:10

Writing $$Y_t=(t,X_t)$$ and $$y=(t,x)$$ the SDE followed by $$Y$$ is \begin{align} dY_t=\beta(Y_t)\,dt+\alpha(Y_t)\,dW_t \end{align} where $$\beta(y)=\begin{pmatrix}1\\b(t,x)\end{pmatrix}$$ ($$b(t,x)$$ being your old drift vector of $$X_t$$) and $$\alpha(y)=\begin{pmatrix}0&0\\0&\sigma \end{pmatrix}$$ Here, $$\sigma$$ being your old diffusion matrix of $$X$$ which was simply a constant identity matrix.
• Yes I believe that makes sense, I should've clarified more I mean a drift that could be along the lines of say $b(t,x) = \cos(tx)$. Feb 1, 2023 at 19:03
• @KurtG. Great answer. I assume a similar trick for turning a time-dependent SDE into one without $t$ works for Lévy-driven SDEs as well, but do you have a reference on how to compute generators for Lévy-driven SDEs? Feb 1, 2023 at 20:30