Preliminaries and Technical Background
Notation. For a vector v\in\mathbb{R}^{d} , we denote its i th component by either v^{i} or [v]^{i} and the Euclidean norm by |v| (or \|v\| ). \rm{diag}(v) denotes the diagonal matrix with i th diagonal entry of v^{i} . \langle\cdot,\cdot\rangle denotes the dot product of two vectors. \mathcal{U}(d) denotes uniform distribution on d dimension, tr denotes trace, and the superscript T denotes transposition. \circ and \odot denote composition and Hadamard product respectively. \nabla and \nabla^{2} denote gradient and Hessian respectively. \mathbb{E} denotes expectation. \overset{(\mathrm{d})}{=} and \overset{\mathrm{(d)}}{\to} denote equivalence and convergence in the sense of distribution respectively. I denotes the identity matrix, and 1_{A} denotes indicator function of the set A .

Stable distributions. Let \alpha\in(0,2] be a stability parameter and d\geq 1 . A random variable X\in\mathbb{R}^{d} is \alpha -stable if X_{1},\,X_{2},\,\ldots are independent copies of X , then n^{-1/\alpha}\sum_{j=1}^{n}X_{j}\overset{(\mathrm{d})}{=}X\text{ for all }n\geq 1 [ 30 ] .
Stable distributions appear as the limiting distribution in the generalized
central limit theorem (CLT) [ 14 ] . The case \alpha=2 corresponds to the Gaussian distribution, while \alpha=1 corresponds to the Cauchy distribution. The p -th moment of a stable random variable is finite if and only if p<\alpha . Here we are interested in the case \alpha\in(1,2) , such that \mathbb{E}[|X|]<\infty and \mathbb{E}[|X|^{2}]=\infty . If X is symmetric, then there exists a scale parameter c>0 and the characteristic function of X is given by \mathbb{E}\left[\exp(i\langle\xi,X\rangle)\right]=\exp(-|c\xi|^{\alpha}) for all \xi\in\mathbb{R}^{d} . If X is a random vector with independent components, then there exists a scale vector \mathbf{c}=(c^{1},\ldots,c^{d})\in\mathbb{R}^{d}_{+} and the characteristic function of X is given by \mathbb{E}\left[\exp(i\langle\xi,X\rangle)\right]=\exp(-\sum_{j=1}^{d}|c^{j}\xi^{j}|^{\alpha}) .

Lévy processes. Lévy processes are stochastic processes (\mathrm{L}_{t})_{t\geq 0} with independent and stationary increments, and are defined as follows [ 2 ] :

For N\in\mathbb{N} and t_{0}<t_{1}<\ldots<t_{N} , the increments (\mathrm{L}_{t_{i}}-\mathrm{L}_{t_{i-1}}) are independent for all i .

For any t>s>0 , (\mathrm{L}_{t}-\mathrm{L}_{s}) and \mathrm{L}_{t-s} have the same distribution.

\mathrm{L}_{t} is continuous in probability, i.e., for all \delta>0 and s\geq 0 , \mathbb{P}(|\mathrm{L}_{t}-\mathrm{L}_{s}|>\delta)\to 0 as t\to s .

Typical examples of Lévy processes include Brownian motion and the \alpha -stable processes. By the Lévy-Khintchine formula, a Lévy process (\mathrm{L}_{t})_{t\geq 0} with \mathrm{L}_{0}=0 is determined by a triplet (b,\Sigma,\nu) for some b\in\mathbb{R}^{d} , \Sigma\in\mathbb{R}^{d\times d} positive semi-definite and a measure \nu on \mathbb{R}^{d}\setminus\{0\} such that \int_{x\neq 0}\min\{1,|x|^{2}\}\nu(dx)<\infty . The characteristic function of \mathrm{L}_{t} is therefore \exp(-t\Psi(\xi)) , with the characteristic exponent \Psi:\mathbb{R}^{d}\to\mathbb{C} defined by

\Psi(\xi)=-i\langle b,\xi\rangle+\frac{1}{2}\langle\xi,\Sigma\xi\rangle+\int_{\mathbb{R}^{d}}\left[1-e^{i\langle x,\xi\rangle}+i\langle x,\xi\rangle 1_{|x|<1}\right] u(dx)

In the Lévy–Itô decomposition of a Lévy process, b denotes a constant drift, \Sigma is the covariance matrix of a Brownian motion and \nu characterises a Lévy jump process, which is modified to be càdlàg (right-continuous and has left limits everywhere) with countably many jumps. In particular, if b=0_{\mathbb{R}^{d}} , \Sigma=0_{\mathbb{R}^{d\times d}} and \nu(dx)=|x|^{-\alpha-1}dx for some \alpha\in(0,2) , then the triplet (0_{\mathbb{R}^{d}},0_{\mathbb{R}^{d\times d}},|x|^{-\alpha-1}dx) gives a symmetric \alpha -stable process, denoted by (\mathrm{L}^{\alpha}_{t})_{t\geq 0} .

Marcus differential equations. Passing to the limit of the driving processes with jumps in an SDE is not trivial, as one needs to be careful with the meaning of integration [ 8 ] . Unfortunately, the common Itô and Stratonovich integration (which are based on Riemann-Stieltjes sums with the integrand evaluated at the left end point and the midpoint of the partition intervals respectively) fail to provide the desired convergence. To this end, we resort to Marcus differential equations [ 26 ] , which in their basic form are given by:

\mathrm{d}X_{t}=b(X_{t})\diamond\mathrm{d}\mathrm{L}_{t},

where \diamond denotes integration in the Marcus sense. Marcus integrals involve sums over infinitely many jumps and transform under the usual laws of calculus [ 2 ] , and thus play a similar role for Lévy processes as the Stratonovich integral for Brownian motion. The solution map of Marcus differential equations is continuous with respect to the driving process under certain variants of the Skorokhod topology.
Note that if b(X_{t}) is constant in X_{t} , then there is no difference in the solutions of the SDE in the sense of Marcus, Itô and Stratonovich. We refer to Appendix B for more details on these.