# Explanation of notation on Levy-Ito decomposition in a paper by El Fatini and Boukanjime?

In El Fatini and Boukanjime "Stochastic analysis of a two delayed epidemic model incorporating Lévy processes with a general non-linear transmission" paper, the first equation of (3) is defined as:

$$ds(t) = \alpha dt + \sigma_1 dW_1(t) + \int_\mathbb{Z} q_1(z)S(t-) \tilde{N}(dt,dz), \tag{i}$$ where $$\tilde{N}(dt,dz)=N(dt,dz)-\nu(dz)dt.$$

1. Can someone explain the notation of $$N(dt,dz)$$ and $$\nu(dz)dt$$? Also why is there one integral in the equation $$(i)$$ but the notation shows integration over two variables, namely $$dt$$ and $$dz$$? In similar papers, I have noticed the same thing too.

2. Can someone give a detailed explanation of the proof of Proposition 2.1 part (1)?

Edit 2: The proof of proposition 2.1 part (1) attached below.

I am confused with: $$V(S(0),I(0))+KT\geq \epsilon U(m),$$ where $$\epsilon \in(0,1)$$.

They let $$m\rightarrow \infty$$, which leads to $$\infty>V(S(0),I(0))+KT=\infty$$.

Shouldn't it be $$V(S(0),I(0))+KT=\infty=\infty \geq \epsilon\cdot\infty = \infty$$? So the inequality actually holds and isn't a contradiction?

• Note: Not everyone has access to that PDF. Oct 3, 2023 at 17:55
• @BrianTung It is not required. The main issue is presented
– Math
Oct 4, 2023 at 10:38

1. Consider a measure $$\mu$$ on some measurable space $$(\mathcal{X}, \Sigma)$$ and a "nice" function $$f$$ on $$\mathcal{X}$$. Then, at least intuitively, the integral of $$f$$ with respect to $$\mu$$ can be thought of as the limit of the "Riemann sums"

$$\sum_{i\in I} f(x_i) \, \mu(E_i),$$

where $$\{E_i\}_{i\in I}$$ is a partition of $$\mathcal{X}$$ and $$x_i \in E_i$$ for each $$i \in I$$.

Hence, similar to how the notation for Riemann integral encapsulates the structure of the Riemann sums, it is reasonable to formalize the above structure and denote the abstract Lebesgue integral of $$f$$ against $$\mu$$ as

$$\int_{\mathcal{X}} f(x) \, \mu(\mathrm{d}x). \tag{1}$$

Here, $$\mathrm{d}x$$ formalizes a "generic, infinitesimally small subregion of $$\mathcal{X}$$" and $$x$$ is a point in that subregion.

2. Now, all the other "differentials" appearing in OP can be interpreted in this viewpoint. For example,

$$\tilde{N}(\mathrm{d}t, \mathrm{d}z) = N(\mathrm{d}t, \mathrm{d}z) - \nu(\mathrm{d}z)\mathrm{d}t \tag{2}$$

equates the measures of a generic, small rectangular subregion $$\mathrm{d}t \times \mathrm{d}z$$.

3. Strictly speaking, $$\text{(i)}$$ employs two slightly different conventions on infinitesimals. In one convention, $$\mathrm{d}t$$ is used as a standalone object, whereas in the other convention, $$\mathrm{d}z$$ is used as the infinitesimal to be integrated out by $$\int_{\mathbb{Z}}$$. Without proper explanation, this mix-up of conventions might possibly lead to confusions.

Anyway, $$\text{(i)}$$ equates two differentials in the variable $$t$$ alone.

• $$\mathrm{d}s(t)$$ in the LHS of $$\text{(i)}$$ stands for the increment of $$s(t)$$ from $$t$$ to $$t+\mathrm{d}t$$.

• The RHS of $$\text{(i)}$$ is the sum of three differentials in $$t$$, namely $$\alpha \, \mathrm{d}t, \qquad \sigma_1 \mathrm{d}W_1(t), \qquad \text{and}\qquad S(t-) \int_{\mathbb{Z}} q_1(z) \, \tilde{N}(\mathrm{d}t, \mathrm{d}z),$$ where only the variable $$z$$ is integrated out in the integral in the last term.

• It is very important to also note that $N$ is itself random, i.e. a random measure. Oct 7, 2023 at 19:14
• @SmallDeviation Can you explain Question 2?
– Leo
Oct 8, 2023 at 11:56