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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?

enter image description here

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

1 Answer 1

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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.

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  • $\begingroup$ It is very important to also note that $N$ is itself random, i.e. a random measure. $\endgroup$ Oct 7, 2023 at 19:14
  • $\begingroup$ @SmallDeviation Can you explain Question 2? $\endgroup$
    – Leo
    Oct 8, 2023 at 11:56
  • $\begingroup$ @Leo I don't have access to the paper. So i can't answer your second question. $\endgroup$ Oct 8, 2023 at 12:10
  • $\begingroup$ @Sangchul Lee, can you explain if you have access to the paper? $\endgroup$
    – Leo
    Oct 8, 2023 at 14:28
  • $\begingroup$ @SmallDeviation I mentioned you by accident, my apologies. $\endgroup$
    – Leo
    Oct 8, 2023 at 14:29

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