Let $X$ be a Levy process and $\mathcal{M}_n$ be the discretely observed maximum of the Levy process such that $$\mathcal{M}_N=\max_{0\leq j\leq N}X_j.$$
We then define $\mathcal{M}_j-X_j=\max(\mathcal{M}_{j-1},X_j)-X_j=\max(0,M_{j−1} − X_{j−1} − (X_j − X_{j−1})).$ The density function $f_j(x)$ of $\mathcal{M}_j-X_j$ can be described as a convolution such that $$f_j(x)=1_{(0,\infty)}(x)\left(c_{j-1}p(-x)+\int_\mathbb{R}f_{j-1}(y)p(y-x)\mathrm{d} y\right),$$
where, $c_{j-1}=1-\int_\mathbb{R}f_{j-1}(y)\mathrm{d}y$ and $p(y)$ is the density function of $X_j-X_{j-1}.$
Can someone explain to me how to get $c_{j-1}p(-x)$ in the first term of the equation? I have no clue to digest it at all.