Let us suppose that we have $U\sim\mathcal{Uniform}[0,T]\perp\tilde{Z}\sim\mathcal{N}\left(0,1\right)$ and $X=\left(U,S_U=S_{0}e^{(r-\frac{\sigma^{2}}{2})U+\sigma\sqrt{U}\tilde{Z}}\right)$ where $S_0,r,\sigma>0$. We can find the joint pdf of X whose expression is defined as : $f_{X}:\left(t,x\right)\rightarrow\frac{1_{[0,T]}(u)}{T}1_{[0,\infty)}(x)\frac{1}{x\sigma\sqrt{t}}\frac{e^{-\frac{1}{2}d_{2}\left(0,t,S_{0},x\right)^{2}}}{\sqrt{2\pi}}$ where : $d_{1}\left(t,T,x,K\right):=\frac{\log(\frac{x}{K})+\left(r+\frac{\sigma^{2}}{T}\right)\left(T-t\right)}{\sigma\sqrt{T-t}}\text{ and }d_{2}\left(t,T,x,K\right):=d_{1}\left(t,T,x,K\right)-\sigma\sqrt{T-t}$

Now considering the function : $\mathcal{N}(x)=\int_{-\infty}^{x}\frac{e^{-x^2/2}}{2\pi}$. Can we say that the following random variable

$H = S_U\mathcal{N}\left(d_{1}\left(U,T,S_U,K\right)\right)$ has a bounded pdf ?

The first thing to notice is that $\forall p\geq1,$ we have $H\in L^p$ since $0<H\leq S_U\in L^p$

More generally are there results that characterize random variables whose pdf are bounded ?



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