All those connectors are digital only. Your monitor has no analogue video inputs. PS2 is an analogue output only, namely composite video. If you have a TV card in your computer you could send the PS2 signal to that and send it out again via your GFX card's output to your monitor.
I haven't seen a standalone box that does composite to (any digital)
They make Composite to HDMI and Component to HDMI converters. However, the cheap ones are pretty junky and the good ones are $50+. You have have to weed through some reviews to see if people have had good experiences with a particular model. Do you not have another TV that you could hook this up to? Most HD TV's have both a Component and Composite inputs. I have my old PS2 hooked up via a $5 set of Component PS2 cables I got on eBay. Looks goods pretty decent when set to widescreen.