Formation of "hot Jupiters" before star ignition I'm watching a BBC special on the formation of solar systems, and they're covering the topic of "hot Jupiters" that exist near their host stars. The prevailing theory, according to this 2015 special, is that the planets form farther out, then migrate in. This addresses the issue of their gas being driven off by the host star. 

Isn't also possible that these planets formed shortly before the ignition of the host star? The mass that will form the star is already mostly collected, and could hold bodies in orbit, but the heat has yet to begin, and wouldn't drive off gas.
Hot Jupiters can and do form before the hot star "ignites", as you put it.

The conventional core-accretion theory, by which gas-giant planets form, requires them to accrete from a disk of gas and dust for about 10 million years. Coincidentally (or perhaps not a coincidence?), the empirically determined lifetimes of circumstellar gas and dust disks around young stars is also about 10 million years.

Meanwhile, the pre-main-sequence lifetimes of newly-born stars (which I define here as the length of time it takes to contract from a large ball of gas down to a much more compact configuration, with a central temperature high enough to initiate hydrogen fusion), range from about 10 million years for stars a little more massive than the Sun to 100 million years or more for stars that are less massive.

I think, though, that the information that you are missing is that there really is no period of time where the "heat has yet to begin" whilst the protostar is surrounded by a gas/dust disk. A protostar is actually more luminous when it is younger and becomes fainter as it contracts towards its main sequence configuration. The energy that powers this luminosity is derived from gravitational potential energy. The inner portions of the gas/dust disk are always too hot to form hot Jupiters in situ and so all current models focus on how hot Jupiters can be "moved" from where they were formed to where they are found now.