auto accident involving a pedestrian in New York City

Your insurance company will defend you and your son if he was an authorized driver. As he was working for his employer at the time of the accident, his employer may also have liability for any injuries caused by his negligence.

There are two basic issues.

1. Was your son negligent (at fault) in terms of his driving? That usually depends on all the facts and circumstances and who is likely to believe the two or more stories that will emerge -- his story (I was driving very slowly, paying attention, my headlights were on, the road was well lit, she darted out between 2 parked cars, etc.)  and the pedestrian's story ("I crossed at the crosswalk, with the light, after carefully looking both ways and this car with the lights out came from out of no where at a fast speed and hit me..."  If he was acting reasonably and his lights were on and she darted out and no reasonable driver could have stopped, there would be no liability. 

2. What were her damages -- not only immediate medical bills and other economic damages (such as loss of wages, torn clothing, costs of rehab and nursing care, etc.) but non-economic damages such as for any residual injury (such as a scar, a limp) and pain and suffering. 

Your insurance company (and his employer's) will either settle the case or pay any judgment up to and including the amount of the liability limit on your policy. If our limit is too low, and his employer lacks un-owned auto liability coverage you may have a problem. 

By the way, in NY any licensed lawyer can issue the summons, which starts the action.