Developing lung capacity as an asthmatic can be a challenge. There are a few exercises you can do that may increase lung volume, but it's important to realize that simply increasing lung volume does not correlate directly with increased aerobic performance. The volume of air you can suck in is only part of the equation. While you would be able to store more air in your lungs, unless you increase the rate of oxygen transport within the lung tissue, you would not actually increase performance, as you would not deliver more oxygen to the working muscle faster.
Many exercisesincluding using blowing up balloons, inspirometers, or even the trusty straw-and-ping pong ballthat target= increasing lung volume work by exercising the intercostals muscles that raise and spread your ribcage, and your diaphragm. They don't necessarily do anything for the number or size of the alveoli (air sacs) within your lungs. To my knowledge, they also won't do anything to alleviate asthma.
Honestly, the asthmatic athletes I've worked with have been better off focusing on finding a good physician who specializes in the treatment of asthma in active populations. Exercise often exacerbates asthma and some physicians are more experienced in treating patients for whom "slowing down" or "finding another activity" isn't an option.