Yes, you can. On the tax side of things, your LLC will appear on your 1040 Sch C or will be an S-Corp that issues you a K-1. Either way, the business is an extension of you. Any income flows through to you. So if your LLC invests in something and has gains, then you have gains.
There's not a lot of point in keeping the assets in the company's name. You are being taxed on the income whether or not you distribute it to yourself. The LLC provides some protection to your personal assets, so it clearly benefits you to convert the income into investments on the personal side of things rather than the business side.
When you're ready to re-invest the proceeds into expanding your business, you would just make a capital contribution back to your business.