Learn Swedish: Värdet på börsen sjunker oaktat att intäkten ökar. - The value on the stock exchange is falling notwithstanding that the revenue is increasing.

Oaktat att means exactly the same thing as trots att (despite that / even though), but it is much more formal. It is frequently used in financial, legal, or administrative contexts, making it a perfect fit for a sentence about the stock market (börsen).

Can I drop the word att and just say ... sjunker oaktat intäkten ökar?

No, you need att here. Because the second half of the sentence is a full clause with its own subject and verb (intäkten ökar), you must use the conjunction oaktat att. If you were only using a noun phrase without a verb, you would drop att and use it as a preposition, for example: oaktat den ökade intäkten (notwithstanding the increased revenue).

Why do we use the preposition på for börsen (the stock exchange)?

In Swedish, you typically use på for markets, financial institutions, and public offices (like på banken or på apoteket). If you said i börsen, it would sound like you are talking about the inside of a physical coin purse, which is the original literal meaning of the word börs.

Could I use minskar (decreases) instead of sjunker (sinks/falls) for the value?