Would an 18th Century Duke have business holdings?

She can own it as a silent owner. Officially there is a man who is not noble, in charge with managing it, and who passes as the owner, but behind that ownership there is a secret contract of financing which shows that the capital (and what big parts of profit) belongs to her.

An English duke in the 18c would not be the public owner or directly involved in managing an import business. There were no formal prohibitions against titled nobility being involved in trade in England (unlike France, where it could jeopardize one's tax privileges to be actively engaged in commercial activities that weren't about agriculture or real estate) -- but the social prohibition was strict. If the duke is going to be making money in the import/export business, he's going to do it through passive investment or by going through third parties, and he will never participate directly in the operation in any way that doesn't allow for plausible deniability should anyone outside the firm catch wind of it. (People might whisper in scandalized tones about where does he get all that money, but they can't pin anything on him and all that money is a bit delicious, isn't it?)

Either that or he's a complete social renegade who doesn't give a shit what anybody else thinks of him, & doesn't care about its impact on anyone else in the family either.

He could definitely own the business, but he was unlikely to officially manage it. Many British noblemen owned coal mines, for example, though they probably never even visited them. And the Duke of Bridgewater developed and owned canals.
The noblemen who did this sort of thing were the ones who ended up rich in the 19th century, as opposed to the land poor ones who had to marry the American dollar princesses. *g*

Lil said:

He could definitely own the business, but he was unlikely to officially manage it. Many British noblemen owned coal mines, for example, though they probably never even visited them. And the Duke of Bridgewater developed and owned canals.

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Yes, mines were a bit of an odd duck as far as business went because traditionally nobles, like any other property-owner, were encouraged to extract wealth from their land however they could -- which traditionally included things like rents, wool, & timber. Mines started out as a small-scale extension of that kind of practice, so investment in coal mines was kind of grandfathered into the acceptable category even when it became a big industrial operation that raked in the £££££. Canals were also a form of property improvement, as were bridges, enclosures, and bog drainage. Making money from your property was fine -- totally different from getting your hands dirty in vulgar trade. Totally!

The history of the 2nd Marquess of Bute may prove useful https://en.wikipedia.org/wiki/John_Crichton-Stuart,_2nd_Marquess_of_Bute though he is rather later than your time period

I'm not an expert on British aristocracy. As I understand it. Tha aristocracy supported the ruler of the day with money and troops to fight wars. In exchange, they were given land and were exempt from paying tax to the crown. The land could be anywhere in the empire. The west indies was a popular location. The land provided income from rents, mineral rites and the like. Some aristocrats did take rent plus a percentage of the revenue earned from the business. So a kind of partnership did exist. Mills, breweries and coal mines were the typical types of business involved. It was the industrial revolution and the rise of the working middle class the brought the aristocrats bankruptcy.

stephenf said:

I'm not an expert on British aristocracy. As I understand it. Tha aristocracy supported the ruler of the day with money and troops to fight wars. In exchange, they were given land and were exempt from paying tax to the crown.

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That was sort of true in the 13c. By the 18c it had not been true for quite some time.