There are two aspects to your question:
First, who is on the deed to the home? From your question it seems that it is jointly owned.
Second: If you file for divorce, the court will have to determine how the property will be divided. If the house is awarded to you in the divorce, then most likely you would have to refinance the mortgage to remove your husband's name from the loan. I have never seen a lender agree to remove one party's name from a loan without doing a new loan.
In the interim, keep paying and you may be able to get a credit in the divisio of assets and debtsd for part of the payments you are making.
Best of luck,
Cindy S. Vova
Law Offices of Cindy S. Vova, P.A.
954-316-3496/561-962-2785
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