All perfect praise be to Allah, The Lord of The Worlds; and may his blessings and peace be upon our Prophet Mohammad and upon all his family and companions.
Offering an Udhiyah (sacrifice in Eid-ul-Adha) on behalf of a deceased person, who hasn`t bequeathed that before his death, is a controversial issue amongst the scholars. We agree with the opinion of the majority of the Muslim scholars from the Hanafites, the Hanbalites, the Malikites and some of the Shafites, who consider it permissible and avails the deceased, Insha`Allah, even if he hasn`t bequeathed that.
Muslim has also reported on the authority of 'Aishah (May Allah Be Pleased with her): "Allah's Messenger (PBUH) ordered a horned ram with black legs, a black belly and black around its eyes, and it was brought for him to sacrifice and he said to 'Aishah (May Allah Be Pleased on him): "Aishah, get the knife." Then he said: "Sharpen it with a stone." When she had done so he took it, then taking it (the ram) he placed it on the ground and slaughtered it. He then said, "Bismillah (In the name of Allah), O Allah, accept it from Muhammad, Muhammad's family and from Muhammad's Ummah (followers)." (Then he sacrificed it.)". It is well known that some of nation of Muhammad are dead, and since He (PBUH) presented the reward of slaughtering that ram for his Ummah, then offering a sacrifice on behalf of a deceased is permissible.
Moreover, many texts of Sharia indicate that the deceased is rewarded for the acts of devotion undertaken on his behalf. For example, it is permissible to make up for the obligatory fast that he had missed or perform Hajj on his behalf. This has been proven through authentic narrations. Since the deceased is rewarded for the fast and the Hajj performed on his behalf, he will be rewarded for the sacrifice offered on his behalf as well. Also, there is a consensus amongst Muslim scholars that the deceased receives the reward of the charity given on his behalf, and the sacrifice is part of this charity. For all of the above reasons, we believe that it is permissible to offer a sacrifice on behalf of the deceased, even if he hasn`t bequeathed that.
Al-Kasani (May Allah bless him) said: "In the Hadith narrated by Muslim the Prophet (PBUH) sacrificed two rams: one on his behalf and the other on behalf of his Ummah. If some of them had died before they had the chance to slaughter a sacrifice, it is permissible to offer the deceased the reward for the acts of obedience performed on his behalf." [Bda`I As-Sanai`,vol.5/pp.72].
As regards the Hanbalites, they stated: "Offering a sacrifice on behalf of a deceased person is better than offering it on behalf of a living one, because of the deceased`s inability and his need for the reward from Allah."{Mat-talib Uli An-Nohaa}.
The Malikites stated: "It is discouraged to offer a sacrifice on behalf of a dead person for fear of getting accused with adulation and showing off. If the deceased had bought a sheep, but died before sacrificing it, then his heirs should slaughter it on his behalf." [Shareh Mokhtasar Khaleel, vol.3/pp.42] by Al-Khorashi.
The accredited opinion of the Shafite jurists is that offering a sacrifice on behalf of the deceased is impermissible. Al-Imam An-Nawawi (May Allah bless him) said: "As regards offering a sacrifice on behalf of the deceased, Abu Hassan Al-Ebbadi considered it permissible because it is a kind of charity and charity on behalf of the deceased is valid and avails him." [Al-Majmou`.vol.8/pp.380]. And Allah Knows Best.