Alexander Palmer repeatedly stabbed Peter Wrighton, 83, as he walked his two dogs in woodland near East Harling, Norfolk, in August.
His injuries were so severe that police initially believed he had been attacked by an animal.
Palmer, 24, denied murder but a jury took just 44 minutes to unanimously convict him at Nottingham Crown Court on Wednesday.
Following the verdict, Mr Wrighton’s family said mental health professionals had “failed” Palmer and they were “shocked, astounded and angered” at some of the evidence related to his care.
Palmer, who showed no emotion as the verdict was delivered, mouthed the words “I love you” to his family as the judge sent him to the cells.
Mr Justice Goose said: “It is inevitable the defendant will receive a sentence of life imprisonment. It will be up to me as to the minimum term he will have to serve.”
Police were initially unaware of Palmer until a psychologist who treated him at RAF Marham contacted officers after reading reports about Mr Wrighton’s murder.
While serving in the Army, he had been the victim of an assault which appeared to trigger problems that required help from mental health professionals.
Palmer told medical staff he had a voice in his head called “Little Alex” who told him to harm people or kill them, prosecutors said.
The voice would tell Palmer to stab the neck or throat of strangers and that dog walkers appeared to be a particular “bugbear” of his, the court heard.
Prosecutor Stephen Spence told the jury: “He appears to have some ill feeling or a grudge towards dog walkers. There were a number of references to attacks to the throat.
“Particularly of note was his desire to kill strangers; dog walkers seemed to be a particular bugbear of his.”
The court was told Palmer’s personalised number plate read “L666 AHP”, and was bought by his mother as a joke because he was a “little devil”.
He left the Army, where he served in a commando regiment, in November 2015.
In a statement read outside court, Mr Wrighton’s daughter Carol Todd described her father as a “lovely, gentle person” who was “quite simply in the wrong place at the wrong time”.
“We have all been struggling to come to terms with losing him in such a way and our mum, his wife of 59 years, has been left bereft and lonely,” she said.
Ms Todd thanked police for their work on the case but added: “The revelations of the evidence relating to the mental health of Alexander Palmer have shocked, astounded and angered us.
“Evidently an intelligent person, he was able to take himself off medication and get himself discharged from care.
“We feel this should not have happened and mental health professionals failed him, his family and our family.
“My mum, brother, myself and our children not only feel grief but anger, as we believe this crime could have been prevented.”
Palmer, of Cringleford, near Norwich, will be sentenced at Nottingham Crown Court on Thursday.