Canon Background
https://awoiaf.westeros.org/index.php/Baelish_

In-Campaign
When their liege lord wed Lysa Tully at Riverrun in 283 AC Baelish was seated next to Theodore near the back of the hall during the feast. After exchanging pleasantries Baelish made a request of Teddy. He explained that his son Petyr was in need of a position and that he could be an asset to House Jasper as its steward. Teddy agreed to speak to his lord father on Baelish’s behalf.

Lord Baelish, and his son, sought an audience with Lord Jasper in the waning days of Robert’s Rebellion. Teddy, Dolins, Calaila, and Rhaella were in attendance. Baelish spoke of the conversation he shared with Teddy at Riverrun, in which the lordling agreed to speak to his father regarding a position in the house for Petyr on Baelish’s behalf. It was mutually agreed that Petyr would act as a probationary steward for House Jasper for a period of no less than one year. Lord Baelish stated that his house had been done a kindness and that he would not forget it.

In April of 284 AC Baelish welcomed two unexpected guests to his table, Lord Teddy and Ser Maelys of House Jasper. The men shared a meal with Baelish’s rather small household and they discussed the recent capture of Ser Gyles, the knight who had been pillaging Baelish’s lands wearing the House Jasper sigil. Baelish thanked them and agreed to have his steward facilitate a raven to Lord Arryn.

He testified at the trial of Ser Gyles at Snownook in 284 AC. He spoke of the damage caused by the rogue knight and alleged that the man acted alone. The trial resulted in Gyles being found guilty and House Lynderly being punished. Following the conclusion of the trial, Baelish and his son were brought to audience withJon Arryn and the Jaspers. Due to Petyr’s merit and competencies as steward of House Jasper Petyr was offered the opportunity to travel to the capital with Jon and Lysa Arryn and serve as an advisor there. Baelish consented to his son accepting the opportunity, knowing it was what he wanted even though it would take Petyr further from home. Teddy also asked Baelish of any ties he has to Braavos. Baelish explained he had visited Braavos as a young man. Teddy asked if he knew anyone there he dealt in rare objects. Baelish had known a man there named Fyrio Alesis who had traded in rare items and oddities at an emporium near the waterfront.

He was present in the Tower of the Hand when Jon Arryn appointed House Jasper as Marshall of the FIngers and the Isles on the day Robert Baratheon wed Cersei Lannister. He was clearly pleased by the appointment. Baelish later attended the tournament at Snownook in 285 AC celebrating House Jasper’s new title. While watching the melee unfold with his son, Teddy Jasper approached. After some small talk Baelish recommended Teddy seek a buyer for his dragon egg in Braavos (a man named Fyrio Alesis).

Baelish’s wife died in 288 AC. House Jasper sent a wagon load of food and one of their better cooks as a way of showing condolences. The gesture was especially meaningful given that the first springtime harvest had recently be sown and winter food supplies were running low.