The word escalate originates from a back-formation from the word escalator (a moving staircase), which itself comes from the French word escalade, meaning an assault with ladders on a fortification.  

The Oxford English Dictionary (OED) lists the earliest evidence for "escalate" as being from 1922, in the publication Granta. 

Escalate used in a sentence: 
"We do not wish to escalate the war."
"In turn, the increased prices caused health costs to escalate in a vicious circle."
"The president decided to escalate the dispute by withdrawing the ambassador." 
"We are trying not to escalate the violence." 
"The conflict has escalated into an all-out war."
"The cold weather has escalated fuel prices."

Just the facts:

ESCALATE Definition & Meaning - Merriam-Webster