The　driver＇s　stochastic　nature　is　one　of　the　important　causes　of　traffic　oscillation.　To　better　describe　the　impact　of　the　driver＇s　stochastic　characteristics　on　car-following　behavior,　we　propose　a　stochastic　full　velocity　difference　model　(SFVDM)　considering　the　stochastic　variation　of　the　desired　velocity.　In　order　to　mitigate　traffic　oscillation　caused　by　driving　stochasticity,　we　further　propose　a　stable　speed　guidance　model　(S-SFVDM)　by　leveraging　vehicle-to-infrastructure　communication.　Stochastic　linear　stability　conditions　are　derived　to　demonstrate　the　prominent　influence　of　the　driver＇s　stochasticity　on　the　stability　of　traffic　flow　and　the　improvement　of　traffic　flow　stability　by　the　proposed　guidance　strategy,　respectively.　We　present　numerical　tests　to　demonstrate　the　effectiveness　of　the　proposed　models.　The　results　show　that　the　SFVDM　can　capture　the　traffic　oscillation　caused　by　the　driver＇s　stochastic　desired　velocity　and　reproduce　the　same　disturbance　growth　pattern　as　in　the　field　experiment.　The　results　also　indicate　that　the　S-SFVDM　can　significantly　expand　the　stable　area　of　traffic　flow　to　decrease　the　negative　impact　on　traffic　flow　stability　caused　by　the　driver＇s　stochastic　nature.