Large deviations for two scaled diffusions
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We formulate large deviations principle (LDP) for diffusion pair (X ɛ ,ξ ɛ )=(X t ɛ ,ξ t ɛ ), where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for (X ɛ ,ν ɛ ) with ν ɛ (dt, dz) being an occupation type measure corresponding to ξ t ɛ . In some sense we obtain a combination of Freidlin–Wentzell’s and Donsker–Varadhan’s results. Our approach relies on the concept of the exponential tightness and Puhalskii’s theorem.
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