Ruin Analysis and Ldp for Cev Model

We give results on the probability of absorption at zero of the diffusion process Xt with X0 = K > 0 and non-Lipschitz diffusion coefficient σx , γ ∈ [ 1 2 , 1): dXt = μXtdt + σX γ t dBt relative to Brownian motion Bt. Our results give information on the time to ruin τ0 = inf{t : Xt = 0}. We show that P (τ0 ≤ T ) > 0 for all T , give the probability of ultimate ruin, and establish asymptotics in Large Deviations Principle (LDP) scale: lim K→∞ 1 K2(1−γ) logP(τ0 ≤ T ) = − 1 σ2 μ (1 − γ)[1 − e−2μ(1−γ)T ] related to a normed family ̆` Xt K )t∈[0,T ] ̄ K→∞ which is in Freidlin-Wentzell’s framework. In spite of the fact a singular and only Hölder continuous diffusion coefficient, providing ruin of Xt, Freidlin-Wentzell’s result remains valid. A proof of that requires additional efforts having an independent interest. In addition, an approximation to the most likely paths to ruin is given.