Some Inequalities Of Diffusion Processes

Let X= (X_t, F_t)_t≥0 be a diffusion process on R given bydX_t=μ(X_tdt)+σ(X_t)dB_t, X₀=x₀,where B= (B_t)_t≥0 a standard Brownian motion starting at zero andμ,σtwo continuous functions on R, andσ(x)＞0 if x≠0. For a nonnegative continuous functionφwe define the functional J =(J_t,F_t)_t≥0 by J_t= integral from n=0 to tφ(X_s)ds, t≥0. Then under suitable conditions we establish the relationship between L^p-norm of sup₀≤t≤Υ|X+_t| and L^p-norm of J_Î¥for all stopping timesÎ¥. In particular, for a Bessel process Z of dimensionδ＞0 starting at zero, we show that the inequalities hold for all 0＜p＜2 and all stopping timesÎ¥, and we also obtain some inequalities for Ornstein-Uhlenbeck process. More specially, we show that for every continuous local martingale M=(M_t, F_t)_t≥0 the inequalities hold for all 0＜p＜∞andμ＞0, where C_p and c_p are some positive constants depending only on p, and H_μ, h_μare the inverses of x→(e^2μx-2_μx-1)/2_μ² and x→(e^-2μx+2_μx-1)/2_μ² on (0,∞), respectively. Finally, as a related problem we establish the L^P-estimate on the ratio where Z is a Bessel process of dimensionδ＞0 andÎ¥is a stopping time. Zhu Bei (Applied mathematics) Supervised by Yan Litan...