The Euler-Maruyama approximations for the CEV model

The CEV model is given by the stochastic differential equation $X_t=X_0+int_0^tmu X_sds+int_0^tsigma (X^+_s)^pdW_s$, $frac{1}{2}le p<1$. It features a non-Lipschitz diffusion coefficient and gets absorbed at zero with a positive probability. We show the weak convergence of Euler-Maruyama approximations $X_t^n$ to the process $X_t$, $0le tle T$, in the Skorokhod metric. We give a new approximation by continuous processes which allows to relax some technical conditions in the proof of weak convergence in cite{HZa} done in terms of discrete time martingale problem. We calculate ruin probabilities as an example of such approximation. We establish that the ruin probability evaluated by simulations is not guaranteed to converge to the theoretical one, because the point zero is a discontinuity point of the limiting distribution. To establish such convergence we use the Levy metric, and also confirm the convergence numerically. Although the result is given for the specific model, our method works in a more general case of non-Lipschitz diffusion with absorbtion.

Bene$check{bf S}$ condition for discontinuous exponential martingale

It is known the Girsanov exponent $mathfrak{z}_t$, being solution of Doleans-Dade equation $ mathfrak{z_t}=1+int_0^talpha(omega,s)dB_s $ generated by Brownian motion $B_t$ and a random process $alpha(omega,t)$ with $int_0^talpha^2(omega,s)ds<infty$ a.s., is the martingale provided that the Bene${rm check{s}}$ condition $$ |alpha(omega,t)|^2le text{rm const.}big[1+sup_{sin[0,t]}B^2_sbig], forall t>0, $$ holds true. In this paper, we show $B_t$ can be replaced by by a homogeneous purely discontinuous square integrable martingale $M_t$ with independent increments and paths from the Skorokhod space $ mathbb{D}_{[0,infty)} $ having positive jumps $triangle M_t$ with $Esum_{sin[0,t]}(triangle M_s)^3<infty$. A function $alpha(omega,t)$ is assumed to be nonnegative and predictable. Under this setting $mathfrak{z}_t$ is the martingale provided that $$ alpha^2(omega,t)le text{rm const.}big[1+sup_{sin[0,t]}M^2_{s-}big], forall t>0. $$ The method of proof differs from the original Bene${rm check{s}}$ one and is compatible for both setting with $B_t$ and $M_t$.

Large Deviations Application to Billingsleys Example

We consider a classical model related to an empirical distribution function $ F_n(t)=frac{1}{n}sum_{k=1}^nI_{{xi_kle t}}$ of $(xi_k)_{ige 1}$ -- i.i.d. sequence of random variables, supported on the interval $[0,1]$, with continuous distribution function $F(t)=mathsf{P}(xi_1le t)$. Applying ``Stopping Time Techniques, we give a proof of Kolmogorovs exponential bound $$ mathsf{P}big(sup_{tin[0,1]}|F_n(t)-F(t)|ge varepsilonbig)le text{const.}e^{-ndelta_varepsilon} $$ conjectured by Kolmogorov in 1943. Using this bound we establish a best possible logarithmic asymptotic of $$ mathsf{P}big(sup_{tin[0,1]}n^alpha|F_n(t)-F(t)|ge varepsilonbig) $$ with rate $ frac{1}{n^{1-2alpha}} $ slower than $frac{1}{n}$ for any $alphainbig(0,{1/2}big)$.