Let $mathfrak{z}$ be a stochastic exponential, i.e., $mathfrak{z}_t=1+int_0^tmathfrak{z}_{s-}dM_s$, of a local martingale $M$ with jumps $triangle M_t>-1$. Then $mathfrak{z}$ is a nonnegative local martingale with $Emathfrak{z}_tle 1$. If $Emathfrak{
z}_T= 1$, then $mathfrak{z}$ is a martingale on the time interval $[0,T]$. Martingale property plays an important role in many applications. It is therefore of interest to give natural and easy verifiable conditions for the martingale property. In this paper, the property $Emathfrak{z}_{_T}=1$ is verified with the so-called linear growth conditions involved in the definition of parameters of $M$, proposed by Girsanov cite{Girs}. These conditions generalize the Bene^s idea, cite{Benes}, and avoid the technology of piece-wise approximation. These conditions are applicable even if Novikov, cite{Novikov}, and Kazamaki, cite{Kaz}, conditions fail. They are effective for Markov processes that explode, Markov processes with jumps and also non Markov processes. Our approach is different to recently published papers cite{CFY} and cite{MiUr}.
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.
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 $M_t$.
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 func
tion $F(t)=mathsf{P}(xi_1le t)$. 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 $alphainbig(0,{1/2}big)$.
Let $sigma(u)$, $uin mathbb{R}$ be an ergodic stationary Markov chain, taking a finite number of values $a_1,...,a_m$, and $b(u)=g(sigma(u))$, where $g$ is a bounded and measurable function. We consider the diffusion type process $$ dX^epsilon_t =
b(X^epsilon_t/epsilon)dt + epsilon^kappasigmabig(X^epsilon_t/epsilonbig)dB_t, tle T $$ subject to $X^epsilon_0=x_0$, where $epsilon$ is a small positive parameter, $B_t$ is a Brownian motion, independent of $sigma$, and $kappa> 0$ is a fixed constant. We show that for $kappa<1/6$, the family ${X^epsilon_t}_{epsilonto 0}$ satisfies the Large Deviations Principle (LDP) of the Freidlin-Wentzell type with the constant drift $mathbf{b}$ and the diffusion $mathbf{a}$, given by $$ mathbf{b}=sumlimits_{i=1}^mdfrac{g(a_i)}{a^2_i}pi_iBig/ sumlimits_{i=1}^mdfrac{1}{a^2_i}pi_i, quad mathbf{a}=1Big/sumlimits_{i=1}^mdfrac{1}{a^2_i}pi_i, $$ where ${pi_1,...,pi_m}$ is the invariant distribution of the chain $sigma(u)$.
The Large Deviations Principle (LDP) is verified for a homogeneous diffusion process with respect to a Brownian motion $B_t$, $$ X^eps_t=x_0+int_0^tb(X^eps_s)ds+ epsint_0^tsigma(X^eps_s)dB_s, $$ where $b(x)$ and $sigma(x)$ are are locally Lipschitz f
unctions with super linear growth. We assume that the drift is directed towards the origin and the growth rates of the drift and diffusion terms are properly balanced. Nonsingularity of $a=sigmasigma^*(x)$ is not required.
The Large Deviation Principle is established for stochastic models defined by past-dependent non linear recursions with small noise. In the Markov case we use the result to obtain an explicit expression for the asymptotics of exit time.
We give asymptotic analysis for probability of absorbtion $mathsf{P}(tau_0le T)$ on the interval $[0,T]$, where $ tau_0=inf{t:X_t=0}$ and $X_t$ is a nonnegative diffusion process relative to Brownian motion $B_t$, dX_t&=mu X_tdt+sigma X^gamma_tdB_t.
X_0&=K>0 Diffusion parameter $sigma x^gamma$, $gammain [{1/2},1)$ is not Lipschitz continuous and assures $mathsf{P}(tau_0>T)>0$. Our main result: $$ limlimits_{Ktoinfty} frac{1}{K^{2(1-gamma)}}logmathsf{P}(tau_{0}le T) =-frac{1}{2E M^2_T}, $$ where $ M_T=int_0^Tsigma(1-gamma)e^{-(1-gamma)mu s}dB_s $. Moreover we describe the most likely path to absorbtion of the normed process $frac{X_t}{K}$ for $Ktoinfty$.
We formulate large deviations principle (LDP) for diffusion pair $(X^epsilon,xi^epsilon)=(X_t^epsilon,xi_t^epsilon)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, th
e LDP is established for $(X^epsilon, u^epsilon)$ with $ u^epsilon(dt,dz)$ being an occupation type measure corresponding to $xi_t^epsilon$. In some sense we obtain a combination of Freidlin-Wentzells and Donsker-Varadhans results. Our approach relies the concept of the exponential tightness and Puhalskiis theorem.
We consider a continuous time version of Cramers theorem with nonnegative summands $ S_t=frac{1}{t}sum_{i:tau_ile t}xi_i, t toinfty, $ where $(tau_i,xi_i)_{ige 1}$ is a sequence of random variables such that $tS_t$ is a random process with independent increments.
