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Register a new userThe Euler-Maruyama approximations for the CEV model
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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.
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The aim of this paper is to study weak and strong convergence of the Euler--Maruyama scheme for a solution of one-dimensional degenerate stochastic differential equation $mathrm{d} X_t=sigma(X_t) mathrm{d} W_t$ with non-sticky condition. For proving this, we first prove that the Euler--Maruyama scheme also satisfies non-sticky condition. As an example, we consider stochastic differential equation $mathrm{d} X_t=|X_t|^{alpha} mathrm{d} W_t$, $alpha in (0,1/2)$ with non-sticky boundary condition and we give some remarks on CEV models in mathematical finance.
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 $Ktoinfty$.
In this paper, the discrete parameter expansion is adopted to investigate the estimation of heat kernel for Euler-Maruyama scheme of SDEs driven by {alpha}-stable noise, which implies krylovs estimate and khasminskiis estimate. As an application, the convergence rate of Euler-Maruyama scheme of a class of multidimensional SDEs with singular drift( in aid of Zvonkins transformation) is obtained.
This paper investigates a partially observable queueing system with $N$ nodes in which each node has a dedicated arrival stream. There is an extra arrival stream to balance the load of the system by routing its customers to the shortest queue. In addition, a reward-cost structure is considered to analyze customers strategic behaviours. The equilibrium and socially optimal strategies are derived for the partially observable mean field limit model. Then, we show that the strategies obtained from the mean field model are good approximations to the model with finite $N$ nodes. Finally, numerical experiments are provided to compare the equilibrium and socially optimal behaviours, including joining probabilities and social benefits for different system parameters.
Based on a discrete version of the Pollaczeck-Khinchine formula, a general method to calculate the ultimate ruin probability in the Gerber-Dickson risk model is provided when claims follow a negative binomial mixture distribution. The result is then extended for claims with a mixed Poisson distribution. The formula obtained allows for some approximation procedures. Several examples are provided along with the numerical evidence of the accuracy of the approximations.
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