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We develop a method of stochastic differential equation to simulate electronacceleration at astrophysical shocks. Our method is based on It\^{o}'sstochastic differential equations coupled with a particle splitting, employinga skew Brownian motion where an asymmetric shock crossing probability isconsidered. Using this code, we perform simulations of electron acceleration atstationary plane parallel shock with various parameter sets, and studied howthe cutoff shape, which is characterized by cutoff shape parameter $a$, changeswith the momentum dependence of the diffusion coefficient $\beta$. In theage-limited cases, we reproduce previous results of other authors,$a\approx2\beta$. In the cooling-limited cases, the analytical expectation$a\approx\beta+1$ is roughly reproduced although we recognize deviations tosome extent. In the case of escape-limited acceleration, numerical result fitsanalytical stationary solution well, but deviates from the previous asymptoticanalytical formula $a\approx\beta$.

Electron acceleration with improved Stochastic Differential Equation method: cutoff shape of electron distribution in test-particle limit