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We establish, under the Cramer exponential moment condition in aneighbourhood of zero, the Extended Large Deviation Principle for the RandomWalk and the Compound Poisson processes in the metric space $\V$ of functionsof finite variation on $[0,\infty)$ with the modified Borovkov metric $\r(f,g)=\r_\B(\hat{f},\hat{g}) $, where $ \hat f(t)= f(t)/(1+t)$, $t\in \R$, and$\r_\B$ is the Borovkov metric. LDP in this space is "more precise" than thatwith the usual metric of uniform convergence on compacts.

Large Deviations for Processes on Half-Line: Random Walk and Compound Poisson