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Telegraph Processes with Random Jumps and Complete Market Models

Nikita Ratanov ()

Methodology and Computing in Applied Probability, 2015, vol. 17, issue 3, 677-695

Abstract: Abstract We propose a new generalisation of jump-telegraph process with variable velocities and jumps. Amplitude of the jumps and velocity values are random, and they depend on the time spent by the process in the previous state of the underlying Markov process. This construction is applied to markets modelling. The distribution densities and the moments satisfy some integral equations of the Volterra type. We use them for characterisation of the equivalent risk-neutral measure and for the expression of historical volatility in various settings. The fundamental equation is derived by similar arguments. Historical volatilities are computed numerically.

Keywords: Inhomogeneous Jump-telegraph process; Dependence on the past; Historical volatility; Compound poisson process; 60J27; 60J75; 60K99; 91G99 (search for similar items in EconPapers)
Date: 2015
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DOI: 10.1007/s11009-013-9388-x

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