On jump-diffusion processes with regime switching: martingale approach

We study jump-diffusion processes with parameters switching at random times. Being motivated by possible applications, we characterise equivalent martingale measures for these processes by means of the relative entropy. The minimal entropy approach is also developed. It is shown that in contrast to the case of L\'evy processes, for this model an Esscher transformation does not produce the minimal relative entropy.Comment: 23 pages, 2 figure

The delayed uncoupled continuous-time random walks do not provide a model for the telegraph equation

It has been alleged in several papers that the so called delayed continuous-time random walks (DCTRWs) provide a model for the one-dimensional telegraph equation at microscopic level. This conclusion, being widespread now, is strange, since the telegraph equation describes phenomena with finite propagation speed, while the velocity of the motion of particles in the DCTRWs is infinite. In this paper we investigate how accurate are the approximations to the DCTRWs provided by the telegraph equation. We show that the diffusion equation, being the correct limit of the DCTRWs, gives better approximations in $L_2$ norm to the DCTRWs than the telegraph equation. We conclude therefore that, first, the DCTRWs do not provide any correct microscopic interpretation of the one-dimensional telegraph equation, and second, the kinetic (exact) model of the telegraph equation is different from the model based on the DCTRWs.Comment: 12 pages, 9 figure

Option Pricing Model Based on Telegraph Processes with Jumps

In this paper we overcome a lacks of Black-Scholes model, i.e. the infinite propagation velocity, the infinitely large asset prices etc. The proposed model is based on the telegraph process with jumps. The option price formula is derived.Telegraph Processes, option pricing

Single integro-differential wave equation for L\'evy walk

The integro-differential wave equation for the probability density function for a classical one-dimensional L\'evy walk with continuous sample paths has been derived. This equation involves a classical wave operator together with memory integrals describing the spatio-temporal coupling of the L\'evy walk. It is valid for any running time PDF and it does not involve any long-time large-scale approximations. It generalizes the well-known telegraph equation obtained from the persistent random walk. Several non-Markovian cases are considered when the particle's velocity alternates at the gamma and power-law distributed random times.Comment: 5 page