debris disks are disks of dust 10s100s au in size that surround main sequence stars of a wide range of stellar types ( a to m ) and ages ( 0.0110 gyr ) . in general ,debris disks are not spatially resolved and are identified in the infrared from the dust thermal emission that results in an excess over the expected stellar values .debris disks surveys carried out with indicate that they contain a few lunar masses of dust and negligible quantities of gas , and that they are present around % of a - type stars ( su _ et al ._ 2006 ) and 1015% of solar - type fgk stars ( bryden _ et al ._ 2006 ; beichman _ et al ._ 2006 ; trilling _ et al ._ 2008 ; hillenbrand _ et al ._ in preparation ; carpenter __ in preparation ) .however , these results are calibration limited because the disks can only be detected at a certain level above the stellar photosphere due to uncertainties in the stellar flux .figure 1 shows examples of some nearby spatially resolved debris disks .  )-pic ( 0.21  m ; heap _ et al ._ , 2000 ) , au - mic ( 1.63  m ; liu , 2004 ) and tw hydra ( 0.21  m ; roberge , weinberger and malumuth , 2005 ) ; ( ) hd 141569 ( 0.460.72  m ; clampin _ et al ._ , 2003 ) ; ( ) fomalhaut ( 0.690.97  m ; kalas _ et al . _ , 2005 ) and -eri ( 850  m ; greaves _ et al ._ , 2005 ) ; ( ) hr4796 ( 18.2  m ; wyatt _ et al ._ , 1999 ) , hd 32297 ( 1.1  m ; schneider , silverstone and hines , 2005 ) and fomalhaut ( 24 and 70  m ; stapelfeldt _ et al ._ , 2004 ) ; ( ) vega ( 850  m ; holland _ et al ._ , 1998 ) , -eri ( 850  m ; greaves _ et al ._ , 1998 ) , fomalhaut ( 450  m ; holland _ et al ._ , 2003 ) , -pic ( 12.3  m ; telesco _ et al ._ , 2005 ) and au - mic ( 0.460.72  m ; krist _ et al ._ , 2005 ) .all images show emission from 10s to 100s of au . ]the term refers to the fact that the dust can not be primordial , because the expected lifetime of the dust grains due to poynting - robertson drag (  where , and are the grain location , radius and density , respectively burns , lamy and soter , 1979 and backman and paresce , 1993 ) and mutual grain collisions (  backman and paresce , 1993 ) is much shorter than the age of the star , which means that the dust is likely being regenerated by planetesimals like the asteroids , kuiper belt objects ( kbos ) and comets in our solar system .indeed , the solar system is filled in with dust .the sources of dust are the asteroids and comets in the inner region and the kbos and interstellar dust in the outer region .the dust produced in the inner region can be seen in scattered light with our naked eyes , either in the zodiacal light on in the coma of comets , and has extensively been observed in thermal emission by space - based observatories ( and ) .evidence of the presence of dust originated in the kuiper belt ( kb ) comes from dust collision events detected by pioneer 10 and 11 beyond the orbit of saturn ( landgraf _ et al ._ , 2002 ) .figure 2 shows the location of the planetesimals in the outer solar system ( left ) and the expected spatial distribution of the dust generated in that region ( right ) .[ fig : sub : a ] ) distribution of planetesimals in the outer solar system ( courtesy of g. williams at the minor planet center ) .the outer circle is the orbit of neptune .( ) distribution of dust in the outer solar system resulting from dynamical simulations of dust particles originated in the kuiper belt ( from moro - martn and malhotra , 2002 ) .the scale is the same as in the previous panel , with the black dot representing the location of neptune .the structure is the result of gravitational perturbations of the giant planets on the orbit of the dust particles ( see sec .3).,title="fig:",width=188 ] [ fig : sub : b ] ) distribution of planetesimals in the outer solar system ( courtesy of g. williams at the minor planet center ) .the outer circle is the orbit of neptune . ( ) distribution of dust in the outer solar system resulting from dynamical simulations of dust particles originated in the kuiper belt ( from moro - martn and malhotra , 2002 ) .the scale is the same as in the previous panel , with the black dot representing the location of neptune .the structure is the result of gravitational perturbations of the giant planets on the orbit of the dust particles ( see sec .3).,title="fig:",width=188 ]  it is important to study the connection between the solar system debris disks and the much brighter extra - solar debris disks because models and observations of the solar system can help us understand the debris disk phenomenon , and vice versa , models and observations of extra - solar debris disks can help us place our solar system into context .it is thought that the solar system was significantly more dusty in the past because both the asteroid belt ( ab ) and the kuiper belt ( kb ) were more densely populated .the system then became progressively less dusty as the planetesimal belts eroded away by mutual planetesimal collisions .evidence of collisional evolution comes from the modeling and observation of the asteroid and kbo size distributions . in the ab ,bottke _ et al . _( 2005 ) showed that the initial size distribution progressively changes from a power - law to the observed wavy distribution , with peaks at km ( leftover from the accretion process ) and m ( marking the transition at which the energy required to catastrophically destroy a particle is determined by self - gravity rather than strength forces ) . in the kb , bernstein _( 2004 ) found that its current size distribution shows a strong break to a shallower distribution at km ( when particles become more susceptible to collisional destruction ) .models show that this collisional evolution likely resulted in the production of large quantities of dust , as it can be seen in figure 3 ( from kenyon and bromley , 2005 ) : in a planetesimal belt , pluto - sized bodies ( km ) excite the eccentricities of the smaller and more abundant 110 km sized planetesimals , triggering collisions and starting a collisional cascade that produces dust and changes the planetesimal size distribution . because the dust production rate is proportional to the number of collisions , and this is proportional to the square of the number of planetesimals , as the planetesimals erode and grind down to dust , the dust production rate decreases and the expected thermal emission from the dust slowly decays with time as .this decay is punctuated by large spikes that are due to particularly large planetesimal collisions happening stochastically .examples of stochastic events in the recent history of the solar system are the fragmentation of the asteroids giving rise to the hirayama and veritas asteroid families ( the latter happening 8.3 myr ago and accounting for 25% of the present zodiacal thermal emission ; dermott _ et al . _ , 2002 ) and the dust bands observed by _ iras _ ( sykes and greenberg , 1986 ) .m dust thermal emission expected from the collisional evolution of two planetesimal belts extending from 0.681.32 au ( dashed line ) and 0.42 au ( solid line ) around a solar type star ( kenyon and bromley , 2005 ) . ]recent surveys carried out by _spitzer / mips _ have enabled the detection of debris disks around hundreds of a - type and solar - type stars with a wide range of ages , showing that the dust emission follows a decay and there is a large variability likely due to individual collisions ( see figure 4 ) , in broad agreement with the results from collisional cascade models ( su _ et al ._ , 2006 ; siegler _ et al ._ , 2007 ) .because solar and extra - solar planetary systems seem to follow similar evolutions , the imaging of debris disks at different evolutionary stages could be equivalent to a solar system `` time machine '' .m for a survey of solar - type ( fgk ) stars .the stars aligned vertically belong to clusters or associations , therefore sharing the same age .the main features are the decay and the large variability found for a given stellar age .a few particularly massive debris disks are labeled .figure from siegler _ et al . _( 2007 ) . ]   as discussed above , there is observational and theoretical evidence that collisional evolution played a role in the evolution of solar and extra - solar debris disks .however , there is also evidence that additional non - collisional processes , likely related to the dynamical depletion of planetesimals that can result from gravitational interactions with massive planets , have also played a major role is disk evolution .  in the solar system, evidence comes from the late heavy bombardment ( lhb , or lunar cataclysm ) , a period of time in the solar system past during which a large number of impact craters in the moon and the terrestrial planets were created ( with an impact rate at earth of the current value ) .this event , dated from lunar samples of impact melt rocks , happened during a very narrow interval of time 3.8 to 4.1 gyr ago ( myr after the formation of the terrestrial planets ) .thereafter , the impact rate decreased exponentially with a time constant ranging from 10100 myr ( chyba , 1990 ) .( 2005 ) compared the impact cratering record and inferred crater size distribution on the moon , mars , venus and mercury , to the size distribution of different asteroidal populations , showing that the lhb lasted myr , the source of the impactors was the main ab , and the mechanism was size independent .the most likely scenario is that the orbital migration of the giant planets caused a resonance sweeping of the ab and as a result many of the asteroidal orbits became unstable , causing a large scale ejection of bodies into planet - crossing orbits ( explaining the observed cratering record ) , and an increased rate of asteroidal collisions that would have been accompanied by the production of large quantities of dust . under this scenario ,the lhb was a single event in the history of the solar system ( strom _ et al ._ , 2005 ) .a handful of extra - solar debris disks observed with _spitzer _ also show evidence of non - collisional evolution ( e.g. bd+20307 , hd 72905 , eta - corvi and hd 69830 ; wyatt _ et al ._ , 2007 ) .a particularly interesting case is that of hd 69830 , a system that harbors three neptune - like planets inside 2au , shows a strong 24 m dust emission indicative of large quantities of warm grains , no 70 m dust emission indicative of a lack of cold dust , and a dust emission spectra with strong features remarkably similar to the spectra of comet hale bopp ( figure 5 beichman _ et al ._ , 2005a ) .( 2007 ) showed that its 24 m emission , seen as an outlier in figure 4 , implies a very high dust production rate that could not possibly have been sustained for the entire lifetime of the star and must therefore be a transient event rather than the results of steady collisional evolution .its transient nature would account for the fact that hd 69830 shows strong silicate emission features indicative of the presence of large quantities of small grains .these grains have a very short lifetime and therefore must have been produced in a recent event ., 2007 suggested that we may be witnessing a lhb - type of event in which icy planetesimals originally located outside the orbit of the planets are scattered into the inner system producing the observed dust .the models and observations described above for both solar and extra - solar systems indicate that in a planetesimal swarm there is collisional evolution that produces dust , triggered by the largest ( pluto - sized ) planetesimals in the swarm , and on top of that , depending on the planetary configuration , there may be drastic dynamical events that produce very significant depletion of planetesimals and an increased rate of planetesimal collisions and dust production .the next section discusses how the presence of planets not only can affect the production of debris dust , but can also sculpt the debris disk by creating a rich diversity of spatial structure .even though the great majority of debris disks observations are spatially unresolved , their structure can be studied in some detail through the spectral energy distribution ( sed ) of the disk because different wavelengths in the sed trace different distances to the star , so that an sed with sufficiently high spectral resolution can be used to constrain roughly the radial distribution of dust . recent _debris disks surveys suggest that debris disks commonly show evidence of the presence of inner cavities , as most systems show 70 m dust emission ( from cold dust ) , but no emission at 24 m ( i.e. , no warm dust ; see e.g. meyer _ et al . _ , 2004 ; beichman _ et al . _ , 2005b ; bryden _ et al ._ , 2006 ; kim _ et al ._ , 2005 ; moro - martn , wolf and malhotra , 2005 ; moro - martn _ et al ._ , 2007a ; hillenbrand _ et al ._ , in preparation ) .high resolution spatially resolved observations have been obtained for a handful of nearby debris disks and indeed these images show the presence of inner cavities together with more complex morphology , like warps , spirals , offsets , brightness asymmetries and clumpy rings ( see figure 1 ) .dynamical simulations of the orbits of dust particles and their parent planetesimal in systems where massive planets are present suggest that this complex morphology could be the result from gravitational perturbations by planets ( e.g. roques _et al . _ , 1994 ; mouillet _ et al . _ , 1997 ; wyatt _ et al . _ , 1999; wyatt , 2005 , 2006 ; liou and zook , 1999 ; moro - martn and malhotra , 2002 , 2003 , 2005 ; moro - martn , wolf and malhotra , 2005 ; kuchner and holman , 2003 ; see moro - martn _ et al . _ , 2007b for a review ) .the basic mechanisms by which the planets can affect the debris disks structure are the following :  * _ ejection by gravitational scattering : _ this process can affect dust particles as they spiral inward under p - r drag , and dust - producing planetesimals , in the case when the planet migrates outwards , resulting in a depletion of dust inside the orbit of the planet ( an inner cavity ) .dynamical simulations show that this process can be very efficient , ejecting % of the particles in the case of a 310 m planet located between 130 au around a solar - type star . * _ trapping in mean motion resonances ( mmr ) with the planet : _ in a system where the dust producing planetesimals are located outside the orbit of the planet , as the dust particle drift inward due to p - r drag they can get trapped in mmrs with the planet .the mmrs are located where the orbital period of the planet is times that of the particle , where and are integers , and . at these locations the particle receives energy from the perturbing planet that can balance the energy loss due to p - r drag , halting the inward motion of the particle and giving rise to planetary resonant rings . due to the geometry of the resonance ,the spatial distribution of material in resonance is asymmetric with respect to the planet , and this can explain the clumpy structure observed in some disks ( figure 1 ) .an example of mmr trapping of kb in the solar system can be seen in figure 2 , where the ring - like structure , the asymmetric clumps along the orbit of neptune , and the clearing of dust at neptune s location are all due to the trapping of particles in mmrs with the planet , while the dust depleted region inside 10 au is due to gravitational scattering by saturn and jupiter .mmrs can also affect the location of the planetesimals and the dust when the planets migrate outward . * _ effects of secular perturbations : _ these are the long - term average of the perturbing forces and act on timescales .1 myr ( see review by wyatt _ et al . _ , 1999 ) .if the planet and the planetesimal disk are not coplanar , the secular perturbations tend to align the orbits and in the process they will create a warp in the disk .if the planet is in an eccentric orbit , the secular perturbations will force an eccentricity on the dust particles , creating an offset in the disk center with respect to the star that can result in a brightness asymmetry .other effects of secular perturbations are spirals and inner gaps .finally , it is important to point out that because the debris disk structure is sensitive to the presence of planets located far from the star , the study of the structure could be used as a potential planet detection method that would be complementary to the well - established radial velocity and transit techniques ( sensitive to close - in planets ) .large surveys of debris disks over a wide range of evolutionary states , enabled by high sensitivity spaced - based ir telescopes like , are starting to provide a `` movie '' of how planetary systems evolve with time . in thisregard , debris disks help us place our solar system into a broader context and vice versa , the study of the solar system , in particular its dynamical history and the characterization of its small body population , sheds light on the physical processes giving rise to the debris disk phenomenon .debris disks surveys , together with the results from planet searches , can help us understand the frequency of planetesimal and planet formation and the diversity of planetary systems , which ultimately addresses one of the most fundamental questions : is the solar system common or rare ?a.m.m . warmly thanks the organizers of the iau symposium no . 249 for their kind invitation and generous finantial support .is under contract with the jet propulsion laboratory ( jpl ) funded by nasa through the michelson fellowship program .jpl is managed for nasa by the california institute of technology .is also supported by the lyman spitzer fellowship at princeton university .

Abstract: this paper emphasizes the connection between solar and extra - solar debris disks : how models and observations of the solar system are helping us understand the debris disk phenomenon , and vice versa , how debris disks are helping us place our solar system into context .


ion - beam surface sputtering ( ibss ) is a versatile technique for producing nanostructures on material surfaces , for high - tech opto - magneto - electronic applications .these nano - structures are formed by self - organisation of surface atoms , as the surface undergoes continuous bombardment by energetic particles , into highly ordered nano - ripples , nano - dots , and nano - holes .their type , size , shape , separation , symmetry / crystallinity , and orientation can be tuned by varying the sputtering conditions such as irradiation time , ion energy , dose , angle of incidence , and substrate temperature .up till now , the level of control attained in the use of this technique for the fabrication of nanostructures defies a fundamental understanding of the phenomenon , though its core mechanisms have been identified , which greatly limits a large - scale application of the technique .this knowledge gap is further compounded by recent ascertainment of a decisive influence of contaminants in the self - organisation process .for instance , metal ( e.g. fe and mo ) atoms may be accidentally sputtered off the ion source , vacuum walls and target clamps , made from such metallic materials , and incorporated in the evolving surface , thus playing an essential role in the surface morphology . on the other hand , some recent experiments reported the production of pure nanostructures by ibss , the morphology of which was uninfluenced by contaminants .thus , it remains unclear whether there is another mechanism which acts differently in the presence or absence of contaminants such that nanostructures are produced anyway , and whether this mechanism plays any role in tuning the characteristics of the nanostructures .until these issues are resolved in both theory and experiments , the potential of ibss as a nano - patterning and nanostructure production technique will remain largely untapped . and, the potential of the nanostructures evolved from the process for super / smart - tech applications that would exploit their properties and tunable characteristics will remain largely unutilised .therefore , the importance of theoretical modeling for investigating the mechanisms involved in the ibss processes as well as their dynamics and interplays can not be over - emphasised in the quest to achieving a precise control over the above - mentioned properties of the nanostructures produced by ibss . andthe current knowledge gap , despite advances in the theory , motivates a closer scrutiny of existing theoretical models and the development of more sensitive ones .the theory has been widely studied in three major theoretical frameworks , namely : molecular dynamics simulations in the classical physics framework , monte carlo simulations in the statistical physics framework , and simulations with continuum models in a mathematical framework of linear and nonlinear partial differential equations .  however , this is an area where the theoretical framework of statistical physics is particularly suited to reveal interesting and surprising correspondence with other non - equilibrium phenomena .a characteristic feature of this framework is the concept of universality , by which seemingly different and diverse phenomena are governed by the same simple statistical mechanical law because oftentimes the apparent complications of the relevant processes mask well - ordered and far less complicated mechanisms which are inconspicuously present and are ultimately responsible for the law ( see refs . and for reviews , and ref . for a similar feature in quantum field theory ) .  hence ,in this article and with subsequent investigations of universality in view , we introduce a discrete model in the framework of statistical physics , for the simulation of ibss , taking cognizance of the core mechanisms in ibss and their influence on the dynamics and self - organisation of surface constituent particles . in this instance , we focus on amorphous surfaces or surfaces amorphised by the ion impacts ; a characteristic of semiconductor materials , and a paradigm of which is si .this model differs from the other statistical models in its semi - empirical nature and material particle conservation .the rest of the article is organised as follows .we provide background material in section [ sec : expttheor ] , where we review the experimental background of ibss in section [ sec : revexpt ] .in section [ sec : revtheor ] , we review a few md , mc , and md - mc hybrid models in the literature in order to provide a background to the existing models .the experiments and models reviewed in this paper are just a few of the common ones and the more recent ones the author has come across , and not exhaustive . in section [ sec : probs ] , we discuss some open questions and issues on the existing models . in section [ sec :model ] , we present our model and discuss its relation to experimental parameters . in section [ sec : results ] , we present and discuss our results , and finally , we round up the paper with a conclusion and suggestions for further work .in this section , we review the experimental features of surfaces subjected to ibss with singly charged ( usually noble gas ) ions with kinetic energies ranging from 0.1 to 10 kev .other singly charged ions have been used in a few cases but are known to produce similar surface morphologies and topographies as the noble gas ions , except for some distortions which are believed to be localised and abnormal [ 4 ] .higher energy apparatus are far more expensive such that the operation costs may outweigh the cost - effectiveness of the ibss technique . besides they lead to more pronounced sputtering yields and surface erosion , and irradiation - induced damage of the target material which may override any mechanism for self - organised structures of interest . reported the 1 kev , normal incidence , ar sputtering of si(1 0 0 ) surfaces with and without incorporation of mo atom contaminants simultaneously sputtered from the target - fixing clamps .nano - dot patterned and smooth surfaces were found in the presence and absence of mo contaminants , respectively .further , it was demonstrated in ref . under contaminant - free conditions that normal incidence sputtering of a si surface with 0.1 to 0.5 kev ar ions smoothens it .  in refs , a systematically controlled contaminant - free environment in the low energy , 200 ev to 1kev ,ar ion sputtering of si surfaces for a wide range of incidence angle was reported .nanostructures were not found for less than a critical angle , nano - ripples with wave vector parallel to the projection of the ion beam direction onto the surface plane , was found for , while nano - ripples with perpendicular wave vector was found for . seems to differ with the target material , for instance larger has been found in ref . for low energy xe sputtering of si targets .the precise dependence of on the target material and sputtering parameters is yet to be ascertained .patterning dynamics is known to increase with , hence , ripples formed near may require longer irradiations times , to be as pronounced as those for which may be more prominent for shorter sputtering times . it should be noted that in all the experiments the sputtered material was not removed from the vacuum chamber but mostly redeposited on the target and re - incorporated into the target material .in the discrete model of hartmann et . the sputtering process is simulated on a square lattice ( surface ) of area , where l is the lateral size , with periodic boundary conditions .an ion is projected from a random position in a plane parallel to the surface plane , travels along a straight path with an inclination angle of to the surface normal and azimuth .on reaching the surface , the ion penetrates the target substrate through a depth along the original trajectory , but now within the target , distributing its energy within an ellipsoid centered on its stopping position d. as a consequence of the energy redistribution , a surface atom at position * r * is eroded , using standard monte carlo simulation techniques , with a metropolis algorithm and probability proportional to the energy , received at * r * and given by   where is the energy of the ion , is the distance of the surface atom from the stopping point of the ion , and is the distance of the atom perpendicular to the ion trajectory . and are the widths of the ellipsoid parallel and perpendicular to the ion trajectory , respectively .this model simulates material redistribution on the surface as a result of the ion impact , and the non - equilibrium thermal interactions of the target , according to a nearest - neighbour solid - on - solid model of surface diffusion in monte carlo simulations of surface growth .other researchers have refined the energy distribution , as given in eq .( 1 ) by the sigmund model , for a better agreement with md simulations and experiments .however , these modifications only give very approximate results of the sputter yield at grazing angles . as a matter of fact ,the modified sigmund theory does not produce zero sputter yields for .al . performed kinetic mc ( kmc ) simulations of ion beam surface sputtering , using green s functions obtained from md simulations , to study the mechanisms of sputter - erosion and surface curvature - dependent sputtering .these green s functions , which are called crater functions , accounts for both sputtering and ion - induced re - organisation of surface particles and are used to determine the response of a surface to bombardment with a broad ion beam .these crater functions differ from the linear response functions studied much earlier , in the pioneering stages of the development of a suitable theory of sputter erosion , by eklund et .al . who concluded that linear response theory is not suitable for the description of the phenomenon of sputter - erosion .the model of yang et .al . is a hybrid md - kmc model which added a kmc arrhenius model , for the simulation of surface diffusion , to existing md model .nietiadi et .al . performed mc and md simulations of the sputtering of spherical objects , such as clusters , nanoparticles , or aerosol particles , with energetic particles ; taking as a prototypical example , the sputtering of a - si clusters with 20 kev ar ions .they employed the mc code of tri3dst with collisional algorithms that were taken from the sputter version trim.sp of the trim ( transport of ions in matter ) code , with recent modifications . among other classical motion ,the propagation of incident ions and their generated recoils in an amorphous medium was simulated as a sequence of binary collisions in a repulsive screened coulomb potential with the kr - c parametrisation , and the interaction of si atoms of the cluster was modelled with the stillinger - weber potential which fits to the ziegler - biersack - littmark potential for small interaction distances .the issue of contamination , arising from secondary sputtering effects , has rarely being addressed by existing discrete models and remains an open question which was a part of the motivation for this new model , and which this model was aimed at addressing by making it semi - empirical .although , contamination is an unwanted anomaly in ibss , it is now been regarded as a new tool to control the properties of the nanostructures as well as for developing new nanostructures .the yield parameter of this model provides an avenue for studying the influence of these contaminants , however , in this first application , we put the contaminant problem aside to first consider the basics . a brief idea of how the model is applicable to the contaminant problemis provided in the suggestion for further work .the issue of the derivation and parametrisation of an adequate equation of motion for an appropriate continuum modeling of the surface height evolution is also still an open question which this model has the potential of solving via simulated surface height evolution data analysis which can provide detailed information about the height evolution as a function of the relevant sputtering parameters , and can therefore provide knowledge of the applicable discrete and continuum equations of motion .the kinetic monte carlo models have a number of parameters for which realistic or practical interpretations are yet unclear , and seem to be so because they are generalised theories which require extensive simulations for fine - tuning the parameters .this new model bypasses some of these parameters by being based mostly on the empirical sputtering yield .more importantly this new model has a direct relation to experimental parameters like type of ion , ion energy , angle of incidence , and target material , through the sputtering yield .the parameters of any suitable model are the type of ion , ion energy , angle of incidence , type of material or material surface binding energy , ion flux or fluence , and substrate temperature .the first four parameters are accounted for by the sputter yield , while the last two are accounted for in the simulation time ( mc steps ) and temperature ( boltzmann energy scale ) , respectively .however , the target ( ambient ) temperature is a floating variable which , though incorporated in the simulation `` temperature '' , is inseparably interwoven with the binding energy in the sense that increasing temperatures merely energise the target atoms and reduce the binding energy ; this model ignores this effect in this first instance , if found necessary in further studies then it might be incorporated .this model therefore employs a sputter yield parameter which encapsulates both the angular and energy dependence .hence this simple model reiterates the dominance of only one mechanism : the modulated erosion .the modulation entails a huge amount of detail but most of these details eventually lead to the same result of the sputter yield ; in other words , a relatively low range of sputter yields account for a wide range of other parameters encompassed in the huge details of the processes leading to the erosion of surface particles .this model significantly differs from other mc models in its incorporation of a re - deposition algorithm , in the mould of simulation models of surface growth . in thiswise , a holistic viewpoint is taken where the evolving surface is considered as a canonical ensemble ( number of ions is negligible , relative to the number of substrate atoms ) , and the relaxation time is assumed to be shorter than the ion arrival times ( the disturbance is localised in a relatively tiny area around the incidence spot and hence insignificant on the overall relaxation of the material ) for approximating the theory with equilibrium statistical physics principles . hence , in this model , surface diffusion is assumed to be a thermally activated integral part of surface re - organisation , and is not considered separately .the canonical assumption may become invalid at high fluences ( long sputtering times ) and small substrate dimensions and thickness , but it nevertheless is a good starting point .this model comprises of an ion beam surface erosion algorithm ( ibsea ) , a redeposition of eroded material algorithm ( rema ) , and a surface reorganization algorithm ( sra ) . in the ibsea ,a surface site is randomly chosen and a number of surface particles in the vicinity of that site are eroded .all eroded particles are same - plane nearest neighbours , except for surface depressions and protrusions , and the erosion process is subjected to periodic boundary conditions . is determined from the table of sputter yield values calculated by using the semi - empirical equation of seah , clifford , green and gilmore for the sputtering yields of monoatomic elemental solids .nevertheless , any more accurate sputter yield formula can be assumed for .if the chosen site is on a depression then , for , the topmost atoms of nearest neighbor sites are eroded instead of the same - plane nearest neighbours in order to avoid overhangs . on the other hand , if the chosen site is a surface protrusion then the occupied sites below the vacant nearest neighbor sites are eroded , in addition to the chosen site , for . in the rema we choose a random direction for re - deposition , and a random re - deposition site between a distance of one lattice parameter and from the chosen erosion site ; with periodic boundary conditions . where is 1.627 , for 1kev ar sputtering of si at incidence , and kg , for si atom . , and reflects that only a small fraction of contributes to the sputtering , the rest merely disturbs the atoms in the vicinity of the impact point through recoils ; we set .the randomness in the range of the ejected particle ( re - deposition site ) reflects local variations in the projection angle of the ejected particle , as well as in the fraction of the ion kinetic energy transferred to the ejected particle .that is , even if the surface atom was ejected at the angle of radians for maximum range , its actual maximum range may still be much less , as a result of its site bonding ( lattice coordination ) or local interactions with its nearest neighbours ; both of which are crucial factors that will determine the fraction of transferred to the ejected particle . in line with the standard approach of aiming for higher lattice coordination in solid - on - solid ( sos ) simulations , to reflect the tendency of atoms to maximize the surface binding energy, we let the ejected atom stick on the re - deposition site or diffuse to its nearest neighbor site , if vacant ; whichever gives the highest coordination number . in the srawe assume a surface atom with single lattice coordination , that is , one with no same - plane nearest neighbour , can diffuse to a vacant nearest neighbor site , where it has a higher lattice coordination . in all cases above ( rema and sra ) where there is selection of , or relocation of a surface atom to , another surface site we implement periodic boundary conditions . with the implementation of re - deposition, the number of particles of the target material is conserved in this model .however , the overall number of particles in the whole system increases due to the incident ions .in all results presented here the simulation time is in monte carlo steps , which is of the order of ion arrival times , and the distances ( e.g. surface height or lateral size ) are in lattice spacing , which are of the order of lattice constants .the system size is 100 100 and the simulation was performed with yield parameter of 1kev ar sputtering of si at 45 incidence .representative figures of the results of the surface topography are as shown in figs .[ fig : surf10 ] and [ fig : surf4k ] . figures [ fig : surf10 ] and [ fig : surf4k ] are for simulation times of 10 and 4000 monte carlo steps , respectively . in both figures ( coloured ), the legend indicates the height ranges .  .[ fig : surf10],scaledwidth=80.0% ]  both figures show rough surfaces with no ripple formation , in agreement with the experimental reports reviewed in section [ sec : revexpt ] .they also show an increase of the surface depths with sputtering time . although , it can be observed from these profiles that roughening increases with sputtering time but this is not a sufficient measure .we studied the roughness by calculation of surface roughness .  .[ fig : surf4k],scaledwidth=80.0% ]  surface roughness , or width , is a very important index for studying the interplay of competing mechanisms in the surface morphology , such as the destabilising effects of sputter erosion and , in this case , eroded particle re - deposition , and the stabilising effect of reorganisation or diffusion of surface atoms .it is defined as   where is the local surface height at position , and is the average surface height at time . is more conveniently calculated in a simulation by using its simplified form :   we present results of ) and in figs .[ fig : height ] and [ fig : width ] respectively .figure [ fig : height ] shows that the average height of the surface profiles remain constant about a zero value for the entire duration of the simulation .this is to be expected since the number of particles of the target material is constant in this model ; that is , as a surface atom is eroded , it is redeposited somewhere else. however , in order to check that this is the only reason , we switched off re - deposition in the simulation and obtained the results shown in figure [ fig : height_noredp ] .  , scaledwidth=80.0% ]  , scaledwidth=80.0% ]  figure [ fig : height_noredp ] shows the expected result that , when there is no re - deposition of eroded material , the average surface height decreases with simulation time .however , the relatively low values of the average height recorded in figs .[ fig : height ] and [ fig : height_noredp ] , when compared with the higher depth values in figs .[ fig : surf10 ] and [ fig : surf4k ] , give a clue as to the importance of the reorganisation process of sra .other works based on another robust model exhibits lower values of the average height which may be a reflection of an enhanced reorganization process .  in figs .[ fig : width ] and [ fig : width_noredp ] we present the results of our calculations of the surface roughness with and without re - deposition of eroded material , respectively .we do not observe a significant difference in the roughness for both cases .we believe this to be an indication of the re - deposition process being a mirror destabilising process of the erosion process , and that the quest for improved lattice coordination by a redeposited particle largely deviates from a filling of depressions generated by the erosion process ., scaledwidth=80.0% ]  , scaledwidth=80.0% ]we introduced a new monte carlo model that is based on semi - empirical sputter yield in ion - solid energetic collisions , and applied it to 1 kev argon ion sputtering of silicon at incidence angle .this model circumvents the complexity of the existing statistical , classical and continuum models , most of which are difficult to relate to real experimental parameters , by its reliance on semi - empirical values of the sputter yield . constrained by this crucial experimental factor , the model addresses the multidimensional nature of other accompanying physical processes stochastically ; thus reducing the complexity of their computation .its simple but realistic algorithms have the potential of being exploited in a hybrid mc - md ( stochastic - classical ) simulation algorithm that will reduce the phase volume traversed in md simulations and allow the hybrid algorithm to simulate the sputtering process up to the timescale of interest in real experiments .we found that this new model , for argon ion irradiation of silicon at , exhibited the experimentally observed features of silicon surfaces under this condition and allows for a way of studying the effect of the different mechanisms of the surface morphology and the nature of their interplay in the dynamics of surface evolution .our study of the average surface height revealed that it was constant when eroded particles were redeposited but varied linearly with simulation time , when eroded particles were not redeposited .our studies also showed that the roughening process was not significantly affected by re - deposition of eroded material .  for further work , yield parameters of other incidence anglescould be adopted to study the mechanisms of formation of ripple and nano - dot topographies . as a matter of fact , it might be more convenient to implement the semi - empirical formula of seah , clifford , green and gilmore directly , if possible , instead of the present form of taking the value of the yield for an ion - target combination as an input from their tables .in addition , the independence of the sputter yield of an ion on concurrent sputtering of the target by contaminants should be established to confirm the assumption , here , that the overall yield is a summation of the yield due to the ion and those due to each contaminant that sputters .then , the erosion stage of this model could be modified to include additional erosion by contaminants for a systematic study of their influence on surface morphology .this could be easily implemented by incorporating an additional empirical yield parameter which captures the yield due to the contaminants but at varying energy , all of which are lower , than the energy of the primary projectile .oyewande , eo .modelling and simulation of surface morphology driven by ion bombardment .institute for theoretical physics .goettingen : university of goettingen , 2006 .pp . 1 - 106 , ph .d. thesis .publication url : https://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0006-b596-c .self - organized nanopatterning of silicon surfaces by ion beam sputtering .munoz - garcia , j , et al ., et al . s.l .: elsevier , 2014 , materials science and engineering r , vol .86 , pp . 1 - 44 . and references therein..  production of ordered and pure si nanodots at grazing ion beam sputtering under concurrent substrate rotation .chowdhury , debasree , ghose , debabrata and satpati , biswarup .2014 , mater .b , vol . 179 , pp . 1 - 5 . and references therein ..  subsurface channeling of kev ions between graphene layers : molecular dynamics simulation .rosandi , yudi , nietiadi , maureen l and m , urbassek herbert .2015 , physical review b , vol .91 , pp . 125441(1 - 7 ) . and references within ..    kinetic monte carlo simulation of self - organized pattern formation induced by ion beam sputtering using crater functions .yang , zhangcan , lively , micheal a and allain , jean p. 2015 , physical review b , vol .075427 ( 1 - 12 ) . and refernces therein ..  propagation of ripples in monte carlo models of sputter - induced surface morphology .oyewande , emmanuel o , hartmann , alexander k and kree , reiner .2005 , physical review b , vol .195405 ( 1 - 8 ) . and references therein..    crater function approach to ion - induced nanoscale pattern formation : craters for flat surfaces are insufficient .harrison , matt p and bradley , mark r. 2014 , physical review b , vol .245401 ( 1 - 10 ) . and references therein..  theory of off - normal incidence ion sputtering of surfaces of type axb1-x and a conformal map method for stochastic continuum models .oyewande , emmanuel o and adeoti , boluwatife b. 2014 , african review of physics , vol . 9 , pp .0024 ( 1 -7 ) . and references therein ..    theory of off - normal incidence ion sputtering of surfaces of type ab and a conformal map method for stochastic continuum models .oyewande , oluwole e and adeoti , boluwatife b. 2014 , african review of physics , vol . 9 , pp .177 - 183 .kinetic monte carlo simulations of ion - induced ripple formation : dependence on flux , temperature , and defect concentration in the linear regime .chason , eric , chan , w l and bharathi , m s. 2006 , physical review b , vol .74 , p. 224103 .tri3dyn - collisional computer simulation of the dynamic evolution of 3-dimensional nanostructures under ion irradiation .moeller , wolfhard .2014 , nucl .instrum . meth .phys . res .23 - 33 .seah , m p , et al . ,et al . surface and nanoanalysis : sputter yield values . national physical laboratory .[ online ] august 29 , 2014 .[ cited : september 29 , 2015 . ]

Abstract: we introduce a new monte carlo model based on a semi - empirical sputter yield parameter in ion - solid energetic collisions . this model circumvents the complexity of the existing statistical , classical and continuum models , most of which are difficult to relate to real experimental parameters , by its semi - empirical nature of direct reliance on the experimental values of the sputter yield . constrained by this crucial experimental factor , the model then addresses the multidimensional nature of other accompanying physical processes stochastically ; thus reducing the complexity of their computation . this model exhibits the experimentally observed features of solid surfaces that evolve under continuous particle irradiation and allows for a way to study the effect of the different mechanisms of the surface morphology and the nature of their interplay in the dynamics of the surface evolution . our study of the average surface height reveals that it is constant when eroded particles are redeposited but varies linearly with simulation time , when eroded particles are not redeposited . our studies also show that the roughening process is not significantly affected by re - deposition of eroded material .


in the fall of 2008 , many investors and analysts witnessed multiple once - in - a - lifetimeevents in a single week , with disastrous consequences in many cases for the financial health of their portfolios .the once - in - a - lifetime designation of the events arises from a calculation of the probabilities of events based on normal statistics .the returns were more than several sigma beyond the expected return , and by normal statistics , thought to be impossible .the difficulty lies in the use of a normal pdf to describe returns .it is known that returns have fat tails , but for mathematical convenience and perhaps force of habit , a normal pdf is often applied .the celebrated black - scholes formula for pricing european options is based on several assumptions , one of which is that the returns are described by brownian motion .the underlying pdf for brownian motion is a normal pdf .  in this paperwe price european options using a log student s -distribution .we call these formulae of the prices of european options gosset formulae , in honour of the author behind the _ nom de plume _ student .we present evidence that stock returns are fit by a student s _ t_-distribution , in agreement with known results .we then demonstrate the fat tailsof the _ t_-distribution .one of the difficulties with the fat tails is that one of the integrals , which is required to price an option , diverges .we present two similar approaches to handle successfully the divergence , and hence find a price for the option .these approaches can be used to price options using pdf s other than the -distribution .however , we restrict our attention to the -distribution and compare prices for options using the gosset and black - scholes formulae . in general ,the gosset formula yields prices for options that are higher than the prices found from the black - scholes formula .this is not unexpected , as the gosset formula gives greater weight to events in the tails .the starting point for the pricing of the european options is the arbitrage theorem .consider an experiment that returns for a bet of unity on outcome of the possible outcomes of the experiment . in short ,the arbitrage theorem states that either the mean return as calculated over the probability of the events , , is zero or there is a betting scheme that leads to a sure win .the betting scheme may allow for negative , zero , and positive bets .puts and calls allow for negative , zero , and positive bets .let be disjoint events of which one and only one must occur .let be the probability of occurrence of event and * _ c _ * be a vector of bets on the outcomes .the bets may be zero , positive , or negative .the gain for a wager of on the outcome given that event occurred is .note that in a typical bet , if ; the player forfeits the wager if event does not occur .the mean gain for a wager on the event is and is the expectation over the , which is obtained as .following the example of de finetti , we recognize the as equations in unknowns . to ensure that no arbitrage exists , the average gain .this is only possible for arbitrary if .if , then the can be selected to yield , on average , . thus to ensure no sure win , .that this condition must hold for all is a statement that there are no lucky numbers or values for assets . in the context of pricing an option on an asset , the events could be the final price of the asset falling in a range .ross shows how to price an option using the arbitrage theorem , assuming that the theorem can be generalized to handle functions rather than discrete outcomes .if the cost of an option is , then by the principle of no - arbitrage and the arbitrage theorem , where the return for a european call option is and the return for a european put option is .the notation means the maximum value of and 0 , with being the strike price and the value of the asset .ross obtains the black - scholes formula by assuming log - normal statistics for the value of the asset and making some assumptions regarding the time value of the option .it is not necessary to solve the black - scholes partial differential equation to obtain the black - scholes formula for the price of an european option .we accept as a starting point that the arbitrage theorem can be generalized from discrete outcomes to hold for functions .this we consider a reasonable starting point .the probability functions and values of assets can always be quantized to have discrete ranges .we price european options assuming log student s _ t_-distributions , the arbitrage theorem , and results from martingale theory .the results from the martingale theory allow determination of the risk neutral price of the option .first we demonstrate that the log student s _ t_-distribution is a reasonable distribution to use to price an option .the fat - tailproperty was one of the earliest observations of asset returns , with published reports dating back to the early 1960 s . when the black - scholes formula was developed in the early 1970 s the shortcomings of the underlying normal assumption were identified and attempts were made to incorporate kurtosis ,see for example and references therein .also in the early 1970 s the student s _ t_-distribution was first used to describe asset returns .there has been considerable work since 1974 comparing the _ t_-distribution and other distributions to empirical data to determine which distribution best fits observations , see for example , and references therein .the _ t_-distribution has been found to provide realistic fits to observed asset returns ( particularly equities and equity indices ) and in many cases provides better fits than other distributions .  in fig .1 we show fits to the observed frequency of occurence of daily return data for the dow jones industrial average ( djia ) ( left ) and the s&p 500 ( right ) equity indices .each plot shows the data with a best fit _ t_-distribution and normal distribution .the _ t_-distribution fits the observed distribution well , particularly in the tails .we have observed similar quality fits for the _ t_-distribution to frequency of occurence of daily returns for individual equities .[ fig : fig1 ]   the fit paramaters with the associated uncertainties , some critical values , and sample descriptive statistics , such as the count of data points , sample mean , and sample standard deviation are given in table i. the critical values are the returns which solve .given that there are 21,186 data points for the djia , it is expected , using the best fit -distribution , that only 2 returns will fall below -0.26 .this is in agreement with the data shown in fig . 1 .  c c c c c & & + parameter & normal & & normal & +  + location & & && + scale & & & & + shape & & 2.65 0.11 & & 3.24 + & 0.043 & 0.26 & 0.036 & 0.19 + & 0.036 & 0.12 & 0.030 & 0.089 + & 0.027 & 0.053 & 0.022 & 0.040 + count & & + & & + & & +  0.05 in figure 2 is a plot of the value of the student s _ t_-distribution and the normal distribution as a function of the number of standard deviations from the true mean .the fat tails of the student s _ t_-distribution are evident .events that are 10 standard deviations from the true value are expected approximately % of the time for the student s _ t_-distribution with a small number of degrees of freedom , but only % of the time for a normal pdf .for comparison , the values for , a normal pdf with a mean of zero and a standard deviation greater than for the other curves is presented .a volatility of the measured volatility is sometimes used to price options .the enhanced volatility effectively gives the normal pdf fatter tails .it can be observed in fig .2 that the curve tracks the student s _ t_-distribution over a greater distance from the true mean .one possible justification for the use of student s -distribution to describe the daily returns for an asset is as follows .if the returns are assumed to be normally distributed with a given short term variance , the variance is taken to be a random variable that varies slowly in time , and is the pdf for the return at time some time in the future , then where is the pdf for the return given and is the pdf for at time . fits to the 22-day volatilities for the djia and for the s&p 500 show that a gamma distribution fits the 22 day volatilities .a gamma distribution with shape parameter is chi - squared with degrees of freedom , and a normal distribution with a variance that is chi - squared distributed is a student s -distribution .a student s -distribution for the pdf of the daily returns is consistent with the data shown in figthe variance of a student s -distribution is  .as tends to infinity , the -distribution tends to a standard normal pdf , which has a variance of one .the enhancement accounts for the range of values that the variance might take .if it is known that the variance does not change in time ( i.e , where is a dirac delta function ) , then the gosset formula presented in this paper reduces to the black - scholes formula . in this respectthe gosset formula extends the black - scholes formula by removing the condition that the variance is constant in time . under this interpretation , the shape parameter is a measure of the belief that the volatility will not change .a large means that one believes that the volatility will not change in time .a small means that one believes that there is a broad range of values that the volatility might take . if the volatility is determined as the standard deviation of independent returns that are normally distributed with an unknown variance , then the maximum value of should equal , which is the number of degrees of freedom in a determination of a sample standard deviation .this maximum value of takes into account the fact that the sample standard deviation is a random variable and has an uncertainty associated with it .figure 3 is a plot of the relative areas in the tails for the same distributions presented in fig .the curves are normalised by the area in the tail of a normal pdf .figure 3 gives the probability of an event for , relative to the probability of the event for for a normal pdf .the probability of an event lying standard deviations from the mean is time mores likely for a process that follows a student s _ t_-distribution with 3 and 5 degrees of freedom than for a process that follows a normal pdf .   the volatility , each normalised by the area in the tail of a normal pdf . ]the volatility , each normalised by the area in the tail of a normal pdf . ]le be the price of a stock at time .let be the strike price at time , where is the time when the option expires .let be the value of the stock where is a random variable .the cost of a european call option , calculated at the time of expiration , is , which is the expectation of the maximum value of .the desired quantity is , which is the value of the option at time .the desired quantity can be obtained from the expected time value of money . if is the risk free rate , then when the risk free rate is assumed to be time independent .this is a standard assumption in the derivation of the black - scholes formula .if is normally distributed , then follows a log normal distribution and the price for the option follows the black - scholes formula .if follows a student s _ t_-distribution , then the pdf for for degrees of freedom is which can be written in terms of as the following integrals are needed to evaluate the price of a call option at the time of expiration , :      note that is a function of , and , and that these variables are not functions of the random variable ( see eq .( 6 ) and the accompanying text ) .the integral in eq.(4 ) , which is for a call option , will be infinite for small ; the exponential term in the numerator will dominate the denominator .viable and physical results are obtained by restricting the upper limit of integration ( i.e. , truncate the pdf ) or by restricting the value of the asset ( i.e. , cap the value ) . it is not physical for the value of the asset to approach infinity .  fora put option this problem does not directly arise as and thus the integrals are evaluated from to .however , the average price of the asset is needed to price an option , and thus the pricing of a put option suffers also from an integral that diverges .the integral that diverges is eq .if the value of the asset is restricted to be , then the offending integral , eq .( 4 ) , can be rewritten as the integral in the last line of eq.(5 ) is the probability that the stock price is greater than some critical value , which we abbreviate to . the last line in eq .( 5 ) gives the expected value of the cost of capping the value of the asset to .  in pricing a call option , one could also truncate the pdf by setting the last line of eq .( 5 ) to zero .that is , one assumes that the pdf is zero for values greater than the critical value .truncation sets an upper limit on the value of the stock : .it is necessary to divide the truncated pdf by to maintain the normalisation of the truncated pdf .this follows from the definition of conditional probability : . in this case , for and equals the null set otherwise . thus if and otherwise .table ii gives critical values and the maximum increase in stock price for a normal pdf and a student s _ t_-distribution with degrees of freedom , volatility , and various value of .the effect of the fat tails of the student s _ t_-distribution are evident . for a confidence level of 99.99% , the critical value for a student s_ t_-distribution with 5 degrees of freedom is 9.678 and the stock price has increased at most , a 4 798% increase . for a normal pdf ,the critical value is 3.719 and there is a 99.99% chance that the stock price has increased at most , a 442% increase .the other interpretation is that there is only a 0.01% chance that the stock price has increased by under the student s _ t_-distribution or under the normal distribution .  a confidence level of ( i.e. , 99.99% ) seems unrealistic .this confidence level is one part in 10,000 and allows for increases of in the price of the asset that may be physically impossible to achieve ( c.f .table iii ) .if the time period is one year , then once in 10,000 years is too large an extrapolation to be meaningful .even seems unrealistic .the best way to price options may be to pick an upper bound ( or confidence level ) for the price of the asset based on insight and agreement between parties , and then calculate the price using a capped distribution .  although a confidence level might be unrealistic or unphysical in terms of expected increase in the stock price and extrapolation , the results presented here show that the methods allow a price for the option to be calculated .the maximum increase in value of the asset is a function of the number of degrees of freedom and the assumed confidence level or critical value .table iii presents the critical values for different values of and degrees of freedom .note that a student s _ t_-distribution with yields a normal pdf and that a _ t_-distribution with is often considered to be a normal pdf .the variable is required to price the option .this value is obtained from the assumption that an option is a fair wager ( i.e. , a martingale ) , as shown next .. at for volatility = 0.4 and selected [ cols="^,^,^,^,^ " , ]   0.05 ina martingale is a mathematical model for a fair bet .a fair bet can be defined as one for which the conditional expectation ( i.e. , given all the knowledge at the time of the bet ) of the payout of the bet should be zero .a martingale is defined such that for all where is the knowledge ( history ) up to time and the knowledge increases as increases . the form a stochastic process .a martingale is non - anticipating because the outcome of a martingale depends only on the past and not on the future .a * * sub**martingale is defined such that and a * * super**martingale is defined such that .a submartingale represents a favourable process ( the expected outcome is favourable ) and a supermatingale represents an unfavourable process ( like a betting at a casino ) .casinos offer submartingales ( the patrons play supermartingales ) and financiers aspire to offer or to play submartingales.  if is a martingale , then , and are all submartingales .the doob decomposition shows that any submartingale is the sum of a martingale plus an increasing process , , where and depends on the knowledge to  .let be the spot price at time for a stock . where is the expected return on a non - dividend paying stock . is from a capital asset pricing model ( capm ) or equivalent . is called the risk premium and is the expected return on the stock , which equals the compensation for the time value of the money invested in the stock plus the compensation for the risk of holding the stock .the time value is separated out since the value of the option is calculated at time , but the value at time 0 , , is desired , and , assuming the risk free rate is known and constant in time .the doob decomposition is used to create a martingale for pricing options on the stock .the drift of the stock is subtracted from the process .this procedure of subtracting the drift from the process also leads to risk neutral pricing , where one is able to price an option without explicit knowledge of the risk premium .  the average value of a stock at time when we wish to price an option is .the average price of the stock at some time later is where is a random variable and the drift is contained in . for a martingale ( a fair wager ) ,one may use the doob decomposition to offset the drift and require .this is equivalent to requiring that the pdf for should be centred about the drift owing to the risk premium .the term takes into account the time value of money and allows for comparison of values at two different points in time .it is perhaps simpler to find the normalization that satisfies the martingale condition than to shift the pdf . for a capped distribution , let be the value of the integral in the middle row of eq .( 6 ) and let be the value of the integral of the last line of eq .( 6 ) , i.e. , . equation ( 6 ) can then be rewritten as and solved for the value of : for a truncated distribution where the pdf is set for and the pdf in the integral that defines ( middle line of eq .( 6 ) ) will have a factor of where is the probability that .this follows from the definition of conditional probability : if and zero if .note that and that is not a function of the random variable .  for a log normal distribution with and , and .thus where is the cumulative distribution function ( cdf ) for the unit normal pdf , . where and .this is the standard form for the black - scholes formula for a european call .note that and the expressions can be rewritten in terms of with and .thus the value for appears correct and it should be possible to price an option assuming a student s _ t_-distribution ( or any other distribution ) using the approach and equations developed in this manuscript .  the price at the time of expiration for a european call , , using a student s _ t_-distribution and capping the value of the asset at a critical value is for a capped asset , is defined by eq .( 8) .  for a truncated _ t_-distribution ,the price of a call is obtained by dropping the last line of eq .( 9 ) , dividing the integral in the first line of eq .( 9 ) by , and using .it is helpful to remember that is defined by some of the integral on the first line of eq .( 9 ) ( see eqs ( 7 ) and ( 6 ) for the definition of z ) and thus is also divided by a factor of for a truncated pdf .the value of the integral in the last line of eq .( 9 ) equals , which we shorten to .  the price for a european put option at time is and equals for both a capped asset and a truncated pdf , where the truncation occurs at .the value of is different for the capped and truncated approaches , and this makes different for the two approaches .  in pricing a put , one could also truncate the distribution ( i.e. , set the lower limit of integration to a critical value and set the pdf to zero for values ) or use a floor at the critical value ( rather than a ceiling as was done for a call ) .the values of and would be modified in a straightforward manner to accomodate the floor or truncation . in this paperwe have chosen to use the full range of the integral in eq .there are six possibilities to calculate an option : two choices for the call ( cap or truncate one of which must be chosen ) and three choices for the put ( floor , truncate , or nothing ) .our goal is to present the approach to pricing an option using a student s -distribution . in this regard ,is it necessary to truncate or to cap the distribution at the upper end .we leave discussion of the merits of a floor and truncation to future work .we consider a formula for the price of an european option that is obtained using either a maximum value for the asset or a truncated student s _ t_-distribution to be a gosset formula .there is little difference in philosophy between options priced with a capped value or with a truncated pdf . in both approachesthe return is assumed to be always less than a critical value . with the capped asset approach , more weightis applied to the maximum value of the asset than for the truncated pdf approach .the truncated pdf is more compact to write as there is not a second , additive term .there is , however , a difference in risk between the two approaches .the truncated pdf approach assumes that the value of the asset will never exceed the critical value .there is no guarantee that the value will not exceed the critical value .the approach of capping the payout of a benefit is common in insurance .capping the payout reduces the cost to purchase the benefit .capping the payout removes the risk associated with the benefit exceeding the critical value .it is known , by design , that the payout will never exceed the cap but it is accepted that there is a non - zero probability that the expense will exceed the benefit .capping the value of the asset underlying a call option removes the risk associated with the value of the asset exceeding a critical value ( assuming that the buyer of the call option agrees to the condition that the payout not exceed the cap value ) , and thus reduces the cost of the option .capping the value is one approach to keep the integrals , which are needed to price an option , finite. the capped gosset formula would be the proper formula to use to price a european option for an asset that is capped in value .put - call parity holds regardless of the underlying pdf .consider where is the cash value of a call evaluated at time , and is the cash value of a put evaluated at time .for a martingale ( a fair wager ) , , and thus . from the time value of money , , then or .one of the confusing features of pricing an option are the time values .typically the option is priced at time for a stock price at time and a strike price at time .one is usually given , and implicitly for time ( i.e. , , and ) , the strike price implicitly for time , ( i.e. , ) , and asked implicitly for and at time ( i.e. , and ) .the volatility is evaluated at time , and is propagated to time using diffusion : . for diffusion ,the variance increases linearly with time . is used in the pricing of the options . for pricing an option with the gosset formula ,it is necessary to specify the shape parameter , which is a statement of belief about the range of values that the volatility might take at time = .figure 4 is comprised of plots of the price of a call option calculated using the gosset formula minus the price of the same option calculated using the black - scholes formula , as a function of the number of degrees of freedom assumed for the student s _ t_-distribution . a capped distribution was used for the gosset formula and confidence levels of 0.99 , 0.999 , and 0.9999 were used . for small numbers of the degrees of freedom and , the gosset formula predicts a price difference  7.12 for a call option with these parameters .the student s _ t_-distribution is considered to be equivalent to the normal distribution for the number of degrees of freedom . the differences between the prices obtained from the gosset formula and the black - scholes formula are between 0.11 for 40 degrees of freedom for confidence levels of 0.99 , 0.999 , and 0.9999 .figure 5 is comprised of similar plots as for fig .4 , except that a truncated distribution was used to calculate the price of the call option with the gosset formula .the price of a call option obtained by using a truncated distribution is less than the price obtained by using a capped distribution . for a confidence level of 99% and ,the gosset formula predicts a value for a call option that is less than the value predicted by the black - scholes formula .this is not surprising given that for large , the student s _ t_-distribution approaches the normal pdf . as the two distributions become similar , any approach that excludes data in the tails will predict a lower value for the option .black - sholes , as a function of for confidence levels of 0.99 , 0.999 , and 0.9999 using a truncated distribution . ]black - sholes , as a function of for confidence levels of 0.99 , 0.999 , and 0.9999 using a truncated distribution . ]the difference in cost of the option between the capped asset approach and the truncated distribution approach owes to contributions from three terms .the term is different for the two approaches and this leads to differences in the integrands and lower limits of integration . for , , , , , , and , the price difference , capped minus truncated , is 3.14 to the difference .the integrand for the truncated distribution is larger than for the capped asset .this contributes 0.10 , for a total difference of 49.00 , , , and year were used .the curve for essentially overwrites the black - scholes curve for the scale used .the difference between the and black - scholes prices were less then $ 0.10 for the range of .figure 9 gives the prices of calls and puts as a function of the price of the asset .the truncated gosset formulae for and , eqs .( 9 ) and ( 10 ) , were used to calculate the prices of the options .a comparison of figs . 8 and 9 shows that the truncated gosset formula predicts lower prices for the options than does the capped gosset formula .the truncated gosset formula ignores events in the tails and thus costs less than the capped gosset formula , which allows for events in the tails but places a limit on the value of the events in the tails .and using a truncated gosset formula with , and for the black - scholes formula . ]   and using a truncated gosset formula with , and for the black - scholes formula . ]in this paper we have presented a method to price european options using a student s _ t_-distribution .we refer to this method as a gosset method and the formulae for pricing options as gosset formulae , in honour of the person who published under the _ nom de plume _ ( or pseudonym ) student .the gosset formula essentially extends the black - scholes formula by removing the constraint that the volatility be constant in time .student s _ t_-distributions have fat tailsand fit the returns from assets better than a normal pdf .the difficulty with pricing options using the student s _ t_-distribution is that the fat tails cause one of the integrals , which is needed to price an option , to diverge .a viable and physically reasonable value for the price of the option can be obtained by either capping the value of the underlying asset or by truncating the distribution .the value for the cap or where the truncation takes place need not be unrealistic .the required integral can be evaluated for a confidence level of 99.99% , which implies a price increase of 4 000% for realistic parameters ( volatility and shape parameter ) of the student s _ t_-distribution .an increase of 4 000% in the value of an asset would be phenomenal and perhaps unphysical .there is little difference in philosophy between the approaches of pricing an option by capping the value of the asset or truncating the pdf .both approaches reduce the cost of the option .however , there can be a large difference in the exposure to risk between the two approaches .   capping the value of the asset ensures , by design and with consent , that the risk associated with the value of the option exceeding the value of the cap does not exist . in the cap approach, it is accepted that the value of the asset might exceed the value of the cap . however , the price of the option is set assuming that the payout will not exceed the value of the cap .provided that the option writer and buyer agree to this approach , then there is no risk associated with a large increase in value of the asset .the buyer gives away the right to extreme returns in exchange for a lower price for the option .if the option writer and buyer do not agree to a cap on the payout , then the approach of capping the value of the asset still holds risk that the value of the asset will exceed the cap value .for the same critical value and parameters for the pdf , the cap approach offers an increased premium for the risk as compared to the approach of truncation of the pdf .this is because the cap approach allows for the possibility that the value of the asset will exceed the value of the cap .the student s _ t_-distribution has an additional parameter ( the number of degrees of freedom , ) as compared to the normal pdf .this parameter is a statement of the belief about the range of values that the volatility might take at the time that the option expires .a large value of is a statement that the volatility will change little in time . in the limitas approaches infinity , the student s -distribution approaches the normal distribution and hence the gosset formula for the price of an option becomes the same as the black - scholes formula .the black - scholes formula is derived on the assumption that the volatility is constant in time and known .a small value of is a statement that the volatility might take one of a broad range of values .fits to the djia for returns over 100 years give a value of .this value of allows for the distribution of volatility that has been observed over the 100 years of returns . in turbulent markets, it would be wise to use a small value for the number of degrees of freedom and a large confidence level .a confidence level is required in the pricing of options using a student s -distribution . in calm markets, it might be acceptable to use a large , of order where is the number of independent samples that are used to calculate the volatility , and a smaller confidence level to match the expectations of the future and tolerance to risk .the flexibility offered by the gosset formulae of choice of approach ( cap or truncate ) , of the number of degrees of freedom to match the fat tails of returns , and of the critical value allow the analyst and investor additional opportunity to use their insight and circumstances to price options .  the approach used to derive the gosset formulae can be adapted to distributions other than the student s _ t_-distribution .this may be of interest in situations and for returns on assets that do not appear to be well described by a student s _ t_-distribution ., from skews to a skewed- : modelling option implied returns by a skewed student - t " , computational intelligence for financial engineering , 2000 .( cifer ) proceedings of the ieee / iafe / informs 2000 conference on .

Abstract: the distribution of the returns for a stock are not well described by a normal probability density function ( pdf ) . student s _ t_-distributions , which have fat tails , are known to fit the distributions of the returns . we present pricing of european call or put options using a log student s _ t_-distribution , which we call a gosset approach in honour of w.s . gosset , the author behind the _ nom de plume _ student . the approach that we present can be used to price european options using other distributions and yields the black - scholes formula for returns described by a normal pdf . keywords : econophysics ; financial risk ; european options ; fat - tailed distributions ; student s -distribution


stuttering equivalence and branching bisimulation were proposed as alternatives to milner s weak bisimulation .they are very close to weak bisimulation , with as essential difference that all states in the mimicking sequence must be related to either the state before or directly after the from the first system .this means that branching bisimulation and stuttering equivalence are slightly stronger notions than weak bisimulation .  in  time algorithm was proposed for stuttering equivalence and branching bisimulation , where is the number of transitions and is the number of states in either a kripke structure ( for stuttering equivalence ) or a labelled transition system ( for branching bisimulation ) .we refer to this algorithm as gv .it is based upon the algorithm for bisimulation equivalence in .both algorithms require space .they calculate for each state whether it is bisimilar to another state .the basic idea of the algorithms of is to partition the set of states into blocks .states that are bisimilar always reside in the same block . whenever there are some states in a block from which a transition is possible to some block and there are other states in from which such a step is not possible , is split accordingly .whenever no splitting is possible anymore , the partition is called stable , and two states are in the same block iff they are bisimilar .there have been some attempts to come up with improvements of gv .the authors of observed that gv only splits a block in two parts at a time .they proposed to split a block in as many parts as possible , reducing moving states and transitions to new blocks . their worst case time andspace complexities are worse than that of gv , especially the space complexity , but in practice this algorithm can outperform gv . in , the space complexity is brought back to .a technique to be performed on graphics processing units based on both gv and  is proposed in  .this improves the required runtime considerably by employing parallelism , but it does not imply any improvement to the single - threaded algorithm .  in an algorithm is proposed for strong bisimulation as an improvement upon the algorithm of . the core idea for this improvementis described as `` process the smaller half ''  . whenever a block is split in two parts the amount of workmust be contributed to the size of the smallest resulting block . in such a case a stateis only involved in the process of splitting if it resides in a block at most half the size of the block it was previously in when involved in splitting .this means that a state can never be involved in more than splittings . as the time used in each state is proportional to the number of incoming or outgoing transitions in that state , the total required time is .  in this paperwe propose the first algorithm for stuttering equivalence and branching bisimulation in which the `` process the smaller half''-technique is used . by doing so, we can finally confirm the conjecture in  that such an improvement of gv is conceivable .moreover , we achieve an even lower complexity , namely , than conjectured in  by applying the technique twice , the second time for handling the presence of inert transitions .first we establish whether a block can be split by combining the approach regarding bottom states from gv with the detection approach in  .subsequently , we use the `` process the smaller half''-technique again to split a block by only traversing transitions in a time proportional to the size of the smallest subblock .as it is not known which of the two subblocks is smallest , the transitions of the two subblocks are processed alternatingly , such that the total processing time can be contributed to the smallest block . for checking behavioural equivalences , applying such a techniqueis entirely new .we are only aware of a similar approach for an algorithm in which the smallest bottom strongly connected component of a graph needs to be found  .the algorithm that we propose is complex .although the basic sketch of the algorithm is relatively straightforward , it heavily relies on auxiliary data structures .for instance , for each transition it is recalled how many other transitions there are from the block where the transition starts to the constellation in which the transition ends .maintaining such auxiliary data structures is a tedious process .therefore , we do not only prove the major steps of the algorithm correct , we also provide a very detailed description of the algorithm , and we ran the implemented algorithm on many thousands of randomly generated test cases , comparing the reductions with the outcomes of existing algorithms .this not only convinced us that the algorithm is correct and correctly implemented , it also allows others to easily reimplement the algorithm .  from a theoretical viewpointour algorithm outperforms its predecessors substantially . buta fair question is whether this also translates into practice .the theoretical bounds are complexity upperbounds , and depending on the transition system , the classical algorithms can be much faster than the upperbound suggests .furthermore , the increased bookkeeping in the new algorithm may be such a burden that all gains are lost .for this reason we compared the practical performance of our algorithm with that of the predecessors , and we found that for practical examples our algorithm can always match the best running times , but especially when the kripke structures and transition systems get large , our algorithm tends to outperform existing algorithms with orders of magnitude .  compared to checking other equivalences the existing algorithms for branching bisimulation / stuttering equivalence were already known to be practically very efficient .this is the reason that they are being used in multiple explicit - state model checkers , such as cadp  , the mcrl2 toolset  and tvt  .in particular they are being used as preprocessing steps for other equivalences ( weak bisimulation , trace based equivalences ) that are much harder to compute . for weak bisimulation recently a algorithm has been devised , but until that time an expensive transitive closure operation of at best was required . using our algorithm as a preprocessing step, the computation time of all other behavioural ` weak ' equivalences can be made faster .we introduce kripke structures and ( divergence - blind ) stuttering equivalence .labelled transitions systems and branching bisimulation will be addressed in section [ sec : bbisim ] .kripke structure a _ kripke structure _ is a four tuple , where   is a finite set of states . is a finite set of atomic propositions . is a total transition relation , i.e. , for each there is an s.t . . is a state labelling .we use for the number of states and for the number of transitions . for a set of states , we write for and , and iff there is some such that .we write and iff it is not the case that , resp ., .divergence - blind stuttering equivalence let be a kripke structure .a symmetric relation is a _ divergence - blind stuttering equivalence _iff for all such that :   .  for all if , then there are for some such that , , , and for all .we say that two states are _ divergence - blind stuttering equivalent _ , notation {}} ] , iff there is a divergence - blind stuttering equivalence relation such that .an important property of divergence - blind stuttering equivalence is that if states on a loop all have the same label then all these states are divergence - blind stuttering equivalent .we define stuttering equivalence in terms of divergence - blind stuttering equivalence using the following kripke structure .stuttering equivalence let be a kripke structure .define the kripke structure where is an atomic proposition not occurring in and is a fresh state not occurring in .furthermore ,   is on a cycle of states all labelled with , or .for all we define and .states are _ stuttering equivalent _ , notation {}} ] iff there is a divergence - blind stuttering equivalence relation on such that .note that an algorithm for divergence - blind stuttering equivalence can also be used to determine stuttering equivalence by employing only a linear time and space transformation .therefore , we only concentrate on an algorithm for divergence - blind stuttering equivalence .our algorithms perform partition refinement of an initial partition containing the set of states .a _ partition _ is a set of non empty subsets such that for all and .each is called a _ block_.we call a transition _ inert w.r.t.  _ iff and are in the same block .we say that a partition _ coincides _ with divergence - blind stuttering equivalence when {}} ] iff there is a block such that .we say that a partition _ respects _ divergence - blind stuttering equivalence iff for all if {}} ] then there is some block such that .the goal of the algorithm is to calculate a partition that coincides with divergence - blind stuttering equivalence .this is done starting with the initial partition consisting of blocks satisfying that if then .note that this initial partition respects divergence - blind stuttering equivalence .we say that a partition is _ cycle - free _ iff there is no state such that  for some for each block .it is easy to make the initial partition cycle - free by merging all states on a cycle in each block into a single state .this preserves divergence - blind stuttering equivalence and can be performed in linear time employing a standard algorithm to find strongly connected components  .the initial partition is refined until it coincides with divergence - blind stuttering equivalence .given a block of the current partition and the union of some of the blocks in the partition , we define note that if , then .the sets and are intended as the new blocks to replace .it is common to split blocks under single blocks , i.e. , corresponding with a single block  .however , as indicated in , it is required to split under the union of some of the blocks in to obtain an algorithm .we refer to such unions as _ constellations_. in section [ sec : constellations ] , we use constellations consisting of more than one block in the splitting .we say that a block is _ unstable _ under iff and  .a partition is _ unstable _ under iff there is at least one which is unstable under .if is not unstable under then it is called _stable under _if is stable under all , then it is simply called stable .a _ refinement _ of under consists of two new blocks and .a partition is a refinement of under iff all unstable blocks have been replaced by new blocks and .the following lemma expresses that if a partition is stable then it coincides with divergence - blind stuttering equivalence .it also says that during refinement , the encountered partitions respect divergence - blind stuttering equivalence and remain cycle - free .   1 .  for all states , if with a block of the partition , is stable , and a refinement of the initial partition , then {}} ] .if respects divergence - blind stuttering equivalence then any refinement of under the union of some of the blocks in also respects it .if is a cycle - free partition , then any refinement of is also cycle - free .  1 .we show that if is a stable partition , the relation is a divergence - blind stuttering equivalence .it is clear that is symmetric .assume .obviously , because and refines the initial partition . forthe second requirement of divergence - blind stuttering equivalence , suppose .there is a block such that .as is stable , it holds for that for some , and .this clearly shows that for all  , and .so , is a divergence - blind stuttering equivalence , and therefore it holds for all states that reside in the same block of that {}} ] .the second part can be proven by reasoning towards a contradiction .let us assume that a partition that is a refinement of under does not respect divergence - blind stuttering equivalence , although does. hence , there are states with {}} ] and a block with and and are in different blocks in . given that is a refinement of under , and ( or vice versa , which can be proven similarly ) . by definition of , there are ( ) and such that .then , either and , but then . or , and since {}} ] , there are ( ) and such that with for all , and .this means that we have , again contradicting that .3 .  if is cycle - free , this property is straightforward , since splitting any block of will not introduce cycles .s. cranen , j.f .groote , j.j.a .keiren , f.p.m .stappers , e.p .de vink , j.w .wesselink , and t.a.c .an overview of the mcrl2 toolset and its recent advances .tacas13 , lncs 7795 , pp .199 - 213 , springer , 2013 .see also ` www.mcrl2.org ` .

Abstract: we provide a new algorithm to determine stuttering equivalence with time complexity , where is the number of states and is the number of transitions of a kripke structure . this algorithm can also be used to determine branching bisimulation in time where is the set of actions in a labelled transition system .   theoretically , our algorithm substantially improves upon existing algorithms which all have time complexity at best . moreover , it has better or equal space complexity . practical results confirm these findings showing that our algorithm can outperform existing algorithms with orders of magnitude , especially when the sizes of the kripke structures are large .   the importance of our algorithm stretches far beyond stuttering equivalence and branching bisimulation . the known algorithms were already far more efficient ( both in space and time ) than most other algorithms to determine behavioural equivalences ( including weak bisimulation ) and therefore it was often used as an essential preprocessing step . this new algorithm makes this use of stuttering equivalence and branching bisimulation even more attractive .