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College of Arts & Sciences

Archil Gulisashvili

Archil Gulisashvili

Professor

Mathematics
555 Morton Hall
gulisash@ohio.edu
740-593-1281

Education

Dr.Sc., Tbilisi State University

Research Interests

Financial Mathematics

  • General stochastic asset price models; classical stochastic volatility models (Hull-White, Stein-Stein, Heston); Gaussian Volterra type stochastic volatility models; fractional and rough models; scaling properties of stochastic volatility models; models with jumps; option pricing theory; asymptotic behavior of stock price distribution densities, option pricing functions, and the implied volatility; moment explosions, geometrical methods in financial mathematics; Heston geometry, large and moderate deviation principles; the G\"{ä}rtner-Ellis theorem. 

Stochastic Processes 

  • Non-homogeneous Markov processes, time-reversal and duality theory for Markov processes, applications of Markov processes to parabolic initial and final value problems. 

Semigroup Theory and Propagator Theory

  • Schrödinger semigroups and Feynman-Kac propagators; non-autonomous Kato classes of functions and measures; smoothing properties of Schrödinger semigroups.

Publications

Books

Large Deviations and Asymptotic Methods in Finance, Springer Proceedings in Mathematics and Statistics, Vol 110, P. K. Friz, J. Gatheral, A. Gulisashvili, A. Jacquier, J. Teichmann (Eds.), Springer International Publishing Switzerland, 2015.

A Gulisashvili, Analytically Tractable Stochastic Stock Price Models, Springer Finance, 2012.

A. Gulisashvili and J. A. van Casteren, Non-Autonomous Kato Classes and Feynman-Kac Propagators, World Scientific, Singapore, 2006.

Book Chapters

(With P. Tankov), Implied volatility of basket options at extreme strikes, in: Large Deviations and Asymptotic Methods in Finance, Springer Proceedings in Mathematics and Statistics, Vol 110, P. K. Friz, J. Gatheral, A. Gulisashvili, A. Jacquier, J. Teichmann (Eds.), Springer International Publishing Switzerland, 2015, pp. 175-212.

(With J. Teichmann), The Gärtner-Ellis theorem, homogenization, and affine processes, in: Large Deviations and Asymptotic Methods in Finance, Springer Proceedings in Mathematics and Statistics, Vol 110, P. K. Friz, J. Gatheral, A. Gulisashvili, A. Jacquier, J. Teichmann (Eds.), Springer International Publishing Switzerland, 2015, pp. 287-320.

(With J. Vives), Asymptotic behavior of the distribution densities in models with jumps, in: M. Zili and D. V. Filatova (eds.), Stochastic Differential Equations and Processes, Springer Proceedings in Mathematics, 7, Springer-Verlag, Berlin Heidelberg, 2011, 237-252.

Free propagators and Feynman-Kac propagators, Seminar of Mathematical Analysis, Proceedings, Universities of Malaga and Seville (Spain), September 2003-June 2004, Daniel Girela Alvarez, Genaro Lopez Acedo, Rafael Villa Carro, eds., Universidad de Sevilla, 2004, 47-64.

Nonautonomous Kato classes and the behavior of backward Feynman-Kac propagators, Analyse stochastique et theorie du potentiel, Saint Priest de Gimel (2002), Association Laplace-Gauss, 2003, 5 pages.

On the Kato classes of distributions and the BMO-classes, in “Differential Equations and Control Theory” (eds. S. Aizikovici and N. Pavel), Lecture Notes in Pure and Applied Mathematics, v. 225, Marcel Dekker, Inc., New York, 2001, 159-176.

(With G. Muraz) Rearrangement-invariant hulls and kernels of the amalgam spaces on Rn, in “Function Spaces” (Hudzik/Skrzypczak, eds), Marcel Dekker, Inc., 2000, 171-180.

Hulls and kernels of function classes on Rn, in “Recent Developments in Complex Analysis and Computer Algebras” (J. Gilbert, J. Kajiwara, and Y. Xu, eds.) Kluwer, Dordrecht, 1999, 81-94.

Rearrangement-invariant hulls of sets, Linear and Complex Analysis, Problem Book, Lecture Notes in Math., 1043, (1984), 642-645.

Estimates for the Pettis integral in interpolation spaces with some applications, Banach Space Theory and its Applications, Proceedings, Bucharest, 1981, Lecture Notes in Math, 991 (1983), 55-76.

Submitted Papers and Working Papers

Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions, submitted for publication, 2018, available on arXiv:1808.00421

(with Ch. Bayer, P. K. Friz, B. Horvath, and B. Stemper) Short-time near-the-money skew in rough fractional volatility models, accepted for publication in Quantitative Finance (2018); available on arXiv:1703.05132

(with P. K. Friz, B. Horvath, and P. Pigato), A unified approach to asymptotics of exotic digitals, working paper, 2017.

(with M. Lagunas, R. Merino, and J. Vives), Higher order approximation of plain vanilla option prices under stochastic volatility models, working paper, 2017.

Beyond the Gärtner-Ellis theorem with applications to time-averages of the Ornstein-Uhlenbeck process, working paper, 2016.

Published Papers

Large deviation principle for Volterra type fractional stochastic volatility models, SIAM Journal on Financial Mathematics, 9 (2018), 1102-1136; available on arXiv:1710.10711

(with F. Viens and X. Zhang) Small-time asymptotics for Gaussian self-similar stochastic volatility models, Applied Mathematics and Optimization (2018). https://doi.org/10.1007/s00245-018-9497-6, 41 p., available on arXiv:1505.05256, 2016.

(with F. Viens and X. Zhang) Extreme-strike asymptotics for general Gaussian stochastic volatility models, Annals of Finance (2018). https://doi.org/10.1007/s10436-018-0338-z, 43 p., available on arXiv:1502.05442v3, 2017.

(with B. Horvath and A. Jacquier) Mass at zero in the uncorrelated SABR model and implied volatility asymptotics, Quantitative Finance, 18 (2018), 1753-1765; available on arXiv:1502.03254

Distance to the line in the Heston model, Journal of Mathematical Analysis and Applications, 450 (2017), 197-228; available on arXiv:1409.6027.

(With P. K. Friz, B. Gess, and S. Riedel), The Jain-Monrad criterion for rough paths and applications to Random Fourier Series and Non-Markovian Hörmander Theory, Annals of Probability, 44 (2016) 684-738.

(With P. Tankov) Tail behavior of sums and differences of log-normal random variables, Bernoulli, Volume 22, Number 1 (February 2016), 444-493.

(with B. Horvath and A. Jacquier), On the probability of hitting the boundary for Brownian motions on the SABR plane, Electronic Communications in Probability, 21, No 75 (2016), 1-13.

(with J. Vives) Asymptotic analysis of stock price densities and implied volatilities in stochastic volatility models with Lévy type jumps, SIAM Journal on Financial Mathematics 6 (2015), pp. 158-188.

Left-wing asymptotics of the implied volatility in the presence of atoms, International Journal of Theoretical and Applied Finance, 18, 1550013 (2015) [25 pages].

(With P. Laurence), The Heston Riemannian distance function, Journal de mathématiques pures et appliquées, 101, (2014) 303-329.

Asymptotic equivalence in Lee’s moment formulas for the implied volatility, asset price models with moment explosions, and Piterbarg’s conjecture, International Journal of Theoretical and Applied Finance, 15, (2012) 1250020 (34 pages).

(With P. K. Friz, S. Gerhold, and S. Sturm), On refined volatility smile expansion in the Heston model, Quantitative Finance, 11, (2011) 1151-1164.

(With E. M. Stein) Asymptotic behavior of the distribution densities in models with stochastic volatility, I , Mathematical Finance, 20 (2010), 447-477.

(With E. M. Stein) Asymptotic behavior of the stock price distribution density and implied volatility in stochastic volatility models, Applied Mathematics and Optimization, 61, (2010), 287-315.

Asymptotic formulas with error estimates for call pricing functions and the implied volatility at extreme strikes, SIAM Journal on Financial Mathematics, 1, (2010) 609-641.

(With E. M. Stein) Implied volatility in the Hull-White model, Mathematical Finance, 19 (2009), 303-327.54. Markov processes, classes of time-dependent measures, and Feynman-Kac propagators, Transactions of the American Mathematical Society, 360 (2008), 4063-4098.

Markov processes, classes of time-dependent measures, and Feynman-Kac propagators, Transactions of the American Mathematical Society, 360 (2008), 4063-4098.

(With E. M. Stein) Asymptotic behavior of the distribution of the stock price in models with stochastic volatility: The Hull-White model, Comptes Rendus Acad Sci. Paris, 343 (2006), 519-523.

Nonautonomous Kato classes of measures and Feynman-Kac propagators, Transactions of the American Mathematical Society, 357 (2005), 4607-4632.

Feynman-Kac propagators and viscosity solutions (with J. van Casteren), Journal of Evolution Equations 5 (2005), 105-121.

On the heat equation with a time-dependent singular potential, Journal of Functional Analysis, 194, No 1 (2002).

Classes of time-dependent measures and the behavior of Feynman-Kac propagators, C. R. Acad. Sci. Paris, Ser. I 334 (2002), 445-449.

(With M. Cwikel) Interpolation on families of characteristic functions, Studia Mathematica, 138 (2000), 209-224.

Sharp estimates in smoothing theorems for Schrödinger semigroups, Journal of Functional Analysis, 170 (2000), 161-187.

Rearrangements of functions on a locally compact abelian group and integrability of the Fourier transform, Journal of Functional Analysis, 146, No 1 (1997), 62-115.

Rearrangement-invariant spaces of functions on LCA groups, Journal of Functional Analysis, 156 (1998), 384-410.

(with M. Kon) Exact smoothing properties of Schrödinger semigroups, American Journal of Mathematics, 118, No 6 (1996), 1215-1248.

(with C.R. MacCluer) Linear chaos in the quantum harmonic oscillator, American Society of Mechanical Engineering, Journal of Dynamic Systems, Measurement, and Control, 118, June 1996, 337-338.

(with R. Merletti and L.R. Lo Conte) Estimation of shape characteristics of surface EMG spectra from time domain data, IEEE Transactions in Biomedical Engineering, 42, No 8 (1995), 769-776.

(with G. Sandri) Quasianalytic function classes and the complex-time diffusion group, Mathematical Models and Methods in Applied Sciences5, No 2 (1995), 167-190.

On the Dimensionality Problem for Classes of Band-Limited Functions, in “The Interaction Between Functional Analysis, Harmonic Analysis, and Probability”, The Department of Mathematics, University of Missouri at Columbia, 1994, p. 8 – abstract.

(with M. Kon) Smoothness of Schrödinger semigroups and eigenfunctions, International Mathematics Research Notices, No 5  (1994), 193-199.

Mean dimension of function classes with Lebesgue measurable spectral sets, Journal of Approximation Theory, 75, No.1 (1993), 44-78.

A capacitary weak type estimate for spherical maximal functions, Abstracts of the AMS, 13, No. 2 (1992), 291.

Mean ε-dimension of classes of functions with spectra in a given set, Doklady of the USSR Academy of Sciences, 317, No.4 (1991), 803-807 (in Russian).

On the trace and extension operators for Besov spaces, Proceedings of the Tbilisi Institute of Mathematics, 95 (1989), 39-55 (in Russian).

Traces of functions belonging to Sobolev and Besov spaces and extensions from subsets of Euclidean space, Zapiski Nauchnyh Seminarov LOMI (Leningrad Inst. Of Math.), 157 (1987), 137-145 (in Russian). English translation in the Journal of Soviet Mathematics, 44, No.6 (1989).

Traces of differentiable functions on subsets of Euclidean space, Zapiski Nauchnyh Seminarov LOMI (Leningrad Inst. of Math.), 149 (1986), 52-66 (in Russian). English translation in the Journal of Soviet Math., 42, No.2 (1988), 1573-1583.

On traces of Besov spaces and the convergence of singular integrals, Reports of the Vekua Institute of Applied Mathematics Seminars (Tbilisi), 3, No.2 (1988), 33-36 (in Russian).

On the traces of differentiable functions on subsets of Euclidean space, Leningrad Inst. of Math. Preprint No 2-85, Leningrad (1985) (in Russian).

Multipliers in Besov spaces, Zapiski Nauchnyh Seminarov LOMI (Leningrad Institute of Mathematics), 135 (1984), 36-50 (in Russian). English translation in the Journal of Soviet Mathematics, 31, No.1 (1985).

Multipliers in Besov spaces and traces of functions on subsets of Euclidean space, Doklady of the USSR Academy of Sciences, 287, No.4 1985), 777-781 (in Russian).

Rearrangements of functions, sign-arrangements and convergence of sequences of operators, Zapiski Nauchnyh Seminarov LOMI (Leningrad Institute of Mathematics), 107 (1982), 46-70 (in Russian). English translation in the Journal of Soviet Mathematics, 36, No.3 (1987), 326-341.

Estimates for the Pettis integral in interpolation spaces and some inverse imbedding theorems, Doklady of the USSR Academy of Sciences, 263, No.4 (1982), 793-798 (in Russian). English translation in the Soviet Mathematics Doklady, 25, No.2 (1982), 428-432.

On the singularities of summable functions, Zapiski Nauchnyh Seminarov LOMI (Leningrad Institute of Mathematics), 113 (1981), 76-96 (in Russian). English translation in the Journal of Soviet Mathematics, 22, No.6 (1983), 1743-1757.

On the convergence of orthogonal series in L1 and almost everywhere, Bull Georgian Academy of Sciences, 102, No.3 (1981), 541-544 (in Russian).

On the behavior of the Fourier coefficients of equimeasurable functions, Doklady of the USSR Academy of Sciences, 250, No.4 (1980), 787-790 (in Russian). English translation in the Soviet Mathematics Doklady, 21, No.1 (1980), 191-194.

On the behavior of the Fourier transform in function spaces, Funktsional’nyi Analiz, 14, No.4 (1980), 71-722 (in Russian).

On the convergence of expansions of rearranged and sign-changed functions, Bulletin of the Georgian Academy of Sciences, 96, No.3 (1979), 537-540 (in Russian).

On the behavior of linear operators on the trajectories of elements, Bulletin of the Georgian Academy of Sciences, 93, No.3 (1979), 549-552 (in Russian).

The individual interpolation theorem, Bulletin of the Georgian Academy of Sciences, 94, No.1 (1979), 33-36 (in Russian).

On the behavior of the Fourier transform of rearranged functions, Bulletin of the Georgian Academy of Sciences, 87, No.2 (1977), 297-299 (in Russian).

(with V.A. Rodin and E.M. Semenov) The Fourier coefficients of summable functions, Matematicheskii Sbornik, 102, No.3 (1977), 362-372 (in Russian).

The interpolation theorem on subsets, Bulletin of the Georgian Academy of Sciences, 88, No.3 (1977), 545-548 (in Russian).

On the behavior of the Fourier coefficients of equimeasurable functions, Studia Mathematica, 54 (1975), 117-129 (in Russian).

An interpolation theorem of weak type and the behavior of the Fourier transform of a function having prescribed Lebesgue sets, Doklady of the USSR Academy of Sciences, 218, No.6 (1974), 1268-1271 (in Russian). English translation in Soviet Mathematics Doklady, 15, No.5 (1974), 1481-1485.

Metric equivalence and Fourier coefficients, Bulletin of the Georgian Academy of Sciences, 94, No.1 (1972), 553-555 (in Russian).

Fourier transforms of monotonic functions and distribution functions, Doklady of the USSR Academy of Sciences, 196, No.6 1971, 1259-1262 (in Russian). English translation in Soviet Mathematics Doklady 12, No.1 (1971), 336-340.

Trigonometric series with monotonically decreasing coefficients and distribution functions, Matematicheskie Zametki, 10, No.1 (1971), 3-10 (in Russian).

Estimates for the distribution functions of trigonometric series with monotonically decreasing coefficients, Bulletin of the Georgian Academy of Sciences, 58, No.1 (1970), 21-24 (in Russian).

More Publications

Book review on K. J. Falconer, The Geometry of Fractal Sets, Cambridge Tracts in Math., Vol. 85, Cambridge: University Press, 1985,– in Algebra and Analysis, 2, No.2 (1990), 249-259 (in Russian).

Book review on David Ruelle, Chaotic Evolution and Strange Attracators, Cambridge: University Press, 1989, Algebra and Analysis, 3, No. 2 (1991). English translation in St. Petersburg Math. J., 3, No. 4 (1992), 923-935.

Book review on Selected Papers of Norman Levinson, Birkhäuser, Boston-Basel-New York, 2 vol., 1998, Annals of the Polish Mathematical Society, s.III, Applied Mathematics, 41 (1999), 55-59.

Conversations with Gustave Choquet about his visit to Poland soon after the Second World War, Analyse stochastique et theorie du potentiel, Saint Priest de Gimel (2002), Association Laplace-Gauss, 2003.

Gustave Choquet rozmawia o swoim pobycie w Polsce po II wojnie światowej, Roczniki Polskiego Towarzystwa Matematycznego, Seria II: Wiadomosci Matematyczne, 39 (2003), 157-164 (translation of the previous item into Polish).

Feynman-Kac propagators and viscosity solutions (with J. A. van Casteren), Centre de Recerca Matematica Preprint Series, Barcelona, Spain, No 572, April 2004, 17 pages.

Markov processes and Feynman-Kac propagators, Centre de Recerca Matematica Preprint Series, Barcelona, Spain, No 573, April 2004, 48 pages.


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