Book contents
References
Published online by Cambridge University Press: 06 April 2020
- Type
- Chapter
- Information
- Mathematics of the Bond Market , pp. 373 - 378Publisher: Cambridge University PressPrint publication year: 2020
References
[1]
Aihara, S. I., Bagchi, A.: Stochastic hyperbolic dynamics for infinite-dimensional forward rates and option pricing (2005), Mathematical Finance 15, 27–47.CrossRefGoogle Scholar
[2]
Applebaum, D.: Lévy Processes and Stochastic Calculus (2004), Cambridge University Press.Google Scholar
[3]
Barski, M. Incompleteness of the bond market with Lévy noise under the physical measure (2015), Advances in Mathematics of Finance, Banach Center Publications 104, 61–84.Google Scholar
[4]
Barski, M., Jakubowski, J., Zabczyk, J.: On incompleteness of bond markets with infinite number of random sources (2011), Mathematical Finance 21, 3, 541–556.CrossRefGoogle Scholar
[5]
Barski, M., Zabczyk, J.: Completeness of bond market driven by Lévy processes (2010), International Journal of Theoretical and Applied Finance 13, 635–656.Google Scholar
[6]
Barski, M., Zabczyk, J.: Heat–Jarrow–Morton–Musiela equation with Lévy perturbation (2012), Journal of Differential Equations 253, 9, 2657–2697.Google Scholar
[7]
Barski, M., Zabczyk, J.: Forward rate models with linear volatilities (2012), Finance and Stochastics 16, 3, 537–560.CrossRefGoogle Scholar
[9]
Barski, M., Zabczyk, J.: On CIR equations with general factors (forthcoming), SIAM Journal on Financial Mathematics.Google Scholar
[10]
Bayraktar, E., Chen, L., Poor, H. V.: Consistency problem for jump diffusion models, arXiv:cs/0501055v1 [cs.IT January 23, 2005.Google Scholar
[11]
Berestycki, J., Döring, L., Mytnik, L., Zambotti, L.: Hitting properties and non-uniqueness for SDE driven by stable processes (2015), Stochastic Processes and Their Applications 150, 918–949.Google Scholar
[13]
Bichteler, K.: Stochastic Integration with Jumps (2011), Cambridge University Press.Google Scholar
[14]
Bichteler, K., Lin, S. J.: On the stochastic Fubini theorem (1995), Stochastics and Stochastic Reports 54, 271–279.CrossRefGoogle Scholar
[15]
Brace, A., Gątarek, D., Musiela, M.: The market model of interest rate dynamics (1997), Mathematical Finance 7, 127–147.Google Scholar
[17]
Björk, T.: Interest rate theory (1996), Lecture Notes in Mathematics 1656, 53–122.Google Scholar
[18]
Björk, T.: On the term structure of discontinuous interest rates (1995), Obozrenie Prikladnoi i Promyshlennoi Matematiki 2, 626–657 (in Russian).Google Scholar
[19]
Björk, T., Di Masi, G., Kabanov, Y., Runggaldier, W.: Towards a general theory of bond markets (1997), Finance and Stochastics 1, 141–174.CrossRefGoogle Scholar
[20]
Björk, T., Kabanov, Y., Runggaldier, W.: Bond market structure in the presence of marked point process (1997), Mathematical Finance 7, 211–239.CrossRefGoogle Scholar
[21]
Borwein, P. B., Erdélyi, T.: Generalizations of Müntz’s theorem via a Remez-type inequality for Müntz spaces (1997), Journal of the American Mathematical Society 10, 2, 327–349.Google Scholar
[22]
Brzeźniak, Z., Peszat, S., Zabczyk, J.: Continuity of stochastic convolutions (2001), Czechoslovak Mathematical Journal 51, 679–684.Google Scholar
[23]
Brzeźniak, Z., Kok, T.: Stochastic evolution equations in Banach spaces and applications to the Heath–Jarrow–Morton–Musiela equations (2018), Finance and Stochastics 22, 4, 959–1006.Google Scholar
[24]
Carmona, R. A., Tehranchi, M. R.: A characterization of hedging portfolios for interest rate contingent claims (2004), Annals of Applied Probability 14, 1267–1294.Google Scholar
[25]
Carmona, R. A., Tehranchi, M. R.: Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective (2006), Springer.Google Scholar
[26]
Chatelain, M., Stricker, C.: On componentwise and vector stochastic integration (1994), Mathematical Finance 4, 57–65.Google Scholar
[27]
Cherny, A. S. Vector stochastic integrals in the fundamental theorem of asset pricing (1998) Proceedings of the Workshop on Mathematical Finance, INRIA, 149–163.Google Scholar
[28]
Cont, R.: Modeling term structure dynamics: An infinite dimensional approach (2005), International Journal of Theoretical and Applied Finance 8, 3, 357–380.Google Scholar
[29]
Cont, R., Tankov, P.: Financial Modelling with Jump Processes (2004), Chapman & Hall.Google Scholar
[30]
Cox, J., Ingersoll, J., Ross, S.: A theory of the term structure of interest rates (1985), Econometrica 53, 385–408.CrossRefGoogle Scholar
[31]
Dawson, D. A., Zenghu, L.: “Skew convolution semigroups and affine Markov processes (2006), Annals of Probability, 34, 1103–1142.Google Scholar
[33]
Da Prato, G., Zabczyk, J. Stochastic Equations in Infinite Dimensions (1992), Cambridge University Press.Google Scholar
[34]
Davis, M. H. A.: The representation of martingales of a jump process (1976), SIAM Journal on Control and Optimization 14, 623–638.Google Scholar
[35]
De Acosta, A.: Exponential moments of vector valued random series and triangular arrays (1980), Annals of Probability 8, 381–389.Google Scholar
[36]
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing (1994), Mathematische Annalen 300, 463–520.Google Scholar
[37]
Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes (1998), Mathematische Annalen 312, 215–250.Google Scholar
[38]
Dellacherie, C.: Intégrales stochastiques par rapport aux processus de Wiener ou de Poisson (1974), (Séminaire de Probabilités, 494. VIII, University of Strasbourg, 1972–1973), Lecture Notes in Mathematics 381, 25–26.CrossRefGoogle Scholar
[39]
Dellacherie, C.: Corrections to Intégrales stochastiques par rapport aux processus de Wiener ou de Poisson (Séminaire de Probabilités, VIII, (University of Strasbourg, Année Universitaire 1972–1973, Lecture Notes in Mathematics, 1974, 381, 25–26). Séminaire de Probabilités, IX (Seconde Partie, University of Strasbourg, Strasbourg, Annés Universitaires 1973–1974 et 1974–1975) (1975), Lecture Notes in Mathematics 465–494.Google Scholar
[40]
De Donno, M., Pratelli, M.: On the use of measured-valued strategies in bond markets (2004), Finance and Stochastics, 8, 87–109.Google Scholar
[41]
De Donno, M.: A note on completeness in large financial markets (2004), Mathematical Finance, 14, 2, 295–315.Google Scholar
[42]
Duffie, D., Filipović, D. and Schachermeyer, W.: Affine processes and applications in finance (2003), Annals of Applied Probability, 13, 3, 984–1053.Google Scholar
[45]
Dybvig, P., Ingersoll, J., Ross, S.: Long forward and zero coupon rates can never fall (1996), Journal of Business, 69, 1–25.Google Scholar
[47]
Eberlein, E., Jacod, J., Raible, S.: Lévy term structure models: No-arbitrage and completeness (2005), Finance and Stochastics, 9, 67–88.Google Scholar
[48]
Eberlein, E., Raible, S.: Term structure models driven by general Lévy processes (1999), Mathematical Finance, 9, 31–53.CrossRefGoogle Scholar
[49]
Eberlein, E., Özkan, F.: The Lévy LIBOR model (2005), Finance and Stochastics, 9, 327–348.Google Scholar
[50]
Ekeland, I., Taflin, E.: A theory of bond portfolios (2005), Annals of Applied Probability, 15, 2, 1260–1305.CrossRefGoogle Scholar
[51]
Feller, W.: An Introduction to Probability Theory and Its Applications (1970), John Willey and Sons.Google Scholar
[53]
Filipović, D.: A general characterization of one factor affine term structure models (2001), Finance and Stochastics, 5, 3, 389–412.Google Scholar
[54]
Filipović, D.: Consistency problems for Heath–Jarrow–Morton interest rate models (2001), Lecture Notes in Mathematics, Vol. 1760.Google Scholar
[55]
Filpović, D.: Separable term structures and the maximal degree problem (2002), Mathematical Finance, 12, 4, 341–349.Google Scholar
[56]
Filipović, D., Tappe, S.: Existence of Lévy term structure models (2008), Finance and Stochastics, 12, 83–115.Google Scholar
[57]
Filipović, D., Tappe, S., Teichmann, J.: Term structure models driven by Wiener process and Poisson measures: Existence and positivity (2010), SIAM Journal on Financial Mathematics, 1, 523–554.CrossRefGoogle Scholar
[58]
Filipović, D., Zabczyk, J.: Markovian term structure models in discrete time (1999), Preprint, Institute of Matematics, Polish Academy of Sciences, 601, 1–12.Google Scholar
[59]
Filipović, D., Zabczyk, J.: Markovian term structure models in discrete time (2002), The Annals of Applied Probability, 12, 710–729.Google Scholar
[60]
Fu, Z., Li, Z.: Stochastic equations of non-negative processes with jumps (2010), Stochastic Processes and Their Applications, 120, 306–330.Google Scholar
[61]
Gantmacher, F. , Krein, M.: Oscillation Matrices and Kernels and Small Vibrations of Mechanical Systems (2002), rev. ed., AMS Chelsea Publishing.Google Scholar
[62]
Gątarek, D.: Some remarks on the market model of interest rates (1996), Control and Cybernetics, 25, 6, 1233–1244.Google Scholar
[63]
Gikhman, I., Skorohod, A.V.: The theory of stochastic processes (1974), Vols. I–II, Springer.Google Scholar
[64]
Glasserman, P., Kou, S. G.: The term structure of simple forward rates with jump risk (2003), Mathematical Finance, 13, 383–410.Google Scholar
[65]
Grzywacz, T., Leżaj, Ł., Trajan, B.: Transition densities of subordinators (2018), arXiv: 1812.06793 [math.Pr].Google Scholar
[66]
Harrison, J. M., Pliska, S. R.: Martingales and stochastic integrals in the theory of continuous trading (1981), Stochastic Processes and Their Applications, 11, 3, 215–260.Google Scholar
[67]
Heath, D., Jarrow, R., Morton, A.: Bond pricing and the term structure of interest rates: A new methodology for contingent claim valuation (1992), Econometrica, 60, 77–105.Google Scholar
[68]
Hubalek, F.: A counterexample involving exponential-affine Laplace transforms (2001), unpublished, TU Vienna.Google Scholar
[69]
Hubalek, F., Klein, I., Teichmann, J.: A general proof of the Dybvig–Ingersoll–Ross theorem: Long forward rates can never fall (2002), Mathematical Finance, 12, 447–451.Google Scholar
[70]
Hunt, P. J., Kennedy, J. E.: Financial Derivatives in Theory and Practice (2005), John Wiley & Sons, Ltd.Google Scholar
[71]
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes (1981), Kodansha, Ltd.Google Scholar
[72]
Ikeda, N., Watanabe, S.: On the uniqueness and non-uniqueness of a class of non-linear equations and explosion problem for branching processes (1970), Journal of the Faculty of Science University Tokyo Sect. I, 17, 187–214.Google Scholar
[73]
Itô, K.: Spectral type of the shift transformation of differential processes with stationary increments (1956), Transactions of the American Mathematical Society, 81, 253–63.Google Scholar
[74]
Jacod, J.: Calcul Stochastique et Problém des Martingales (1979), Lecture Notes in Mathematics 714, Springer.Google Scholar
[75]
Jacod, J., Shiryaev, A. N.: Limit Theorems for Stochastic Processes (2002), Springer.Google Scholar
[76]
Jakubowski, J., Zabczyk, J.: Exponential moments for HJM models with jumps (2007), Finance and Stochastics, 11, 429–445.Google Scholar
[77]
Jarrow, R. A., Madan, D. B.: A characterization of complete security markets on a Brownian filtration (1991), Mathematical Finance, 1, 31–43.Google Scholar
[78]
Jeanblanc, K., Yor, M., Chesney, M.: Mathematical Methods for Financial Markeets (2009), Springer.Google Scholar
[79]
Kallenberg, O.: Foundations of Modern Probability (2001), 2nd ed., Springer-Verlag.Google Scholar
[81]
Kawazu, K., Watanabe, S.: Branching processes with immigration and related topics (1971), Theory of Probability and Its Applications 16, 36–54.Google Scholar
[82]
Kinney, J. H.: “Continuity properties of sample functions of Markov processes (1953), Transactions of the American Mathematical Society, 74, 280–302.Google Scholar
[84]
Kunita, H.: Representation of martingales with jumps and applications to mathematical finance (2004), Advanced Studies in Pure Mathematics 41, Stochastic Analysis and Related Topics, 209–232.Google Scholar
[85]
Kunita, H., Watanabe, S.: On square-integrable martingales (1967), Nagoya Mathematical Journal 30, 209–245.Google Scholar
[86]
Kwapień, S., Marcus, M. B., Rosiński, J.: Two results on continuity and boundness of stochastic convolutions (2006), Annales de l’Institut Henri Poincare 42, 553–566.CrossRefGoogle Scholar
[87]
Kyprianou, A. E., Pardo, J. C.: Continuous-state branching processes and self-similarity (2008), Journal of Applied Probability, 45, 1140–1160.Google Scholar
[88]
Li, Z., Mytnik, L.: Strong solutions for stochastic differential equations with jumps (2011), Annales de l’Institut Henri Poincaré Probability et Statistics, 47, 1055–1087.Google Scholar
[89]
Liptser, R. S., Shiryaev, A. N.: Statistics of Random Processes, I: General Theory, vol. 2 (2001), 2nd ed., Springer-Verlag.Google Scholar
[90]
Marinelli, C.: Local well-posedness of Musiela’s SPDE with Lévy noise (2010), Mathematical Finance 20, 341–363.Google Scholar
[91]
Milian, A.: Comparison theorems for stochastic evolution equation (2002), Stochastics and Stochastics Reports, 72, 79–108.Google Scholar
[92]
Métivier, M.: Semimartingales: A Course on Stochastic Processes (1982), Walter de Gruyter.Google Scholar
[93]
Mikulevicius, R., Rozovskii, B. L.: Normalized stochastic integrals in topological vector spaces (1998), Séminaire de probabilités XXXII (Lecture Notes in Mathematics), Springer, 137–165.Google Scholar
[94]
Mikulevicius, R., Rozovskii, B. L.: Martingales problems for stochastic PDEs (1999), Stochastic Partial Differential Equations: Six Perspectives, ed. R. Carmona, B. Rozovski (Mathematical Surveys and Monographs, 64), American Mathematical Society, 243–326.Google Scholar
[95]
Morton, A.: Arbitrage and martingales (1989), PhD dissertation, Cornell University.Google Scholar
[96]
Müntz, C. H.: Über den Approximationssatz von Weierstrass (1914). In H. A. Schwarz’s Festschrift, 303–312.Google Scholar
[97]
Musiela, M.: Stochastic PDEs and term structure models (1993), Journées International de Finance, IGR-AFFI, La Baule.Google Scholar
[98]
Musiela, M., Gołdys, B.: Infinite dimensional diffusions, Kolmogorov equations and interest rate models (2001), Option Pricing, Interest Rates and Risk Management, Handbook of Mathematical Finance, ed. Jouini, E., Cvitanic, J., Musiela, M., Cambridge University Press, 314–335.Google Scholar
[99]
Özkan, F.: Lévy processes in credit risk and market models (2002), Ph thesis, Freiburg University.Google Scholar
[100]
Peszat, S., Zabczyk, J.: Stochastic Partial Differential Equations with Lévy Noise (2007), Cambridge University Press.Google Scholar
[101]
Peszat, S., Zabczyk, J.: Heath–Jarrow–Morton–Musiela equation of bond market (2007), www.impan.pl/Preprints/p677.pdf.Google Scholar
[102]
Protter, P.: Stochastic Integration and Differential Equations (2005), Springer.Google Scholar
[103]
Pham, H.: A predictable decomposition in infinite asset model with jumps: Application to hedging and optimal investment (2003), Stochastics and Stochastic Reports, 5, 343–368.Google Scholar
[105]
Raible, S.: Lévy processes in finance: Theory, numerics and empirical facts (2000), PhD thesis, Freiburg University.Google Scholar
[106]
Rogers, L. C. G., Williams, D.: Diffusions, Markov Processes and Martingales (2000), Cambridge University Press.Google Scholar
[107]
Ringer, N., Tehranchi, M.: Optimal portfolio choice in the bond market (2006), Finance and Stochastics, 10, 4, 553–573.Google Scholar
[108]
Rosiński, J.: Remarks on strong exponential integrability of vector-valued random series and triangular arrays (1995), Annals of Probability, 23, 464–473.Google Scholar
[109]
Rusinek, A.: Properties of solutions to the Musiela equation with Lévy noise (2011), PhD dissertation, Institute of Mathematics, Polish Academy of Sciences.Google Scholar
[110]
Rusinek, A.: A note on HJMM models on a space of square integrable functions (2012), manuscript.Google Scholar
[111]
Rusinek, A.: Mean reversion for HJMM forward rate models (2010), Advances in Applied Probability, 42, 371–391.Google Scholar
[112]
Rusinek, A.: Invariant measures for forward rate HJM model with Lèvy noise (2006), Preprint IMPAN 669, www.impan.pl/Preprints/p669.pdf.Google Scholar
[113]
Rusinek, A.: Forward rate models on the space of square integrable functions (2016), Rocznik Naukowy WZ Ciechanów, Vol. X, 1–4.Google Scholar
[114]
Sato, K. I.: Lévy Processes and Infinite Divisible Distributions (1999), Cambridge University Press.Google Scholar
[115]
Schachermayer, W.: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time (1992), Insurance: Mathematics and Economics, 11, 4, 249–257.Google Scholar
[116]
Shiryaev, A. N.: Essentials of Stochastic Finance, Facts, Models, Theory (1999), vol. 3, World Scientific Publishing.Google Scholar
[119]
Taflin, E.: Bond market completeness and attainable contingent claims (2005), Finance and Stochastics, 9, 429–452.Google Scholar
[120]
Taflin, E.: Generalized integrands and bond portfolios: Pitfalls and counterexamples (2011), Annals of Applied Probability, 21, 266–282.Google Scholar
[121]
Vasiček, O.: An equilibrium characterization of the term structure (1977), Journal of Financial Economics, 5, 177–188.Google Scholar
[123]
Zabczyk, J.: Mathematical Control Theory: An Introduction (1992), Birkhäuser (2nd ed. forthcoming in 2020).Google Scholar
[124]
Zanzotto, P. A.: On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion, Stochastic Processes and Their Applications (1997), 68, 209–228.Google Scholar
[125]
Zongfei, F., Zenghu, L.: “Stochastic equations of non-negative processes with jumps (2010), Stochastic Processes and Their Applications, 120, 306–330.Google Scholar