메뉴 건너뛰기




Volumn 48, Issue 2, 2012, Pages 518-550

A Milstein-type scheme without Lévy area terms for SDEs driven by fractional Brownian motion

Author keywords

Approximation schemes; Fractional Brownian motion; L vy area

Indexed keywords


EID: 84874933937     PISSN: 02460203     EISSN: None     Source Type: Journal    
DOI: 10.1214/10-AIHP392     Document Type: Article
Times cited : (68)

References (47)
  • 1
    • 0346718762 scopus 로고    scopus 로고
    • Stochastic integration with respect to the fractional brownian motion
    • E. Alòs and D. Nualart. Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (2003) 129-152.
    • (2003) Stoch. Stoch. Rep. , vol.75 , pp. 129-152
    • Alòs, E.1    Nualart, D.2
  • 2
    • 34047148984 scopus 로고    scopus 로고
    • Operators associated with a stochastic differential equation driven by fractional brownian motions
    • F. Baudoin and L. Coutin. Operators associated with a stochastic differential equation driven by fractional Brownian motions. Stochastic Process. Appl. 117 (2007) 550-574.
    • (2007) Stochastic Process. Appl. , vol.117 , pp. 550-574
    • Baudoin, F.1    Coutin, L.2
  • 3
    • 52949087905 scopus 로고    scopus 로고
    • Pricing by hedging and no-arbitrage beyond semimartingales
    • C. Bender, T. Sottinen and E. Valkeila. Pricing by hedging and no-arbitrage beyond semimartingales. Finance Stoch. 12 (2008) 441-468.
    • (2008) Finance Stoch. , vol.12 , pp. 441-468
    • Bender, C.1    Sottinen, T.2    Valkeila, E.3
  • 4
    • 0042050466 scopus 로고
    • A law of large numbers for the maximum in a stationary Gaussian sequence
    • S. M. Berman. A law of large numbers for the maximum in a stationary Gaussian sequence. Ann. Math. Statist. 33 (1962) 93-97.
    • (1962) Ann. Math. Statist. , vol.33 , pp. 93-97
    • Berman, S.M.1
  • 5
    • 17444428635 scopus 로고    scopus 로고
    • A note on wick products and the fractional black-scholes model
    • T. Björk and H. Hult. A note on Wick products and the fractional Black-Scholes model. Finance Stoch. 9 (2005) 197-209.
    • (2005) Finance Stoch. , vol.9 , pp. 197-209
    • Björk, T.1    Hult, H.2
  • 6
    • 62549127896 scopus 로고    scopus 로고
    • Non-degeneracy of wiener functionals arising from rough differential equations
    • T. Cass, P. Friz and N. Victoir. Non-degeneracy of Wiener functionals arising from rough differential equations. Trans. Amer. Math. Soc. 361 (2009) 3359-3371.
    • (2009) Trans. Amer. Math. Soc. , vol.361 , pp. 3359-3371
    • Cass, T.1    Friz, P.2    Victoir, N.3
  • 7
    • 34548852567 scopus 로고    scopus 로고
    • Fractal dimension estimation via spectral distribution function and its application to physiological signals
    • S. Chang, S. Li, M. Chiang, S. Hu and M. Hsyu. Fractal dimension estimation via spectral distribution function and its application to physiological signals. IEEE Trans. Biol. Eng. 54 (2007) 1895-1898.
    • (2007) IEEE Trans. Biol. Eng. , vol.54 , pp. 1895-1898
    • Chang, S.1    Li, S.2    Chiang, M.3    Hu, S.4    Hsyu, M.5
  • 8
    • 84879314212 scopus 로고    scopus 로고
    • Power variation analysis of some integral long-memory processes
    • F. E. Benth et al. (Eds). Abel Symposia 2. Springer, Berlin
    • J. M. Corcuera. Power variation analysis of some integral long-memory processes. In Stochastic Analysis and Applications 219-234. F. E. Benth et al. (Eds). Abel Symposia 2. Springer, Berlin, 2007.
    • (2007) Stochastic Analysis and Applications , pp. 219-234
    • Corcuera, J.M.1
  • 9
    • 0036002985 scopus 로고    scopus 로고
    • Stochastic rough path analysis and fractional brownian motion
    • L. Coutin and Z. Qian. Stochastic rough path analysis and fractional Brownian motion. Probab. Theory Related. Fields 122 (2002) 108-140.
    • (2002) Probab. Theory Related. Fields , vol.122 , pp. 108-140
    • Coutin, L.1    Qian, Z.2
  • 10
    • 0009040449 scopus 로고
    • Stock price returns and the joseph effect: A fractional version of the black-scholes model
    • E. Bolthausen et al. (Eds). Prog. Probab. 36. Birkhäuser, Basel
    • N. J. Cutland, P. E. Kopp and W. Willinger. Stock price returns and the Joseph effect: A fractional version of the Black-Scholes model. In Seminar on Stochastic Analysis, Random Fields and Applications 327-351. E. Bolthausen et al. (Eds). Prog. Probab. 36. Birkhäuser, Basel, 1995.
    • (1995) Seminar on Stochastic Analysis, Random Fields and Applications , pp. 327-351
    • Cutland, N.J.1    Kopp, P.E.2    Willinger, W.3
  • 11
    • 62649139100 scopus 로고    scopus 로고
    • Differential equations driven by rough paths: An approach via discrete approximation
    • A. Davie. Differential equations driven by rough paths: An approach via discrete approximation. Appl. Math. Res. Express 2 (2007) 1-40.
    • (2007) Appl. Math. Res. Express , vol.2 , pp. 1-40
    • Davie, A.1
  • 12
    • 77955845466 scopus 로고    scopus 로고
    • Flow properties of differential equations driven by fractional brownian motion
    • P. H. Baxendale et al. (Eds). Interdiscip. Math. Sci. 2. World Sci. Publ., Hackensack, NJ
    • L. Decreusefond and D. Nualart. Flow properties of differential equations driven by fractional Brownian motion. In Stochastic Differential Equations: Theory and Applications 249-262. P. H. Baxendale et al. (Eds). Interdiscip. Math. Sci. 2. World Sci. Publ., Hackensack, NJ, 2007.
    • (2007) Stochastic Differential Equations: Theory and Applications , pp. 249-262
    • Decreusefond, L.1    Nualart, D.2
  • 14
    • 34547972571 scopus 로고    scopus 로고
    • Transient noise simulation: Modeling and simulation of 1/f-noise
    • K. Antreich et al. (Eds). Int. Ser. Numer. Math. 146. Birkhäuser, Basel
    • G. Denk, D. Meintrup and S. Schäffler. Transient noise simulation: Modeling and simulation of 1/f-noise. In Modeling, Simulation, and Optimization of Integrated Circuits 251-267. K. Antreich et al. (Eds). Int. Ser. Numer. Math. 146. Birkhäuser, Basel, 2001.
    • (2001) Modeling, Simulation, and Optimization of Integrated Circuits , pp. 251-267
    • Denk, G.1    Meintrup, D.2    Schäffler, S.3
  • 15
    • 34547988243 scopus 로고    scopus 로고
    • Modelling and simulation of transient noise in circuit simulation
    • G. Denk and R. Winkler. Modelling and simulation of transient noise in circuit simulation. Math. Comput. Model. Dyn. Syst. 13 (2007) 383-394.
    • (2007) Math. Comput. Model. Dyn. Syst. , vol.13 , pp. 383-394
    • Denk, G.1    Winkler, R.2
  • 16
    • 33749665074 scopus 로고    scopus 로고
    • Curvilinear integrals along enriched paths
    • D. Feyel and A. de La Pradelle. Curvilinear integrals along enriched paths. Electron. J. Probab. 11 (2006) 860-892.
    • (2006) Electron. J. Probab. , vol.11 , pp. 860-892
    • Feyel, D.1    De La Pradelle, A.2
  • 18
    • 70449519425 scopus 로고    scopus 로고
    • Discretization of the attractor of a system driven by fractional brownian motion
    • M. J. Garrido Atienza, P. E. Kloeden and A. Neuenkirch. Discretization of the attractor of a system driven by fractional Brownian motion. Appl. Math. Optim. 60 (2009) 151-172.
    • (2009) Appl. Math. Optim. , vol.60 , pp. 151-172
    • Garrido Atienza, M.J.1    Kloeden, P.E.2    Neuenkirch, A.3
  • 19
    • 33745022624 scopus 로고    scopus 로고
    • No arbitrage under transaction costs, with fractional brownian motion and beyond
    • P. Guasoni. No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16 (2006) 569-582.
    • (2006) Math. Finance , vol.16 , pp. 569-582
    • Guasoni, P.1
  • 20
    • 4344654665 scopus 로고    scopus 로고
    • Controlling rough paths
    • M. Gubinelli. Controlling rough paths. J. Funct. Anal. 216 (2004) 86-140.
    • (2004) J. Funct. Anal. , vol.216 , pp. 86-140
    • Gubinelli, M.1
  • 21
    • 73549099331 scopus 로고    scopus 로고
    • Ramification of rough paths
    • M. Gubinelli. Ramification of rough paths. J. Differential Equations 248 (2010) 693-721.
    • (2010) J. Differential Equations , vol.248 , pp. 693-721
    • Gubinelli, M.1
  • 22
    • 77953669175 scopus 로고    scopus 로고
    • Rough evolution equations
    • M. Gubinelli and S. Tindel. Rough evolution equations. Ann. Probab. 38 (2010) 1-75.
    • (2010) Ann. Probab. , vol.38 , pp. 1-75
    • Gubinelli, M.1    Tindel, S.2
  • 23
    • 51549086483 scopus 로고    scopus 로고
    • Ergodic theory for SDEs with extrinsic memory
    • M. Hairer and A. Ohashi. Ergodic theory for SDEs with extrinsic memory. Ann. Probab. 35 (2007) 1950-1977.
    • (2007) Ann. Probab. , vol.35 , pp. 1950-1977
    • Hairer, M.1    Ohashi, A.2
  • 24
    • 70449678769 scopus 로고    scopus 로고
    • Rough path analysis via fractional calculus
    • Y. Hu and D. Nualart. Rough path analysis via fractional calculus. Trans. Amer. Math. Soc. 361 (2009) 2689-2718.
    • (2009) Trans. Amer. Math. Soc. , vol.361 , pp. 2689-2718
    • Hu, Y.1    Nualart, D.2
  • 25
    • 0348173778 scopus 로고    scopus 로고
    • On convergence of the uniform norms for Gaussian processes and linear approximation problems
    • J. Hüsler, V. Piterbarg and O. Seleznjev. On convergence of the uniform norms for Gaussian processes and linear approximation problems. Ann. Appl. Probab. 13 (2003) 1615-1653.
    • (2003) Ann. Appl. Probab. , vol.13 , pp. 1615-1653
    • Hüsler, J.1    Piterbarg, V.2    Seleznjev, O.3
  • 26
    • 84894226734 scopus 로고    scopus 로고
    • Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global lipschitz coefficients
    • A. Jentzen, P. E. Kloeden and A. Neuenkirch. Pathwise approximation of stochastic differential equations on domains: Higher order convergence rates without global Lipschitz coefficients. Numer. Math. 112 (2009) 41-64.
    • (2009) Numer. Math. , vol.112 , pp. 41-64
    • Jentzen, A.1    Kloeden, P.E.2    Neuenkirch, A.3
  • 27
    • 80052377200 scopus 로고    scopus 로고
    • Multilevel Monte Carlo for stochastic differential equations with additive fractional noise
    • P. E. Kloeden, A. Neuenkirch and R. Pavani. Multilevel Monte Carlo for stochastic differential equations with additive fractional noise. Ann. Oper. Res. 189 (2011) 255-276.
    • (2011) Ann. Oper. Res. , vol.189 , pp. 255-276
    • Kloeden, P.E.1    Neuenkirch, A.2    Pavani, R.3
  • 29
    • 54949102951 scopus 로고    scopus 로고
    • Stochastic modeling in nanoscale physics: Subdiffusion within proteins
    • S. Kou. Stochastic modeling in nanoscale physics: Subdiffusion within proteins. Ann. Appl. Statist. 2 (2008) 501-535.
    • (2008) Ann. Appl. Statist. , vol.2 , pp. 501-535
    • Kou, S.1
  • 31
    • 52149107530 scopus 로고    scopus 로고
    • The rate of convergence for euler approximations of solutions of stochastic differential equations driven by fractional brownian motion
    • Y. Mishura and G. Shevchenko. The rate of convergence for Euler approximations of solutions of stochastic differential equations driven by fractional Brownian motion. Stochastics 80 (2008) 489-511.
    • (2008) Stochastics , vol.80 , pp. 489-511
    • Mishura, Y.1    Shevchenko, G.2
  • 32
    • 56549118742 scopus 로고    scopus 로고
    • Minimal errors for strong and weak approximation of stochastic differential equations
    • A. Keller et al. (Eds). Springer, Berlin
    • T. Müller-Gronbach and K. Ritter. Minimal errors for strong and weak approximation of stochastic differential equations. In Monte Carlo and Quasi-Monte Carlo Methods 2006 53-82. A. Keller et al. (Eds). Springer, Berlin, 2008.
    • (2008) Monte Carlo and Quasi-Monte Carlo Methods 2006 , pp. 53-82
    • Müller-Gronbach, T.1    Ritter, K.2
  • 33
    • 55649094460 scopus 로고    scopus 로고
    • Optimal pointwise approximation of stochastic differential equations driven by fractional brownian motion
    • A. Neuenkirch. Optimal pointwise approximation of stochastic differential equations driven by fractional Brownian motion. Stochastic Process. Appl. 118 (2008) 2294-2333.
    • (2008) Stochastic Process. Appl. , vol.118 , pp. 2294-2333
    • Neuenkirch, A.1
  • 34
    • 35548963865 scopus 로고    scopus 로고
    • Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional brownian motion
    • A. Neuenkirch and I. Nourdin. Exact rate of convergence of some approximation schemes associated to SDEs driven by a fractional Brownian motion. J. Theoret. Probab. 20 (2007) 871-899.
    • (2007) J. Theoret. Probab. , vol.20 , pp. 871-899
    • Neuenkirch, A.1    Nourdin, I.2
  • 38
    • 51849102908 scopus 로고    scopus 로고
    • A simple theory for the study of SDEs driven by a fractional brownian motion, in dimension one
    • Lecture Notes in Math. 1934. Springer, Berlin
    • I. Nourdin. A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one. In Sém. Probab. XLI 181-197. Lecture Notes in Math. 1934. Springer, Berlin, 2008.
    • (2008) Sém. Probab. XLI , pp. 181-197
    • Nourdin, I.1
  • 39
    • 44049089660 scopus 로고    scopus 로고
    • Correcting newton-cotes integrals by lévy areas
    • I. Nourdin and T. Simon. Correcting Newton-Cotes integrals by Lévy areas. Bernoulli 13 (2007) 695-711.
    • (2007) Bernoulli , vol.13 , pp. 695-711
    • Nourdin, I.1    Simon, T.2
  • 41
    • 0038771348 scopus 로고    scopus 로고
    • Differential equationsrtfgh driven by fractional brownian motion
    • D. Nualart and A. Rǎşcanu. Differential equationsrtfgh driven by fractional Brownian motion. Collect. Math. 53 (2002) 55-81.
    • (2002) Collect. Math. , vol.53 , pp. 55-81
    • Nualart, D.1    Rǎşcanu, A.2
  • 42
    • 15844396187 scopus 로고    scopus 로고
    • Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth
    • D. Odde, E. Tanaka, S. Hawkins and H. Buettner. Stochastic dynamics of the nerve growth cone and its microtubules during neurite outgrowth. Biotechnol. Bioeng. 50 (1996) 452-461.
    • (1996) Biotechnol. Bioeng. , vol.50 , pp. 452-461
    • Odde, D.1    Tanaka, E.2    Hawkins, S.3    Buettner, H.4
  • 43
    • 79953703984 scopus 로고    scopus 로고
    • The rough path associated to the multidimensional analytic fBm with any hurst parameter
    • S. Tindel and J. Unterberger. The rough path associated to the multidimensional analytic fBm with any Hurst parameter. Collect. Math. 62 (2011) 197-223.
    • (2011) Collect. Math. , vol.62 , pp. 197-223
    • Tindel, S.1    Unterberger, J.2
  • 44
    • 66149191161 scopus 로고    scopus 로고
    • Stochastic calculus for fractional brownian motion with hurst exponent H > 1/4: A rough path method by analytic extension
    • J. Unterberger. Stochastic calculus for fractional Brownian motion with Hurst exponent H > 1/4: A rough path method by analytic extension. Ann. Probab. 37 (2009) 565-614.
    • (2009) Ann. Probab. , vol.37 , pp. 565-614
    • Unterberger, J.1
  • 45
    • 23044532516 scopus 로고    scopus 로고
    • On a functional limit result for increments of a fractional brownian motion
    • W. Wang. On a functional limit result for increments of a fractional Brownian motion. Acta Math. Hung. 93 (2001) 153-170.
    • (2001) Acta Math. Hung. , vol.93 , pp. 153-170
    • Wang, W.1
  • 47
    • 0038290919 scopus 로고    scopus 로고
    • Integration with respect to fractal functions and stochastic calculus I
    • M. Zähle. Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Related Fields 111 (1998) 333-374.
    • (1998) Probab. Theory Related Fields , vol.111 , pp. 333-374
    • Zähle, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.