Sensitivities via rough paths
ESAIM: Probability and Statistics, Tome 19 (2015), pp. 515-543.

Motivated by a problematic coming from mathematical finance, the paper deals with existing and additional results on the continuity and the differentiability of the Itô map associated to rough differential equations. These regularity results together with the Malliavin calculus are applied to the sensitivities analysis of stochastic differential equations driven by multidimensional Gaussian processes with continuous paths as the fractional Brownian motion. The well-known results on greeks in the Itô stochastic calculus framework are extended to stochastic differential equations driven by a Gaussian process which is not a semi-martingale.

Reçu le :
DOI : 10.1051/ps/2015001
Classification : 60H10
Mots clés : Rough paths, Rough differential equations, Malliavin calculus, sensitivities, mathematical finance, Gaussian processes 
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     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2015001/}
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Marie, Nicolas. Sensitivities via rough paths. ESAIM: Probability and Statistics, Tome 19 (2015), pp. 515-543. doi : 10.1051/ps/2015001. http://www.numdam.org/articles/10.1051/ps/2015001/

N.H. Bingham and R. Kiesel, Risk-Neutral Valuation. Pricing and Hedging of Financial Derivatives. Springer Finance XVIII. Springer (2004). | MR | Zbl

H. Cartan, Cours de calcul différentiel. Méthodes, Hermann (2007). | Zbl

T. Cass, C. Litterer and T. Lyons, Integrability Estimates for Gaussian Rough Differential Equations. Preprint: (2011). | arXiv | MR | Zbl

P. Cheridito, Regularizing Fractional Brownian Motion with a View towards Stock Price Modeling. Ph.D. thesis, Université de Zürich (2001). | MR

L. Decreusefond and A. Ustunel, Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1999) 177–214. | MR | Zbl

T. Dieker, Simulation of Fractional Brownian Motion. Master thesis, University of Twente (2004).

Y. El Khatib and N. Privault, Computations of Greeks in markets with jumps via the Malliavin calculus. Finance Stoch. 8 (2004) 161–179. | MR | Zbl

E. Fournié, J-M. Lasry, J. Lebuchoux, P.-L. Lions and N. Touzi, Applications of Malliavin calculus to Monte-Carlo methods in finance. Finance Stoch. 3 (1999) 391–412. | MR | Zbl

P. Friz and N. Victoir, Differential equations driven by Gaussian signals. Ann. Inst. Henri Poincaré, Probab. Stat. 46 (2010) 369–341. | MR | Zbl

P. Friz and N. Victoir, Multidimensional Stochastic Processes as Rough Paths: Theory and Applications. Vol. 120 of Cambridge Stud. Appl. Math. Cambridge University Press, Cambridge (2010). | MR | Zbl

E. Gobet and R. Münos, Sensitivity analysis using Itô–Malliavin calculus and martingales, and application to stochastic optimal control. SIAM J. Control Optim. 43 (2005) 1676–1713. | MR | Zbl

H. Kunita, Stochastic Flows and Stochastic Differential Equations. Vol. 24 of Cambridge Stud. Appl. Math.. Cambridge University Press, Cambridge (1997). | MR | Zbl

D. Lamberton and B. Lapeyre, Introduction au calcul stochastique appliqué`a la finance, 2nd edition. Math. Appl. Ellipses (1997). | MR

A. Lejay, Controlled differential equations as Young integrals: A simple approach. J. Differ. Equ. 248 (2010) 1777–1798. | MR | Zbl

T. Lyons, Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998) 215–310. | MR | Zbl

T. Lyons and Z. Qian, System Control and Rough Paths. Oxford University Press (2002). | MR | Zbl

P. Malliavin and A. Thalmaier, Stochastic Calculus of Variations in Mathematical Finance. Springer Finance. Springer-Verlag, Berlin (2006). | MR | Zbl

B.B. Mandelbrot and J.W. Van Ness, Fractional Brownian motion, fractional noises and applications. SIAM Rev. 10 (1968) 422–437. | MR | Zbl

N. Marie, A generalized mean-reverting equation and applications. ESAIM: PS 18 (2014) 799–828. | MR | Zbl

N. Marie, Trajectoires rugueuses, processus gaussiens et applications. Probability [math.PR]. University Paul Sabatier, Toulouse III (2012) HAL: tel-00783931v2.

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. | MR | Zbl

D. Nualart, The Malliavin Calculus and Related Topics, 2nd edition. Probab. Appl. Springer-Verlag, Berlin (2006). | MR

N. Privault and X. Wei, A Malliavin calculus approach to sensitivity in insurance. Insurance: Math. Econ. 35 (2004) 679–690. | MR | Zbl

L.C.G. Rogers, Arbitrage with fractional Brownian motion. Math. Finance 7 (1997) 95–105. | MR | Zbl

S. Samko, A. Kilbas and O. Marichev, Fractional Integrals and Derivatives. Gordon and Breach Science (1993). | MR | Zbl

J. Teichmann, Calculating the Greeks by cubature formulas. Proc. Roy. Soc. London A 462 (2006) 647–670. | MR | Zbl

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