Fractional smoothness of functionals of diffusion processes under a change of measure

Stefan Geiss (University of Innsbruck)
Emmanuel Gobet (École Polytechnique)

Abstract


Let $v:[0,T]\times {\mathbf R}^d \to {\mathbf R}$ be the solution of the parabolic backward equation $$\partial_t v + (1/2) \sum_{i,j} [\sigma \sigma^\top]_{i,j} \partial_{x_i}\partial_{x_j}v+ \sum_{i} b_i \partial_{x_i}v + kv =0$$ with terminal condition $g$, where the coefficients are time-and state-dependent, and satisfy certain regularity assumptions. Let $X = (X_t)_{t\in [0,T]}$ be the associated ${\mathbf R}^d$-valued diffusion process on some appropriate $(\Omega,{\mathcal F},{\mathbb Q})$. For $p\in [2,\infty)$ and a measure $d{\mathbb P}=\lambda_T d{\mathbb Q}$, where $\lambda_T$ satisfies the Muckenhoupt condition $A_p$, we relate the behavior of \[  \|g(X_T)-{\mathbf E}_{\mathbb P}(g(X_T)|{\mathcal F}_t) \|_{L_p({\mathbb P})}, \quad     \|\nabla v(t,X_t)  \|_{L_p({\mathbb P})},           \quad    \|D^2 v(t,X_t)  \|_{L_p({\mathbb P})} \]to each other, where $D^2v:=(\partial_{x_i}\partial_{x_j}v)_{i,j}$ is the Hessian matrix.


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Pages: 1-14

Publication Date: June 13, 2014

DOI: 10.1214/ECP.v19-2786

References

  • Avikainen, Rainer. On irregular functionals of SDEs and the Euler scheme. Finance Stoch. 13 (2009), no. 3, 381--401. MR2519837
  • Bonami, Aline; Lépingle, Dominique. Fonction maximale et variation quadratique des martingales en présence d'un poids. (French) Séminaire de Probabilités, XIII (Univ. Strasbourg, Strasbourg, 1977/78), pp. 294--306, Lecture Notes in Math., 721, Springer, Berlin, 1979. MR0544802
  • Friedman, Avner. Partial differential equations of parabolic type. Prentice-Hall, Inc., Englewood Cliffs, N.J. 1964 xiv+347 pp. MR0181836
  • Geiss, Christel; Geiss, Stefan. On approximation of a class of stochastic integrals and interpolation. Stoch. Stoch. Rep. 76 (2004), no. 4, 339--362. MR2075477
  • Geiss, Christel; Geiss, Stefan; Gobet, Emmanuel. Generalized fractional smoothness and $L_ p$-variation of BSDEs with non-Lipschitz terminal condition. Stochastic Process. Appl. 122 (2012), no. 5, 2078--2116. MR2921973
  • Geiss, Stefan; Hujo, Mika. Interpolation and approximation in $L_ 2(\gamma)$. J. Approx. Theory 144 (2007), no. 2, 213--232. MR2293387
  • Gobet, Emmanuel; Munos, Rémi. Sensitivity analysis using Itô-Malliavin calculus and martingales, and application to stochastic optimal control. SIAM J. Control Optim. 43 (2005), no. 5, 1676--1713 (electronic). MR2137498
  • Gobet, Emmanuel; Makhlouf, Azmi. ${\bf L}_ 2$-time regularity of BSDEs with irregular terminal functions. Stochastic Process. Appl. 120 (2010), no. 7, 1105--1132. MR2639740
  • Gobet, Emmanuel; Makhlouf, Azmi. The tracking error rate of the delta-gamma hedging strategy. Math. Finance 22 (2012), no. 2, 277--309. MR2897386
  • Gobet, Emmanuel; Temam, Emmanuel. Discrete time hedging errors for options with irregular payoffs. Finance Stoch. 5 (2001), no. 3, 357--367. MR1849426
  • Geiss, Stefan; Toivola, Anni. Weak convergence of error processes in discretizations of stochastic integrals and Besov spaces. Bernoulli 15 (2009), no. 4, 925--954. MR2597578
  • S. Geiss and A. Toivola. On fractional smoothness and L_p-approximation on the Wiener space. arXiv:1206.5415, 2012. To appear in Annals Prob. as "On fractional smoothness and L_p-approximation on the Gaussian space".
  • Imkeller, Peter; Dos Reis, Goncalo. Path regularity and explicit convergence rate for BSDE with truncated quadratic growth. Stochastic Process. Appl. 120 (2010), no. 3, 348--379. MR2584898
  • Izumisawa, M.; Kazamaki, N. Weighted norm inequalities for martingales. Tôhoku Math. J. (2) 29 (1977), no. 1, 115--124. MR0436313
  • Kazamaki, Norihiko. Continuous exponential martingales and BMO. Lecture Notes in Mathematics, 1579. Springer-Verlag, Berlin, 1994. viii+91 pp. ISBN: 3-540-58042-5 MR1299529
  • Kunita, Hiroshi. Stochastic flows and stochastic differential equations. Reprint of the 1990 original. Cambridge Studies in Advanced Mathematics, 24. Cambridge University Press, Cambridge, 1997. xiv+346 pp. ISBN: 0-521-35050-6; 0-521-59925-3 MR1472487
  • Protter, Philip E. Stochastic integration and differential equations. Second edition. Applications of Mathematics (New York), 21. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2004. xiv+415 pp. ISBN: 3-540-00313-4 MR2020294


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