## The Annals of Applied Probability

### Numerical simulation of quadratic BSDEs

#### Abstract

This article deals with the numerical approximation of Markovian backward stochastic differential equations (BSDEs) with generators of quadratic growth with respect to $z$ and bounded terminal conditions. We first study a slight modification of the classical dynamic programming equation arising from the time-discretization of BSDEs. By using a linearization argument and BMO martingales tools, we obtain a comparison theorem, a priori estimates and stability results for the solution of this scheme. Then we provide a control on the time-discretization error of order $\frac{1}{2}-\varepsilon$ for all $\varepsilon>0$. In the last part, we give a fully implementable algorithm for quadratic BSDEs based on quantization and illustrate our convergence results with numerical examples.

#### Article information

Source
Ann. Appl. Probab., Volume 26, Number 1 (2016), 262-304.

Dates
Revised: September 2014
First available in Project Euclid: 5 January 2016

https://projecteuclid.org/euclid.aoap/1452003239

Digital Object Identifier
doi:10.1214/14-AAP1090

Mathematical Reviews number (MathSciNet)
MR3449318

Zentralblatt MATH identifier
1334.60129

#### Citation

Chassagneux, Jean-François; Richou, Adrien. Numerical simulation of quadratic BSDEs. Ann. Appl. Probab. 26 (2016), no. 1, 262--304. doi:10.1214/14-AAP1090. https://projecteuclid.org/euclid.aoap/1452003239

#### References

• [1] Ankirchner, S., Imkeller, P. and Dos Reis, G. (2007). Classical and variational differentiability of BSDEs with quadratic growth. Electron. J. Probab. 12 1418–1453 (electronic).
• [2] Bally, V., Pagès, G. and Printems, J. (2005). A quantization tree method for pricing and hedging multidimensional American options. Math. Finance 15 119–168.
• [3] Barrieu, P. and El Karoui, N. (2013). Monotone stability of quadratic semimartingales with applications to unbounded general quadratic BSDEs. Ann. Probab. 41 1831–1863.
• [4] Bender, C. and Steiner, J. (2012). Least-squares Monte Carlo for backward sdes. In Numerical Methods in Finance (R. A. Carmona, P. Del Moral, P. Hu and N. Oudjane, eds.). Springer Proceedings in Mathematics 12 257–289. Springer, Berlin.
• [5] Bouchard, B. and Touzi, N. (2004). Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations. Stochastic Process. Appl. 111 175–206.
• [6] Briand, P. and Confortola, F. (2008). BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces. Stochastic Process. Appl. 118 818–838.
• [7] Briand, P. and Elie, R. (2013). A simple constructive approach to quadratic BSDEs with or without delay. Stochastic Process. Appl. 123 2921–2939.
• [8] Briand, P. and Hu, Y. (2006). BSDE with quadratic growth and unbounded terminal value. Probab. Theory Related Fields 136 604–618.
• [9] Briand, P. and Hu, Y. (2008). Quadratic BSDEs with convex generators and unbounded terminal conditions. Probab. Theory Related Fields 141 543–567.
• [10] Chassagneux, J.-F. (2014). Linear multistep schemes for BSDEs. SIAM J. Numer. Anal. 52 2815–2836.
• [11] Chassagneux, J.-F. and Crisan, D. (2014). Runge–Kutta schemes for backward stochastic differential equations. Ann. Appl. Probab. 24 679–720.
• [12] Chassagneux, J.-F. and Richou, A. (2014). Numerical stability analysis of the Euler scheme for BSDEs. SIAM J. Numer. Anal. 53 1172–1193.
• [13] Cheridito, P. and Stadje, M. (2012). Existence, minimality and approximation of solutions to BSDEs with convex drivers. Stochastic Process. Appl. 122 1540–1565.
• [14] Cheridito, P. and Stadje, M. (2013). BS$\Delta$Es and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness. Bernoulli 19 1047–1085.
• [15] Chevance, D. (1997). Numerical methods for backward stochastic differential equations. In Numerical Methods in Finance. Publ. Newton Inst. 232–244. Cambridge Univ. Press, Cambridge.
• [16] Delarue, F. and Guatteri, G. (2006). Weak existence and uniqueness for forward–backward SDEs. Stochastic Process. Appl. 116 1712–1742.
• [17] Delarue, F. and Menozzi, S. (2006). A forward–backward stochastic algorithm for quasi-linear PDEs. Ann. Appl. Probab. 16 140–184.
• [18] Delarue, F. and Menozzi, S. (2008). An interpolated stochastic algorithm for quasi-linear PDEs. Math. Comp. 77 125–158 (electronic).
• [19] Delbaen, F., Hu, Y. and Richou, A. (2013). On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions: The critical case. Preprint. Available at arXiv:1303.4859v1.
• [20] Delbaen, F., Hu, Y. and Richou, A. (2011). On the uniqueness of solutions to quadratic BSDEs with convex generators and unbounded terminal conditions. Ann. Inst. Henri Poincaré Probab. Stat. 47 559–574.
• [21] Dellacherie, C. and Meyer, P.-A. (1980). Probabilités et Potentiel. Chapitres V à VIII, Revised ed. Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics]. Théorie des martingales 1385. Hermann, Paris.
• [22] Gobet, E. and Labart, C. (2007). Error expansion for the discretization of backward stochastic differential equations. Stochastic Process. Appl. 117 803–829.
• [23] Gobet, E., Lemor, J.-P. and Warin, X. (2005). A regression-based Monte Carlo method to solve backward stochastic differential equations. Ann. Appl. Probab. 15 2172–2202.
• [24] Gobet, E. and Makhlouf, A. (2010). $L_{2}$-time regularity of BSDEs with irregular terminal functions. Stochastic Process. Appl. 120 1105–1132.
• [25] Gobet, E. and Turkedjiev, P. Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions. HAL-00642685.
• [26] Graf, S. and Luschgy, H. (2000). Foundations of Quantization for Probability Distributions. Lecture Notes in Math. 1730. Springer, Berlin.
• [27] Hu, Y., Imkeller, P. and Müller, M. (2005). Utility maximization in incomplete markets. Ann. Appl. Probab. 15 1691–1712.
• [28] Imkeller, P. and Dos Reis, G. (2010). Path regularity and explicit convergence rate for BSDE with truncated quadratic growth. Stochastic Process. Appl. 120 348–379.
• [29] Imkeller, P., Dos Reis, G. and Zhang, J. (2010). Results on numerics for FBSDE with drivers of quadratic growth. In Contemporary Quantitative Finance (A. Chiarella and C. Novikov, eds.) 159–182. Springer, Berlin.
• [30] Izumisawa, M., Sekiguchi, T. and Shiota, Y. (1979). Remark on a characterization of BMO-martingales. Tôhoku Math. J. (2) 31 281–284.
• [31] Kazamaki, N. (1979). A sufficient condition for the uniform integrability of exponential martingales. Math. Rep. Toyama Univ. 2 1–11.
• [32] Kazamaki, N. (1994). Continuous Exponential Martingales and BMO. Lecture Notes in Math. 1579. Springer, Berlin.
• [33] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Applications of Mathematics (New York) 23. Springer, Berlin.
• [34] Kobylanski, M. (2000). Backward stochastic differential equations and partial differential equations with quadratic growth. Ann. Probab. 28 558–602.
• [35] Lepeltier, J.-P. and San Martín, J. (1998). Existence for BSDE with superlinear-quadratic coefficient. Stoch. Stoch. Rep. 63 227–240.
• [36] Mania, M. and Schweizer, M. (2005). Dynamic exponential utility indifference valuation. Ann. Appl. Probab. 15 2113–2143.
• [37] Pagès, G., Pham, H. and Printems, J. (2004). Optimal quantization methods and applications to numerical problems in finance. In Handbook of Computational and Numerical Methods in Finance 253–297. Birkhäuser, Boston, MA.
• [38] Pardoux, É. and Peng, S. G. (1990). Adapted solution of a backward stochastic differential equation. Systems Control Lett. 14 55–61.
• [39] Richou, A. (2011). Numerical simulation of BSDEs with drivers of quadratic growth. Ann. Appl. Probab. 21 1933–1964.
• [40] Richou, A. (2012). Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition. Stochastic Process. Appl. 122 3173–3208.
• [41] Rouge, R. and El Karoui, N. (2000). Pricing via utility maximization and entropy. Math. Finance 10 259–276. INFORMS Applied Probability Conference (Ulm, 1999).
• [42] Tevzadze, R. (2008). Solvability of backward stochastic differential equations with quadratic growth. Stochastic Process. Appl. 118 503–515.
• [43] Zhang, J. (2004). A numerical scheme for BSDEs. Ann. Appl. Probab. 14 459–488.