nach oben

Erschienen in:

2013 | OriginalPaper | Buchkapitel

12. Backward Stochastic Differential Equations

verfasst von : Prof. Stéphane Crépey

Erschienen in: Financial Modeling

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We saw in Chap. 4 that the problem of pricing and hedging financial derivatives can be modeled in terms of (possibly reflected) backward stochastic differential equations (BSDEs) or, equivalently in the Markovian setup, by partial integro-differential equations or variational inequalities (PIDEs or PDEs for short). Also, Chaps. 10 and 11 just provided thorough illustrations of the abilities of simulation/regression numerical schemes for solving high-dimensional pricing equations: very large systems of partial differential equations in Chap. 10 and Markov chain related systems of ODEs in Chap. 11.

Now that we experimented the power of the theory, let’s dig into it. The next few chapters provides a thorough mathematical treatment of the BSDEs and PDEs that are of fundamental importance for our approach. More precisely, Chaps. 12 to 14 develop, within a rigorous mathematical framework, the connection between backward stochastic differential equations and partial differential equations. This is done in a jump-diffusion setting with regime switching, which covers all the models considered in the book. To start with, Chap. 12 establishes the well-posedness of a Markovian reflected BSDE in a rather generic jump-diffusion model with regime switching, denoted by (X,N), which covers all the models considered in this book. In standard applications, the main component of the model, in which the payoffs of a derivative are expressed, is X. The other model component N can be used to represent a pricing regime, which may also be viewed as a degenerate form of stochastic volatility. More standard diffusive forms of stochastic volatility may also be accounted for in X. The presence of jumps in X is motivated by the empirical evidence of the short-term volatility smile in the market. In credit and counterparty risk modeling, the main model component (the one which drives the cash flows) is the Markov-chain-like-component N, representing a vector of default status and/or credit ratings of reference obligors; a jump-diffusion-like-component X can be used to represent the evolution of economic variables modulating the dynamics of N. Frailty and default contagion are accounted for by the coupled interaction between N and X.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Simulation/Regression Pricing Schemes in Pure Jump Setups

Nächstes Kapitel Analytic Approach

Typically finite or Euclidean.

See Sect. 4.1.

Or Hilbert spaces, in the cases of L ², \(\mathcal{H}^{2}_{d}\) or \(\mathcal{H}_{\mu}^{2}\).

See Definition 4.1.12.

The notation R2BSDE refers to the fact that there are two barriers involved; it has nothing to do with any notion of second-order BSDEs such as in [247].

The “D” in RDBSDE stands for “delayed”.

Boolean-valued processes.

Recalling \(\theta \in\mathcal {T}_{\vartheta }\), so that θ≥ϑ and \(\overline{U}_{\theta }=U_{\theta }\).

Simply reflected BSDEs can be treated without quasimartingale conditions; see [112].

Strictly speaking, the operator \({{\mathcal{G}}}\) is the generator of the time-extended process \((t,\mathcal{X}_{t})\).

Note that the initial time is t here instead of 0 in Sect. 12.1. Superscripts ^t are thus added, where needed, to the notation of Sect. 12.1.

Recall our notational conventions such as “g ⁱ(t,x,…)≡g(t,x,i,…)”.

A discrete sum in the case of w.

Except for slightly more general RIBSDEs to be introduced of Sect. 14.2.

See Theorem 5 on p. 25 of Brémaud [56].

See e.g. Jeanblanc et al. [159].

See (12.27) for the definition of \(|\widetilde{v} - \widetilde {v}^{\prime}|^{t}_{s}\).

Recall (12.28) and (12.68) for the definitions of \(\widetilde{g}\) and \(\widehat {g}\).

Otherwise more general but less constructive representations can be given in terms of Malliavin calculus [96, 214].

Albanese, C., Campolieti, G., Carr, P., & Lipton, A. (2001). Black–Scholes goes hypergeometric. Risk, December 2001, 99–103.

20.

Barles, G., Buckdahn, R., & Pardoux, E. (1997). Backward stochastic differential equations and integral-partial differential equations. Stochastics & Stochastics Reports, 60, 57–83. MathSciNetMATHCrossRef

25.

Becherer, D., & Schweizer, M. (2005). Classical solutions to reaction–diffusion systems for hedging with interacting Itô and point processes. The Annals of Applied Probability, 15, 1111–1144. MathSciNetMATHCrossRef

30.

Bielecki, T., Brigo, D., & Crépey, S. (2013). Counterparty risk modeling—collateralization, funding and hedging. Taylor & Francis (in preparation).

36.

Bielecki, T. R., Crépey, S., Jeanblanc, M., & Rutkowski, M. (2007). Valuation of basket credit derivatives in the credit migrations environment. In J. Birge & V. Linetsky (Eds.), Handbook of financial engineering. Amsterdam: Elsevier.

42.

Bielecki, T. R., Jakubowski, J., Vidozzi, A., & Vidozzi, L. (2008). Study of dependence for some stochastic processes. Stochastic Analysis and Applications, 26(4), 903–924. MathSciNetMATHCrossRef

48.

Boel, R., Varaiya, P., & Wong, E. (1975). Martingales on jump processes: I Representation results. SIAM Journal on Control and Optimization, 13(5), 999–1021. MathSciNetMATHCrossRef

49.

Boel, R., Varaiya, P., & Wong, E. (1975). Martingales on jump processes: II Applications. SIAM Journal on Control and Optimization, 13(5), 1022–1061. MathSciNetMATHCrossRef

56.

Brémaud, P. (1981). Point processes and queues, martingale dynamics. New York: Springer. MATHCrossRef

65.

Chassagneux, J.-F., & Crépey, S. (2012). Doubly reflected BSDEs with call protection and their approximation. ESAIM: Probability and Statistics (in revision).

80.

Crépey, S. (2013). Bilateral counterparty risk under funding constraints—Part I: Pricing. Mathematical Finance (online December 2012, doi:10.1111/mafi.12004).

81.

Crépey, S. (2013). Bilateral counterparty risk under funding constraints—Part II: CVA. Mathematical Finance (online December 2012, doi:10.1111/mafi.12005).

87.

Crépey, S., & Matoussi, A. (2008). Reflected and doubly reflected BSDEs with jumps: a priori estimates and comparison principle. The Annals of Applied Probability, 18(5), 2041–2069. MathSciNetMATHCrossRef

90.

Cvitanic, J., & Karatzas, I. (1996). Backward stochastic differential equations with reflection and Dynkin games. Annals of Probability, 24(4), 2024–2056. MathSciNetMATHCrossRef

91.

Darling, R., & Pardoux, E. (1997). Backward SDE with random terminal time and applications to semilinear elliptic PDE. Annals of Probability, 25, 1135–1159. MathSciNetMATHCrossRef

95.

Dellacherie, C., & Meyer, P.-A. (1982). Probabilities and potential. Amsterdam: North-Holland. MATH

96.

Delong, L. (2013). Backward stochastic differential equations with jumps and their actuarial and financial applications. Springer (forthcoming in the EAA Series).

106.

Dynkin, E. B. (1965). Markov processes. Berlin: Springer. MATH

112.

El Karoui, N., Kapoudjian, E., Pardoux, C., Peng, S., & Quenez, M.-C. (1997). Reflected solutions of backward SDEs, and related obstacle problems for PDEs. Annals of Probability, 25, 702–737. MathSciNetMATHCrossRef

114.

El Karoui, N., Peng, S., & Quenez, M.-C. (1997). Backward stochastic differential equations in finance. Mathematical Finance, 7, 1–71. MathSciNetMATHCrossRef

119.

Ethier, H., & Kurtz, T. (1986). Markov processes. Characterization and convergence. New York: Wiley-Interscience. MATHCrossRef

126.

Freidlin, M. (1996). Markov processes and differential equations: asymptotic problems. Basel: Birkhäuser. MATHCrossRef

139.

Hamadène, S., & Hassani, M. (2006). BSDEs with two reflecting barriers driven by a Brownian motion and an independent Poisson noise and related Dynkin game. Electronic Journal of Probability, 11, 121–145. MathSciNetCrossRef

140.

Hamadène, S., & Ouknine, Y. (2003). Reflected backward stochastic differential equation with jumps and random obstacle. Electronic Journal of Probability, 8, 1–20. MathSciNetCrossRef

149.

Ikeda, N., & Watanabe, S. (1989). Stochastic differential equations and diffusion processes (2nd ed.). Amsterdam: North-Holland. MATH

153.

Jacod, J., & Shiryaev, A. (2003). Limit theorems for stochastic processes. Berlin: Springer. MATHCrossRef

159.

Jeanblanc, M., Yor, M., & Chesney, M. (2010). Mathematical methods for financial markets. Berlin: Springer.

167.

Kunita, H. (2010). Itô’s stochastic calculus: its surprising power for applications. Stochastic Processes and Their Applications, 120, 622–652. MathSciNetMATHCrossRef

168.

Kunita, H., & Watanabe, T. (1963). Notes on transformations of Markov processes connected with multiplicative functionals. Memoirs of the Faculty of Science. Kyushu University. Series A. Mathematics, 17, 181–191. MathSciNetMATHCrossRef

170.

Kushner, H., & Dupuis, B. (1992). Numerical methods for stochastic control problems in continuous time. Berlin: Springer. MATHCrossRef

186.

Lepeltier, J.-P., & Maingueneau, M. (1984). Le jeu de Dynkin en théorie générale sans l’hypothèse de Mokobodski. Stochastics, 13, 25–44. MathSciNetMATHCrossRef

193.

Lipton, A. (2002). Assets with jumps. Risk, September 2002, 149–153.

194.

Lipton, A., & Mc Ghee, W. (2002). Universal barriers. Risk, May 2002, 81–85.

211.

Nikeghbali, A. (2006). An essay on the general theory of stochastic processes. Probability Surveys, 3, 345–412. MathSciNetMATHCrossRef

214.

Nualart, D. (1995). The Malliavin calculus and related topics. New York: Springer. MATH

217.

Palmowski, Z., & Rolski, T. (2002). A technique for exponential change of measure for Markov processes. Bernoulli, 8(6), 767–785. MathSciNetMATH

219.

Pardoux, E., Pradeilles, F., & Rao, Z. (1997). Probabilistic interpretation of systems of semilinear PDEs. Annales de l’Institut Henri Poincaré, série Probabilités–Statistiques, 33, 467–490. MathSciNetMATHCrossRef

228.

Protter, P. E. (2004). Stochastic integration and differential equations (2nd ed.). Berlin: Springer. MATH

238.

Royer, M. (1996). Backward stochastic differential equations with jumps and related nonlinear expectations. Stochastic Processes and Their Applications, 13(3), 293–317.

247.

Soner, M., Touzi, N., & Zhang, J. (2012). Wellposedness of second order backward SDEs. Probability Theory and Related Fields, 153, 149–190. MathSciNetMATHCrossRef

Titel: Backward Stochastic Differential Equations
verfasst von: Prof. Stéphane Crépey
Verlag: Springer Berlin Heidelberg
Buch: Financial Modeling
Print ISBN: 978-3-642-37112-7

Electronic ISBN: 978-3-642-37113-4

Copyright-Jahr: 2013
DOI: https://doi.org/10.1007/978-3-642-37113-4_12

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner