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2013 | OriginalPaper | Chapter

12. Backward Stochastic Differential Equations

Author : Prof. Stéphane Crépey

Published in: Financial Modeling

Publisher: Springer Berlin Heidelberg

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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.

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previous chapter Simulation/Regression Pricing Schemes in Pure Jump Setups

next chapter 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].

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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 \(\widehat {g}\).

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Title: Backward Stochastic Differential Equations
Author: Prof. Stéphane Crépey
Publisher: Springer Berlin Heidelberg
Book: Financial Modeling
Print ISBN: 978-3-642-37112-7

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

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

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