Backward Stochastic Differential Equations
Backward Stochastic Differential Equations
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DOI：
10.1007/978-3-642-37113-4_12
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发表时间：
2013
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影响因子：
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通讯作者：
S. Crépey
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作者：
S. Crépey
我们在Chapter中看到 4 金融衍生品的定价和套期保值问题可以用（可能反映）倒向随机微分方程（BSDEs）建模，或者等价地用马尔可夫模型，用偏积分微分方程或变分不等式（PIDEs或PDEs）建模。还有，Chaps。 10 和 11 刚刚提供了模拟/回归数值方案解决高维定价方程的能力的彻底说明：在第章中的偏微分方程的非常大的系统。 10 和马尔可夫链相关的常微分方程系统。 11 既然我们已经试验了这个理论的威力，那么让我们深入研究它。接下来的几章将提供对我们的方法至关重要的倒向随机微分方程和偏微分方程的彻底的数学处理。更确切地说，第12章至 14 在一个严格的数学框架内，建立倒向随机微分方程和偏微分方程之间的联系。这是在一个跳跃扩散设置与政权切换，其中涵盖了所有的模型考虑在书中。首先，第12章在一个相当一般的带区域转换的跳跃-扩散模型（记为（X，N））中建立了马尔可夫反射边界元的适定性，该模型涵盖了本书中考虑的所有模型。在标准应用中，模型的主要组成部分是X，其中表示衍生工具的收益。另一个模型分量N可以用来表示一个定价机制，它也可以被看作是随机波动率的退化形式。随机波动率的更标准的扩散形式也可以在X中解释。X跳跃的存在是由市场短期波动微笑的经验证据驱动的。在信用和交易对手风险建模中，模型的主要成分（驱动现金流的成分）是马尔可夫链成分N，代表参考债务人的违约状态和/或信用评级的向量;跳跃扩散成分X可以用来代表调节N动态的经济变量的演变。脆弱性和违约传染是由NandX之间的耦合相互作用来解释的。
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, isX. The other model componentNcan 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 inX. The presence of jumps inXis 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-componentN, representing a vector of default status and/or credit ratings of reference obligors; a jump-diffusion-like-componentXcan be used to represent the evolution of economic variables modulating the dynamics ofN. Frailty and default contagion are accounted for by the coupled interaction betweenNandX.