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Backward Stochastic Differential Equations and Related Control Problems

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Encyclopedia of Systems and Control

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Synonyms

BSDE

Definition

A typical real valued backward stochastic differential equation defined on a time interval [0, T] and driven by a d-dim. Brownian motion B is

$$\left \{\begin{array}{rl} dY _{t} =& - f(t,Y _{t},Z_{t})dt + Z_{t}dB_{t}, \\ Y _{T} =&\xi , \end{array} \right .$$

or its integral form

$$Y _{t} =\xi +\displaystyle\int _{t}^{T}f(s,\omega ,Y _{ s},Z_{s})ds -\displaystyle\int _{t}^{T}Z_{ s}dB_{s},$$
(1)

where ξ is a given random variable depending on the (canonical) Brownian path B t (ω) =  ω(t) on [0, T], f(t, ω, y, z) is a given function of the time t, the Brownian path ω on [0, t], and the pair of variables \((y,z) \in \mathbb{R}^{m} \times \mathbb{R}^{m\times d}\)​. A solution of this BSDE is a pair of stochastic processes (Y t , Z t ), the solution of the above equation, on [0, T] satisfying the following constraint: for each t, the value of Y t (ω), Z t (ω) depends only on the Brownian path ω on [0, t]. Notice that, because of this constraint, the extra freedom Z...

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Author information

Authors and Affiliations

  1. Shandong University, Jinan, 27 Shanda south road, 250100, Shandong Province, China

    Dr. Shige Peng

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  1. Dr. Shige Peng
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Correspondence to Dr. Shige Peng .

Editor information

Editors and Affiliations

  1. Electrical and Computer Engineering, Boston University, Boston, Massachusetts, USA

    John Baillieul

  2. Automation and Control Solutions, Honeywell, Golden Valley, Minnesota, USA

    Tariq Samad

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Peng, D. (2014). Backward Stochastic Differential Equations and Related Control Problems. In: Baillieul, J., Samad, T. (eds) Encyclopedia of Systems and Control. Springer, London. https://doi.org/10.1007/978-1-4471-5102-9_234-1

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  • DOI: https://doi.org/10.1007/978-1-4471-5102-9_234-1

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