Stochastic Control of Jump Diffusions

  • Bernt ØksendalEmail author
  • Agnès Sulem
Part of the Universitext book series (UTX)


Fix a domain \(\mathcal S\subset \mathbb {R}^k\) (our solvency region) and let \(Y(t)=Y^{(u)}(t)\) be a stochastic process of the form
$$\begin{aligned} \mathrm{d}Y(t)&= b(Y(t), u(t))\mathrm{d}t+\sigma (Y(t),u(t))\mathrm{d}B(t) \nonumber \\&\quad +\int _{\mathbb {R}^\ell } \gamma (Y(t^-), u(t^-),\zeta )\bar{N}(\mathrm{d}t,\mathrm{d}\zeta ),\quad Y(0)=y\in \mathbb {R}^k, \end{aligned}$$
$$ b:\mathbb {R}^k\times U\rightarrow \mathbb {R}^k,\quad \sigma :\mathbb {R}^\ell \times U\rightarrow \mathbb {R}^{k\times m},\quad \hbox {and}\quad \gamma :\mathbb {R}^k\times U\times \mathbb {R}^\ell \rightarrow \mathbb {R}^{k\times \ell } $$
are given functions, \(U\subset \mathbb {R}^p\) is a given set. The process \(u(t)=u(t,\omega ):[0,\infty )\times \varOmega \rightarrow U\) is our control process , assumed to be càdlàg and adapted. We call \(Y(t)=Y^{(u)}(t)\) a controlled jump diffusion .

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsloOsloNorway
  2. 2.Inria Research Center of ParisParisFrance

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