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On maximum principle of near-optimality for diffusions with jumps, with application to consumption-investment problem. (English) Zbl 1261.49004

Based on I. Ekeland’s variational principle [J. Math. Anal. Appl. 47, 324–353 (1974; Zbl 0286.49015)], a weak Pontryagin’s maximum principle [A. Bensoussan, “Lectures on stochastic control”, Lect. Notes Math. 972, 40–62 (1982; Zbl 0505.93079)] of near-optimality [X. Y. Zhou, SIAM J. Control Optimization 36, No. 3, 929–947 (1998; Zbl 0914.93073)] is proved for diffusions with jumps \[ \begin{cases} dx_t = f(t,x_t,u_t)dt + \sigma (t,x_t,u_t)dW_t+ \int_{\Lambda} g(t,x_{t^{-}},u_t, \theta) N(d \theta ,dt), \;\Lambda \subset \mathbb R^k\\ u_0 = \xi, \end{cases} \] where \(N(d \theta,dt)\) is a Poisson martingale measure and \((W_{t})_ {t \in [0,T]}\) is a Brownian motion. The control variable \( u= (u_t)\) takes values in a compact convex set. An application to the finance problem of consumption-investment is provided.

MSC:

49K45 Optimality conditions for problems involving randomness
49J55 Existence of optimal solutions to problems involving randomness
93E20 Optimal stochastic control
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
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