The maximum principle in optimal control of systems driven by martingale measures

Labed, Saloua; Mezerdi, Brahim

doi:10.16929/as/2017.1095.94

Cited by 2 publications

(1 citation statement)

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“…In the classical risk neutral case, and when the control set A is non-convex and the diffusion coefficient depends on the control, first and second order adjoint equations are needed to characterize the optimal control, see e.g. the classic work by Peng [59] and more recent results for relaxed controls in [9,49]. The issues resulting from the need for second order expansions are exacerbated in the risk aware setting, where the second order expansions will also require us to compute second order functional derivatives of the risk function ρ.…”

Section: Definition 22mentioning

confidence: 99%

Risk aware minimum principle for optimal control of stochastic differential equations

Isohätälä¹,

Haskell²

2018

Preprint

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We present a probabilistic formulation of risk aware optimal control problems for stochastic differential equations. Risk awareness is in our framework captured by objective functions in which the risk neutral expectation is replaced by a risk function, a nonlinear functional of random variables that account for the controller's risk preferences. We state and prove a risk aware minimum principle that is a parsimonious generalization of the well-known risk neutral, stochastic Pontryagin's minimum principle. As our main results we give necessary and also sufficient conditions for optimality of control processes taking values on probability measures defined on a given action space. We show that remarkably, going from the risk neutral to the risk aware case, the minimum principle is simply modified by the introduction of one additional real-valued stochastic process that acts as a risk adjustment factor for given cost rate and terminal cost functions. This adjustment process is explicitly given as the expectation, conditional on the filtration at the given time, of an appropriately defined functional derivative of the risk function evaluated at the random total cost. For our results we rely on the Fréchet differentiability of the risk function, and for completeness, we prove under mild assumptions the existence of Fréchet derivatives of some common risk functions. We give a simple application of the results for a portfolio allocation problem and show that the risk awareness of the objective function gives rise to a risk premium term that is characterized by the risk adjustment process described above. This suggests uses of our results in e.g. pricing of risk modeled by generic risk functions in financial applications.

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Section: Definition 22mentioning

confidence: 99%