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We study the problem of optimal inside control of an SPDE (a stochastic evolution equation) driven by a Brownian motion and a Poisson random measure. Our optimal control problem is new in two ways: (i) The controller has access to inside information, i.e. access to information about a future state of the system, (ii) The integro-differential operator of the SPDE might depend on the control. In the first part of the paper, we formulate a sufficient and a necessary maximum principle for this type of control problem, in two cases: (1) When the control is allowed to depend both on time t and on the space variable x. (2) When the control is not allowed to depend on x. In the second part of the paper, we apply the results above to the problem of optimal control of an SDE system when the inside controller has only noisy observations of the state of the system. Using results from nonlinear filtering, we transform this noisy observation SDE inside control problem into a full observation SPDE insider control problem. The results are illustrated by explicit examples.
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 functi
We deal with the problem of parameter estimation in stochastic differential equations (SDEs) in a partially observed framework. We aim to design a method working for both elliptic and hypoelliptic SDEs, the latters being characterized by degenerate d
In this paper, we introduce the concept of Developmental Partial Differential Equation (DPDE), which consists of a Partial Differential Equation (PDE) on a time-varying manifold with complete coupling between the PDE and the manifolds evolution. In o
A Dynkin game is considered for stochastic differential equations with random coefficients. We first apply Qiu and Tangs maximum principle for backward stochastic partial differential equations to generalize Krylov estimate for the distribution of a
In this article, we propose a new unifying framework for the investigation of multi-agent control problems in the mean-field setting. Our approach is based on a new definition of differential inclusions for continuity equations formulated in the Wass