Controlled diffusion processes

By Vivek Shripad Borkar. 2005

649 citations recorded by the source corpus.

This article gives an overview of the developments in controlled diffusion processes, emphasizing key results regarding existence of optimal controls and their characterization via dynamic programming for a variety of cost criteria and structural assumptions. Stochastic maximum principle and control under partial observations (equivalently, control of nonlinear filters) are also discussed. Several other related topics are briefly sketched.

Subfields: Computational Theory and Mathematics, Finance

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Citations in the corpus

A stochastic differential reinsurance game (2010, background): “Then, with other essential assumptions, G(x; π2) satisfies the following Hamilton–Jacobi–Bellman (HJB) equation (see [5] or [7]):”
Ergodic Control of Diffusion Processes (2012, background): “For a merely measurable diffusion matrix, even in the non-degenerate case only the existence of a weak solution is available, the uniqueness may not hold [26] (except in one and two dimensions – see [35], pp.”
Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations (2016, background): “4 in [19]) as follows: there exists a constant N depending on the dimension d and on R through the ellipticity constants of the diffusion matrix and the sup-norm of the drift on the set BR ×BR, where BR = {x ∈ R : |x| ≤ R}, so that E1(γn,R > T ) ∫ T”
Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions (2021, positive): “Following [31], we also define the class of step control processes for the problem starting at t ∈ [0, T ]: A step := { α ∈ A : there exist n ∈ N and t = t0 < t1 < · · · < tn−1 < tn = T such that αs = αti , ∀ s ∈ [ti, ti+1), i = 0, .”
A “maximum principle for semicontinuous functions” applicable to integro-partial differential equations (2006, positive): “, [15, 16, 28, 30]): There are constants K1,Kx ≥ 0, a function ρ ≥ 0, and a modulus of continuity ω, such that the following statements hold for every x, y ∈ R , α ∈ A, β ∈ B and z ∈ R \ {0}:”
CONTINUOUS DEPENDENCE ESTIMATES FOR VISCOSITY SOLUTIONS OF INTEGRO-PDES (2005, positive): “(A2’) These assumptions have been used in Krylov [37] (but see Remark 4.”
Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions (2007, neutral): “The coefficients a β ij can be computed from the coefficients of the stochastic differential equation (see [26] [40] for more details).”
Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients (2014, background): “6) and generalized Itô’s formula (see [19], page 122, Theorem 1), we have (4.”
On some recent aspects of stochastic control and their applications (2005, background): “We mention among them [56], [68], [61], [15], [41] or [89].”
On Ergodic Measures for McKean-Vlasov Stochastic Equations (2006, methodology): “using Skorokhod’s technique (see [17] or [11]), quite similarly to the passage from (13) to (14) indeed.”
Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation (2006, background): “The existence and uniqueness of (1) in the sense of viscosity solution can be found in, for example, [12], [18], [32], [46], and [27] for C(1,2)-solution if σ0 > 0 (see also in [30] for elliptic cases).”