Invariance of stochastic control systems with deterministic arguments

Abstract

We prove that a closed set K of a finite-dimensional space is invariant under the stochastic control system dX ¼ bðX ; vðtÞÞ dt þ sðX ; vðtÞÞ dW ðtÞ; vðtÞAU;if and only if it is invariant under the deterministic control system with two controlsDs j ðx; vðtÞÞs j ðx; vðtÞÞ þ sðx; vðtÞÞuðtÞ; uðtÞAH 1 ; vðtÞAU:This extends the well-known result of stochastic differential equations to stochastic control systems. Furthermore, we ask only C 1;1 regularity of the diffusion s instead of the usual assumption sAC 2 :In this way our result is new even for stochastic differential equations. The arguments of the proof are based on estimates between solutions of the stochastic control system with time independent controls and families of solutions fx o ðÁÞg oAO to the deterministic control system x 0 ¼ sðx; v o Þu o ðtÞ; u o ðtÞAH 1 with appropriately chosen controls u o ðtÞ and v o AU: r

Invariant measures associated to degenerate elliptic operators

“…Another approach to the problem was proposed in [7], using viscosity solutions of a second order Hamilton-Jacobi equation. Later on, in [5,6], second order jets were used to study invariance and viability, while in [12] the second and third author applied the Stratonovich drift to give first order necessary and sufficient conditions for the invariance of an arbitrary closed set for a stochastic control system. Then, using the distance function, in [10,14] a condition similar to (1.2) was shown to be necessary and sufficient for the invariance of closed convex sets, while in [11] a sufficient condition for the invariance of the interior was derived.…”

Almost sure properties of controlled diffusions and worst case properties of deterministic systems

“…Therefore the invariance theory of [18] does not give any sufficient condition of viability in this case.…”

“…Different from the invariance studied in [18], both these properties involve a conflict of the controller against the disturbance, and they are satisfied if the controller can win it. For the deterministic system (DS) this is modelled as a two-person zero-sum differential game.…”

“…In the recent paper [18] Da Prato and Frankowska showed that a closed set K is invariant for (CSDE) (i.e., it is almost surely invariant for all stochastic controls α t , that is, functions in A progressively measurable with respect to B t ) if and only if it is invariant for (DS) (i.e., for all α…”

Dynamical Allocation Method of Emission Rights of Pollutants by Viability Constraints Under Tychastic Uncertainty

“…Regarding viability and capturability issues, the stochastic case is a (too) special particular case of tychastic viability (see [33] and [15]). A characterization of stochastic viability and invariance in terms of stochastic tangent cones has been carried over in [11][12][13], and, in terms of distance functions, in [27][28][29] among many other studies in this direction.…”