Sufficient Conditions for Terminal Invariance of Stochastic Jump Diffusion Systems

Abstract

Sufficient conditions for the terminal invariance of nonlinear dynamic stochastic controlled systems (jump diffusions) are formulated and proved. The jump component has the form of an integral over a random Poisson measure. The parameters of this measure (the intensity and distribution of the values of jumps) are assumed to change over time. The conditions of invariance with respect to perturbations for a given initial state and also the conditions of absolute invariance (which ensure the constancy of a terminal criterion for any initial state) are proposed. The results are applied to a number of model examples, which include the numerical simulation and analytical study of the designed terminally invariant dynamic systems.

Appendices

Appendix

Before proceeding to the substantiation of the results obtained in this paper, present the following proposition without proof.

Proposition 1

Due to Definitions 1and 2and the notations (5)–(10), for any initial condition (t₀, x₀) ∈ B₀, any function φ(⋅) ∈ Φ, and any time instant t ∈ [t₀, t_F], the Itô formula [9, 10, 13, 18]

$$\begin{array}{ccc}d\varphi (t,x(t))=L(t,x(t),u(t,\gamma (x(t),v(t))),v(t))dt\\ +S(t,x(t),u(t,\gamma (x(t),v(t))),v(t))dw(t)\\ +\mathop{\int}\limits_{{{\mathbb{R}}}^{r}}\Gamma (t,x({t}^{-}),u(t,\gamma (x({t}^{-}),v)),v)\hat{\mu }(dt\times dv)\end{array}$$

(A.1)

or equivalently,

$$\begin{array}{cccc}\varphi (t,x(t))=\varphi ({t}_{0},{x}_{0})+\mathop{\int}\limits_{{t}_{0}}^{t}L(s,x(s),u(s,\gamma (x(s),v(s))),v(s))ds\\ +\mathop{\int}\limits_{{t}_{0}}^{t}S(s,x(s),u(s,\gamma (x(s),v(s))),v(s))dw(s)\\ +\mathop{\sum }\limits_{j=1}^{l}\mathop{\sum }\limits_{k=1}^{{\tilde{P}}_{j}({t}_{0},t)}{\Gamma }^{(j)}\left({t}_{k}^{(j)},x\left({t}_{k}^{(j)-}\right),u\left({t}_{k}^{(j)},\gamma \left(x\left({t}_{k}^{(j)-}\right),v\left({t}_{k}^{(j)}\right)\right)\right),v\left({t}_{k}^{(j)}\right)\right)\\ -\mathop{\int}\limits_{{t}_{0}}^{t}\int_{\Theta }\Gamma (s,x({s}^{-}),u(s,\gamma (x({s}^{-}),v)),v)\Pi (s,dv)ds,\end{array}$$

(A.2)

holds with probability 1 on the set D(t₀, x₀).

The rationale for Proposition 1 can be found in [10, 13]; its rigorous proof, in [18]. The following proofs are based on this proposition and, in the rest of their parts, practically repeat the ones from [7]. Therefore, they are presented here in a rather short form.

Proof of Theorem 1. Let a point (t₀, x₀) ∈ B₀ be fixed. Due to conditions (ii)–(iv) of the theorem, the Itô formula (A.2) leads to the equality

$$\varphi (t,x(t))=\varphi ({t}_{0},{x}_{0})+\mathop{\int}\limits_{{t}_{0}}^{t}\eta (s)ds,$$

holding almost surely on D(t₀, x₀). In particular, for t = t_F, condition (i) implies the relation (11). As a result, by Definition 3 the system (1) is invariant with respect to perturbations.

Proof of Theorem 2. Let a point (t₀, x₀) ∈ B₀ be arbitrary. Due to conditions (iii)–(v) of the theorem, the Itô formula (A.1) leads to the equality

$$d\varphi (t,x(t))=\frac{\eta (t)-A\varphi (t,x(t))}{{t}_{F}-t}dt,$$

holding almost surely on D(B₀) and almost everywhere on [t_S, t_F]. This equality represents an ordinary differential equation for the function φ^*(t): = φ(t, x(t)) with an arbitrary initial condition φ^*(t₀) = φ(t₀, x₀) at an arbitrary initial point t₀. This equation has the solution [7] of the form

$${\varphi }^{* }(t)={({t}_{F}-t)}^{A}\left[\frac{\varphi ({t}_{0},{x}_{0})}{{({t}_{F}-{t}_{0})}^{A}}+\mathop{\int}\limits_{{t}_{0}}^{t}\frac{\eta (s)}{{({t}_{F}-s)}^{A+1}}ds\right]$$

on the interval [t₀, t_F). It can be demonstrated (see [7, Lemma 1]) that for t = t_F, due to condition (i) of the theorem, this solution admits of the continuous extension

$$\varphi ({t}_{F},x({t}_{F}))={\varphi }^{* }({t}_{F}^{-})=\frac{\eta ({t}_{F})}{A}.$$

Hence, condition (ii) implies the relation (12), and the system (1) is absolutely invariant by Definition 4.

Funding

This work was supported in part by the Russian Foundation for Basic Research, project no. 20-08-00400.