Approximation of General Stochastic Hybrid Systems by Switching Diffusions with Random Hybrid Jumps

In this work we propose an approximation scheme to transform a general stochastic hybrid system (SHS) into a SHS without forced transitions due to spatial guards. Such switching mechanisms are replaced by spontaneous transitions with state-dependent transition intensities (jump rates). The resulting switching diffusion process with random hybrid jumps is shown to converge in distribution to the original stochastic hybrid system execution. The obtained approximation can be useful for various purposes such as, on the computational side, simulation and reachability analysis, as well as for the theoretical investigation of the model. More generally, it is suggested that SHS which are endowed exclusively with random jumping events are simpler than those that present spatial forcing transitions. In the opening of this work, the general SHS model is presented, a few of its basic properties are discussed, and the concept of generator is introduced. The second part of the paper describes the approximation procedure, introduces the new SHS model, and proves, under some assumptions, its weak convergence to the original system. We describe the general stochastic hybrid system model introduced in [1]. Definition 1 (General Stochastic Hybrid System). A General Stochastic Hybrid System (GSHS) is a collection Sg = (Q, n,A,B, Γ,R , Λ,R, π), where – Q = {q1, q2, . . . , qm},m ∈ N, is a countable set of discrete modes; – n : Q → N is a map such that, for q ∈ Q, the continuous state space is the Euclidean space R. The hybrid state space is then S = ∪q∈Q{q} ×R; – A = {a(q, ·) : R → R, q ∈ Q} is a collection of drift terms; – B = {b(q, ·) : R → Rn(q)×n(q), q ∈ Q} is a collection of diffusion terms; – Γ = ∪q∈Q{q}×Γq ⊂ S, where Γq = ∪q′ 6=q∈Qγqq′ is a closed set composed of m− 1 disjoint guard sets γqq′ causing forced transitions from q to q′ 6= q; – R : B(Rn(·)) × Q × Γ → [0, 1] is the reset stochastic kernel associated with Γ . Specifically, R (·|q′, (q, x)) is a probability measure concentrated on Rn(q) \ Γq′ , which describes the probabilistic reset of the continuous state when a jump from mode q to q′ occurs from x ∈ γqq′ ; – Λ : S\Γ×Q → R is the transition intensity function governing spontaneous transitions. Specifically, for any q 6= q′ ∈ Q, λqq′(x) := Λ((q, x), q′) is the jump rate from mode q to mode q′ when x ∈ R \ Γq; – R : B(Rn(·)) × Q × S \ Γ → [0, 1] is the reset stochastic kernel associated with Λ. In particular, RΛ(·|q′, (q, x)) is a probability measure concentrated on Rn(q) \ Γq′ that describes the probabilistic reset of the continuous state when a jump from mode q to q′ occurs from x ∈ R \ Γq; – π : B(S)→ [0, 1] is a measure on S \Γ describing the initial distribution. u t In the definition above, B(S) denotes the σ-field on S. Assumption 1 (on the system dynamics) 1. The drift and diffusion terms a(q, ·) and b(q, ·), q ∈ Q, are bounded and uniformly Lipschitz continuous. 2. The jump rate function Λ : S\Γ×Q → R satisfies the following conditions: – it is measurable and bounded; – for any q, q′ ∈ Q, q 6= q′, and any sample path ω(t), t ≥ 0, of the process solving the SDE in q, initialized at x ∈ R \ Γq, there exists qq′(x) > 0 such that λqq′(ω(t)) is integrable over [0, qq′(x)). 3. For all C ∈ B(Rn(·)), R (C|·) and R(C ′|·) are measurable. 4. For any execution associated with π = δs, s ∈ S \Γ , the expected value of the number of jumps within the time interval [0, t] is bounded for all t ≥ 0. u t Intuitively, Assumption 1.1 guarantees the existence and uniqueness of the n(q)-dimensional solution to the SDE associated with q ∈ Q, dv(t) = a(q,v(t))dt+ b(q,v(t))dwq(t), where wq is a n(q)-dimensional standard Wiener process. The semantic definition of the GSHS Sg, given via the notion of execution (a stochastic process {s(t) = (q(t),x(t)), t ≥ 0}, with values in S, solution of Sg), can be done as in [1]. Note that a sample-path of a GSHS execution is a right-continuous, S-valued function on [0,∞), with left-limits on (0,∞) (càdlàg). Furthermore, the following property holds. Proposition 1. Consider a GSHS Sg. Under assumptions 1.1-1.2-1.3-1.4, the execution s(t), t ≥ 0, of Sg is a càdlàg strong Markov process. u t It is interesting to associate to the set of real-valued functions f , acting on Markov processes defined on a Borel space, a strong generator L, and a weaker, yet more general, extended generator [2]. Denote with C b (S) the class of real-valued, twice continuously differentiable and bounded functions on S. Let ∂f(q,x) ∂x a(q, x) = ∑n(q) i=1 ∂f(q,x) ∂xi ai(q, x) be the Lie derivative of f(q, ·) along a(q, ·), and Hf (q, x) = [∂2f(q,x) ∂xi∂xj ] i,j=1,2,...,n(q) be the Hessian of f(q, ·). Proposition 2 (Extended Generator of Sg). The extended generator Lg : D(Lg)→ Bb(S) associated with the executions of Sg is, for s = (q, x) ∈ S \ Γ : Lgf(s) = Lgf(s) + IS\Γ (s) ∑ q′∈Q,q′ 6=q λqq′(x) ∫ Rn(q) ( f((q′, z))− f(s) ) RΛ(dz|q′, s), where Lgf(s) = ∑ q∈Q ∂f(q,x) ∂x a(q, x) + 1 2Tr ( b(q, x)b(q, x)Hf (q, x) ) . The domain D(Lg) of Lg is the set of functions f ∈ C b (S) satisfying the condition: f(s) = ∑ q′∈Q,q′ 6=q ∫ Rn(q) f((q ′, z))R (dz|q′, s), s ∈ Γ . Consider the GSHS system Sg in Definition 1. The guard set of Sg within mode q ∈ Q is made up of γqq′ ⊂ R, q′ ∈ Q, q′ 6= q. Assume that each set γqq′ can be expressed as a zero sub-level set of a continuous function hqq′ : R → R: γqq′ = {x ∈ R : hqq′(x) ≤ 0}. Pick a small enough δ > 0, and by the continuity of hqq′ , introduce the sets γ−δ qq′ = {x ∈ R n(q) : hqq′(x) ≤ −δ} ⊆ γqq′ ⊆ γ qq′ = {x ∈ R : hqq′(x) ≤ δ}. For any q ∈ Q, define the set of functions λqq′ : R → R, q′ ∈ Q, q′ 6= q, λqq′(x) =   1 d(x, γ−δ qq′) − 1 sup y:hqq′ (y)=δ d(y, γ−δ qq′) ∧  1 sup y:hqq′ (y)=0 d(y, γ−δ qq′)  , x ∈ γ qq′