On the Approximation of Markov Processes by Compound Poisson Processes

P.J. Fitzsimmons

My intention in this note is to present a simple approach to the approximation of a continuous time Markov process by “compound Poisson” (or “pure jump”) processes. Such approximations have been discussed recently in quite general contexts in [5] and [4]. Our setting is broader than that found in these references, and our proof more direct. A related, but much deeper, approximation of diffusion processes in Euclidean space is the subject of [6].

Let X = (X_{t}, P^{x}) be a simple Markov process with right-continuous left-limited (rcll) sample paths, state space E, transition semigroup (P_{t})_{t≥0}, and resolvent (U^{α})_{α>0}. Thus, the action of P_{t} on abounded Borel function f : E → R is given by

(1)  P_{t}f(x) = P^{x}[f(X_{t})],  t ≥ 0, x ∈ E,

where P^{x} is the law of X under the initial condition X_{0} = x; and

(2)  U^{α}f(x) := P^{x} [∫_{0}^{∞} e^{−αt}f(X_{t}) dt] = ∫_{0}^{∞} e^{−αt}P_{t}f(x) dt.

So as not to obscure the main idea, we assume for simplicity that E is homeomorphic to a Borel subset of a compact metric space, and that P_{t}f is Borel measurable whenever the bounded function f is so. From this it follows that U^{α} also preserves Borel measurability.

For each positive integer n, the kernel K_{n}(x, dy) := nU^{n}(x, dy) is a probability kernel on E; evidently K_{n}(x, ·) is the P^{x}-distribution of X_{T}, where T is independent of X and exponentially distributed with rate n. Let (Y^{n}(k) : k = 0, 1, 2, …) be a Markov chain on E with one-step transition kernel K_{n}. Let Π^{n} = (Π^{n}(t))_{t≥0} be a Poisson process of rate n, independent of Y^{n}, and define a continuous-time pure jump Markov process by

(3)  Z_{t}^{n} := Y^{n}(Π^{n}(t)),  t ≥ 0.

The construction of such processes is discussed in detail in section of [2]. (In particlar, Z^{n} is a strong Markov process.) The following result is proved in [5] for reversible Markov processes and in [4] for Hunt processes.

Theorem. If Y^{n}(0) = x, then the sequence {Z^{n}}_{n≥1} of rcll processes converges weakly to X (under P^{x}) as n → ∞.

Proof. Fix x ∈ E; in the following discussion, the law P^{x} will govern X. Let Λ^{n} = (Λ^{n}(t))_{t≥0} be a Poisson process of rate n, independent of X, and let

(4)  0 = S_{0}^{n} < S_{1}^{n} < S_{2}^{n} < ⋯ < S_{k}^{n} < ⋯

denote the jump times of Λ^{n}. Define

(5)  X_{t}^{n} = X_{S_{k}^{n}} for S_{k}^{n} ≤ t < S_{k+1}^{n} (k = 0, 1, 2, …),

and notice that

(6)  Y^{n}(k) := X_{S_{k}^{n}}^{n}  k = 0, 1, 2, …,

is a realization of the Markov chain described before the statement of the Theorem. We can (and do) assume that the rate n Poisson process Π^{n} of the theorem is independent of X and of Λ^{n}. Let

(7)  0 = T_{0}^{n} < T_{1}^{n} < T_{2}^{n} < ⋯ < T_{k}^{n} < ⋯

denote the jump times of Π^{n}. Finally, let A^{n}(t) be the increasing piece-wise linear process that interpolates the points (T_{k}^{n}, S_{k}^{n}), k = 0, 1, 2, . . .. From (3) and (6) we deduce that Z_{t}^{n} admits the representation

(8)  Z_{t}^{n} = X_{A^{n}(t)}^{n}.

The convergence in distribution of Z^{n} to X will follow from two observations:

(i) Because the “points” of Λ^{n} become dense (in probability) as n → ∞ and because X has rcll paths, X^{n} converges in probability to X, the convergence being uniform on compact time intervals. That is,

(9)  lim_{n} P^{x} [ sup_{0≤s≤t} d(X_{s}, X_{s}^{n}) > ϵ] = 0  ∀ϵ > 0.

(In fact, if we take (as we may) Λ^{n} to be Λ^{(1)} + Λ^{(2)} + ⋯ + Λ^{(n)} where the {Λ^{(k)}}_{k≥1} is a sequence of independent unit-rate Poisson processes, then sup_{0≤s≤t} d(X_{s}, X_{s}^{n}) → 0 almost surely as n → ∞.)

(ii) A^{n}(t) converges to t in probability as n → ∞, and again the convergence is uniform on compacts:

(10)  lim_{n} P^{x} [ sup_{0≤s≤t} |A^{n}(s) − s| > ϵ] = 0  ∀ϵ > 0.

To see this notice that the related process

(11)  B^{n}(t) := S_{k}^{n} for T_{k}^{n} ≤ t < T_{k+1}^{n} (k = 0, 1, 2, …)

is a subordinator [1] with Lévy exponent

(12)  φ_{n}(λ) = −\frac{1}{t} log P^{x}(exp(−λB^{n}(t))) = n ∫_{0}^{∞}(1 − e^{−λz})ne^{−nz} dz = λ\frac{n}{n + λ}.

Thus, for fixed t > 0, B^{n}(t) converges to t in probability as n → ∞. t. The monotonicity of t ↦ B^{n}(t) and the continuity of its limit allow us to conclude that the convergence is uniform on compacts (in probability). The asserted convergence of A^{n} now follows by a simple bracketing argument.

Feeding (i) and (ii) into the representation (8) we see that, for each t > 0,

lim_{n} P^{x}(d_{[0,t]}(Z^{n}, X) > ϵ) = 0 ∀ϵ > 0,

where d_{[0,t]} is the Skorokhod distance in the path space D([0, t], E); see [3]. In particular, Z^{n} converges in distribution (in the Skorokhod topology) to X.

References

[1] Bertoin, J.: Lévy Processes. Cambridge University Press, Cambridge, 1996

[2] Blumenthal, R.M. and Getoor, R.K.: Markov Processes and Potential Theory. Academic Press, New York-London, 1968.

[3] Ethier, S.N. and Kurtz, T.G.: Markov processes. Characterization and convergence. Wiley, New York, 1986.

[4] Ma, Z.-M., Röckner, M., and Sun, W.: Approximation of Hunt process by multivariate Poisson processes. Preprint, 1999.

[5] Ma, Z.-M., Röckner, M., and Zhang, T-S.: Approximation of arbitrary Dirichlet processes by Markov chains. Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 1–22.

[6] Stroock, D. and Zheng, W.A.: Markov chain approximations to symmetric diffusions. Ann. Inst. H. Poincaré Probab. Statist. 33 (1997), 619–649.