Weak convergence for a type of conditional expectation: application to the inference for a class ofasset price models

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1 Nonlinear Analysis 6 (25) Weak convergence for a type of conditional expectation: application to the inference for a class ofasset price models Michael A. Kouritzin a, Yong Zeng b, a Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Al, Canada T6G 2G1 b Department of Mathematics and Statistics, University of Missouri at Kansas City, Kansas City, MO 6411, USA Received 12 March 24; accepted 1 July 24 Abstract We prove weak convergence ofa type ofconditional expectation, which provides a straightforward proofofgoggin s Theorem and further proves the consistency of(integrated) likelihood, posterior, and Bayes factor for a class of transactional asset price models recently developed. 24 Elsevier Ltd. All rights reserved. Keywords: Conditional expectation; Filtering; Counting process 1. Introduction The weak convergence of (X N,Y N ) on a probability space (Ω N, F N,P N ) to (X, Y ) on (Ω, F,P) does not necessarily imply that E P N [F(X N ) Y N E[F(X) Y, even for bounded, continuous functions, F. However, Goggin [2 showed that the limit ofthe conditional expectations can be determined ifit is possible to make an absolutely continuous change ofprobability measure, from P N to Q N with L N (X N,Y N ) = dp N /dq N being the Radon Nikodym derivative, so that X N and Y N are independent under Q N. Goggin s main condition guaranteeing the weak convergence ofthe conditional expectation is We thank Thomas G. Kurtz for a productive conversation. Corresponding author. Tel.: addresses: mkouritz@math.ualberta.ca (M.A. Kouritzin), zeng@mendota.umkc.edu (Y. Zeng) X/$ - see front matter 24 Elsevier Ltd. All rights reserved. doi:1.116/j.na

2 232 M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) that (X N,Y N,L N (X N,Y N )) under Q N converges in distribution to (X, Y, L(X, Y )) under Q. Under the same conditions ofgoggin [2, we provide a short proofofthe weak convergence ofthis type ofconditional expectation: E QN [F(X N )L N (X N,Y N ) Y N E Q [F (X)L(X, Y ) Y as N, establishing Goggin s theorem. We note that Crimaldi and Pratelli in [7 at about the same time independently proved a similar result. We note that E Q [F (X)L(X, Y ) Y for suitably chosen F solves the Duncan Mortensen Zakai (unnormalized filtering) equation in the classical filtering problem. Hence, weak convergence ofthis type ofconditional expectation can be used in proving the consistency ofsome numerical algorithms for computing Duncan Mortensen Zakai equation. However, this paper focuses on another important application to the statistical inference for a class of partially observed models ofasset price for transaction data recently proposed in [6. The main appeal ofthe class ofmodels is that they are framed as a filtering problem with counting process observations. Then, E Q [F (X)L(X, Y ) Y relates to the continuous-time likelihood (or integrated likelihood in Bayesian statistics) ofthe model, which is specified by the unnormalized filtering equation. The Markov chain approximation method can be applied to develop recursive algorithms based on the unnormalized filtering equation to compute the continuous-time likelihood. Therefore, weak convergence ofthis type ofconditional expectation is useful in proving the consistency of the recursive algorithms for likelihoods. Furthermore, together with the continuous mapping theorem, weak convergence ofthis type ofconditional expectation can also prove the consistency ofthe recursive algorithms for computing posterior and Bayes factor, which is the ratio ofthe integrated likelihoods ofthe two models considered in model selection. Section 2 presents and proves the main theorem with two corollaries. The first corollary is Goggin s theorem and is related to the consistency ofposterior. The second relates to the consistency of Bayes factor. It should be noted that the results in Section 2 are fairly general, because they are for the random variables in the complete, separable metric spaces. To present an application, Section 3 first reviews the class ofmodels proposed in [6, then proves the consistencies ofthe likelihoods, posterior and Bayes factor. 2. The main result Theorem 1. Suppose that S 1 and S 2 are complete, separable metric spaces and that (X N, Y N ), N = 1, 2,..., and (X, Y ) are S 1 S 2 -valued random variables defined on the probability spaces (Ω N, F N,P N ) and (Ω, F,P), respectively. Suppose that {(X N,Y N )} converges in distribution to (X, Y ), that P N >Q N on σ(x N,Y N ) with dp N /dq N = L N (X N,Y N ), and that Q N (Q) is a probability measure on F N (F) such that X N, Y N (X, Y ) are independent under Q N (Q). Suppose that the Q N -distribution of (X N,Y N,L N (X N,Y N )) converges weakly to the Q-distribution of (X, Y, L(X, Y )), where E Q [L(X, Y )=1. Then, the following hold: (i) P >Q on σ(x, Y ) and dp/dq = L(X, Y ); (ii) for every bounded continuous function F : S 1 R, E QN [F(X N )L N (X N,Y N ) Y N converges weakly to E Q [F (X)L(X, Y ) Y as N.

3 M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) Proof. Part (i) is from part (i) of Theorem 2.1 in [2. The following is the proof for part (ii). Pick ε >. Then, as in [2, there exists a bounded, continuous function L ε, w.l.o.g. strictly positive, such that E Q [ L(X, Y ) L ε (X, Y ) < ε. Let Z N = L N (X N,Y N ) and Z = L(X, Y ). Using Skorohod s representation, we can find a probability space ( ˆΩ, ˆF, ˆQ) where ( ˆX N, Ŷ N, Ẑ N ) and ( ˆX, Ŷ,Ẑ) are defined with the following three properties: for each N, ( ˆX N, Ŷ N, Ẑ N ) has the same distribution as (X N,Y N,L N (X N,Y N )) under Q N ; ( ˆX, Ŷ,Ẑ)has the same distribution as (X, Y, L(X, Y )) under Q; and ( ˆX N, Ŷ N, Ẑ N ) ( ˆX, Ŷ,Ẑ) ˆQ-a.s. Suppose F C. We observe that [ [ E ˆQ E ˆQ F( ˆX [ N )Ẑ N Ŷ N E ˆQ F( ˆX)Ẑ Ŷ CE ˆQ [ Ẑ L ε ( ˆX, Ŷ) + CE ˆQ [ Ẑ N L ε ( ˆX N, Ŷ N ) + E ˆQ [ E ˆQ [ F( ˆX N )L ε ( ˆX N, Ŷ N ) Ŷ N E ˆQ [ F( ˆX)L ε ( ˆX, Ŷ) Ŷ. (1) Now, it suffices to prove the right-hand side of the above inequality can be arbitrary small for large enough N. The first term is less than Cε. For the second term, we observe [ E ˆQ L ε ( ˆX [ N, Ŷ N ) Ẑ N E ˆQ L ε ( ˆX N, Ŷ N ) L ε ( ˆX, [ Ŷ) + E ˆQ L ε ( ˆX, [ Ŷ) Ẑ + E ˆQ Ẑ Ẑ N. Since ( ˆX N, Ŷ N ) ( ˆX, Ŷ) ( ˆQ-a.s., and L ε is bounded and continuous, the first expectation is less than ε for large N. The second expectation is less than ε by the assumption on L ε. Since Ẑ N Ẑ ˆQ-a.s., and Ẑ N d ˆQ Ẑ d ˆQ (since all the integrals are identically 1), the last expectation is less than ε for large N. Therefore, the second term of(1) is less than 3Cε. Noting that both F and L ε are bounded and continuous real variables, one finds that the mapping: (y, μ) F(x)L ε (x, y)μ(dx) is continuous. Therefore, noting that (Ŷ N, μ N ) (Ŷ,μ), where μ N = ˆQ 1 ( ˆX N ) and μ = ˆQ 1 ( ˆX), we obtain that the third term of(1) is less than ε for large N. Theorem 1 relates to the consistency oflikelihood in the application ofsection 3. The following corollary is Goggin Theorem and relates to the consistency of posterior in the application ofsection 3. We use the notation, X N X (X ε X), to mean X N (X ε ) converges weakly to X in the Skorohod topology as N (ε ). Corollary 1. Under the same conditions of Theorem 1 and the assumption that Q{E Q [L (X, Y ) Y =} =, for every bounded continuous function F : S 1 R, we have: E P N [F(X N ) Y N converges weakly to E P [F(X) Y as N. Proof. Bayes theorem, Theorem 1 and continuous mapping theorem imply that E P N [F(X N ) Y N = EQN [F(X N )L N (X N,Y N ) Y N E QN [L N (X N,Y N ) Y N = E P [F(X) Y. EQ [F (X)L(X, Y ) Y E Q [L(X, Y ) Y

4 234 M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) Corollary 2 relates to the consistency ofbayes factor in the application ofsection 3. Corollary 2. Suppose that for k=1, 2, S 1 and S 2 are complete, separable metric spaces, and that (X N k,yn k ) ((X k,y k )) on (Ω N k, FN k,pn k ) ((Ω k, F k,p k )) satisfies the same conditions of Theorem 1. Suppose that Q k {E Q k[l k (X k,y k ) Y k =}= for k = 1, 2. Define and q N 1 (F 1) = EQN 1 [F 1 (X N 1 )LN 1 (XN 1,YN 1 ) Y N 1 E QN 2 [L N 2 (XN 2,YN 2 ) Y N 2 q N 2 (F 2) = EQN 2 [F 2 (X N 2 )LN 2 (XN 2,YN 2 ) Y N 2 E QN 1 [L N 1 (XN 1,YN 1 ) Y N 1. Define q 1 (F 1 ) and q 2 (F 2 ) similarly. Then, for every bounded continuous function F k : S 1 R, (q1 N (F 1), q2 N (F 2)) converges weakly to (q 1 (F 1 ), q 2 (F 2 )). Proof. By the proofoftheorem 1, ( [ E QN 1 F1 (X1 N )LN 1 (XN 1,YN 1 ) Y 1 N,E Q N [ 2 F2 (X2 N )LN 2 (X2 N,YN 2 ) Y 2 N ) ( [ [ ) E Q 1 F1 (X 1 )L 1 (X 1,Y 1 ) Y 1,E Q 2 F2 (X 2 )L 2 (X 2,Y 2 ) Y 2, and the continuous mapping theorem implies that [ E QN 1 [F 1 (X q1 N 1 N )LN 1 (XN 1,Y 1 N ) Y 1 N (F E 1) q2 N (F = E QN 2 [L N 2 2) (XN 2,Y 2 N ) Y 2 N Q 1 [F 1 (X 1 )L 1 (X 1,Y 1 ) Y 1 E Q 2 [L 2 (X 2,Y 2 ) Y 2 = E QN 2 [F 2 (X N 2 )LN 2 (XN 2,Y N 2 ) Y N 2 E QN 1 [L N 1 (XN 1,Y N 1 ) Y N 1 [ q1 (F 1 ). q 2 (F 2 ) E Q 2 [F 2 (X 2 )L 2 (X 2,Y 2 ) Y 2 E Q 1 [L 1 (X 1,Y 1 ) Y 1 3. Application to a class of asset price models In this section, we first review the class ofpartially-observed models for transactional asset price in [6 and their continuous-time likelihoods, posterior and Bayes factors. Then, we apply the theorem and the two corollaries in Section 2 to prove the consistency ofthe likelihoods, posterior and Bayes factors The class of models In trade-by-trade level, the asset price does not move continuously as the common asset price models such as geometric Brownian motion or jump-diffusion processes suggest, but move level-by-level due to price discreteness. When we view the prices according to price level, we model the prices ofan asset as a collection ofcounting processes in the

5 following form: M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) N 1 ( λ 1(θ(s), X(s), s) ds) N 2 ( Y(t)= λ 2(θ(s), X(s), s) ds). N n (, (2) t λ n(θ(s), X(s), s) ds) where Y j (t)=n j ( λ j (θ(s), X(s), s) ds) is the counting process recording the cumulative number oftrades that have occurred at the jth price level (denoted by y j ) up to time t. Moreover, θ(t) is a vector ofparameters in the model and X(t) is the intrinsic value process, which cannot be observed directly, but can be partially observed through Y. Suppose that P is the probability measure of (θ,x, Y). We invoke five mild assumptions on the model. Assumption 1. {N j } n j=1 are unit Poisson processes under measure P. Assumption 2. (θ, X), N 1,N 2,...,N n are independent under measure P. Assumption 3. The total intensity process, a(θ,x,t), is uniformly bounded above, namely, there exists a positive constant, C, such that <a(θ,x,t) C for all t> and all (θ,x). Remark 1. Assumption 3 is for technical reasons. These three assumptions imply that there exists a reference measure Q and that after a suitable change of measure to Q, (θ, X), Y 1,...,Y n become independent, and Y 1,Y 2,...,Y n become unit Poisson processes. Also, the independence in Assumption 2 implies the probability to have any simultaneous jumps is zero. Assumption 4. The intensity, λ j (θ,x,t) = a(θ,x,t)p(y j x), where a(θ,x,t) is the total trading intensity at time t and p(y j x) is the transition probability from x to y j, the jth price level. Remark 2. This assumption imposes a desirable structure for the intensities of the model. It means that the total trading intensity a(θ(t), X(t), t) determines the overall rate oftrade occurrence at time t and p(y j x) determines the proportional intensity oftrade at the price level, y j, when the value is x. Note that p(y j x) models how the trading noise enters the price process. Assumption 5. (θ,x)is the unique cadlag solution ofa martingale problem for a generator A such that M f (t) = f(θ(t), X(t)) Af(θ(s), X(s)) ds is a F θ,x t -martingale for f D(A), where F θ,x t X(s)) s t. is the σ-algebra generated by (θ(s),

6 236 M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) Remark 3. The martingale problem and the generator approach (see [1) provide a powerful tool for the characterization of Markov processes. Assumption 5 includes all relevant stochastic processes such as diffusion and jump-diffusion processes. Remark 4. Under this representation, (θ(t), X(t)) becomes the signal process, which cannot be observed directly, but can be partially observed through the counting processes, Y(t), corrupted by trading noise, which is modeled by p(y j x) (see [6 for more about trading noise). Hence, (θ,x, Y) is framed as a filtering problem with counting process observations Foundations of statistical inference The continuous-time joint likelihood The probability measure P of (θ,x, Y) can be written as P = P θ,x P y θ,x, where P θ,x is the probability measure for (θ,x) such that M f (t) in Assumption 5 is a F θ,x t - martingale, and P y θ,x is the conditional probability measure for Y given (θ,x). Under P, Y depends on (θ,x). Recall from Remark 1 that there exists a reference measure Q such that under Q, (θ,x)and Y become independent, and Y 1,Y 2,...,Y n become unit Poisson processes. Therefore, Q can be decomposed as Q = P θ,x Q y, where Q y is the product probability measure for n independent unit Poisson processes. Then, the Radon Nikodym derivative ofthe model, L(t), that is the continuous-time joint likelihood of (θ,x, Y), is, L(t) = dp dq (t) = dp θ,x (t) dp y θ,x (t) = dp y θ,x (t) dp θ,x dq y dq y n { = exp log λ k (θ(s ), X(s ), s ) dy k (s) k=1 [λ k (θ(s), X(s), s) 1 ds }. (3) The continuous-time likelihoods of Y However, X cannot be observed. To obtain the integrated likelihood ofthe model, which is the marginal likelihood of Y, we may integrate L(t) on (θ,x), or equivalently, write the integrated likelihood in terms ofconditional expectation. Let F Y t = σ{( Y(s)) s t} be the available information up to time t. Definition 1. Let (f, t) = E Q [f(θ(t), X(t))L(t) F Y t. If (θ(), X()) is fixed, then the likelihood of Y is E Q [L(t) F Y = (1,t). Ifa prior is assumed on (θ(), X()), then the integrated (or marginal) likelihood of Y is also (1,t).

7 The continuous-time posterior M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) Definition 2. Let π t be the conditional distribution of (θ(t), X(t)) given F Y t. Definition 3. π(f, t) = E P [f(θ(t), X(t)) F Y t = f(θ,x)π t (dθ, dx). Ifa prior is assumed on (θ(), X()), then π t becomes the continuous-time posterior, which is determined by π(f, t) for all continuous and bounded f Continuous-time Bayes factors For the class ofmicromovement models, we suppose that Model k is denoted by (θ,x, Y ) for k =1, 2. Denote the joint likelihood of (θ,x, Y ) by L (t), which is given by Eq. (3). Denote k (f k,t)= E Q [f k (θ (t), X (t))l Y (t) F t. Then, the integrated likelihood of Y is k (1,t), for Model k. In general, the Bayes factor of Model 2 over Model 1, B 21, is defined as the ratio of integrated likelihoods ofmodel 2 over Model 1, i.e. B 21 (t) = 2 (1,t)/ 1 (1,t). Suppose B 21 has been calculated. Then, we can interpret it using the table furnished by Kass and Raftery [3 (a survey paper ofbayes factor) as guideline. Similarly, we can define B 12. Obviously, B 12 B 21 = 1. Definition 4. Define the filter ratio processes: q 1 (f 1,t)= 1 (f 1,t) 2 (1,t), and q 2(f 2,t)= 2 (f 2,t) 1 (1,t). Remark 5. Observe that the Bayes factors, B 12 (t) = q 1 (1,t)and B 21 (t) = q 2 (1,t). In summary, (f, t), which determines the likelihood or integrated likelihood, is characterized by the unnormalized filtering equation. π(f, t), which determines the posterior, is characterized by the normalized filtering equation, and is the optimum filter in the sense of least mean square error. q k (f k,t), k = 1, 2, which determines the Bayes factors, is characterized by the system ofevolution equations. The filtering equations for (f, t) and π(f, t) are derived in [6 and the system ofevolution equations for q k (f k,t), k = 1, 2, is derived in [ Consistency of the likelihoods, posterior and Bayes factors To compute them for statistical inference, one constructs recursive algorithms to approximate them. Zeng [6,4 applied Markov chain approximation methods to construct recursive algorithms for computing the posterior and Bayes estimates (the Bayes factors). One basic requirement for the recursive algorithms is consistency: The approximate versions (i.e. the likelihoods, posterior and Bayes factors), computed by the recursive algorithms, must converge to the true ones. The following theorem proved by Theorem 1 and Corollaries 1 and 2 provides the theoretical foundation for consistency.

8 238 M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) Let (θ ε Y,X ε ) be an approximation of (θ,x ). Then, we define N 1 ( λ 1(θ ε (s), X ε (s), s) ds) N 2 ( ε (t) = λ 2(θ ε (s), X ε (s), s) ds). N n (, (4) t λ n(θ ε (s), X ε (s), s) ds) Y set F (( ) ) ε t = σ( Y ε (s), s t), and take L ε (t) = L θ ε (s), X ε (s), Y ε (s). s t Suppose that (θ ε,x ε, Y ε ) lives on (Ω ε, F ε,p ε ), with Assumptions 1 5. Then, there also exists a reference measure Q ε with similar properties. Next, we define the approximate versions that the recursive algorithms compute. Definition 5. For k = 1, 2, let ε,k (f k,t)= E Q ε ( ) [f k π ε,k (f k,t)= E P ε θ ε (t), X ε x (t) q ε,2 (f 2,t)= ε,2 (f 2,t)/ ε,1 (1,t). [f k ( ), θ ε (t), X ε x (t) L Y ε (t) F ε t Y F ε t, q ε,1 (f 1,t)= ε,1 (f 1,t)/ ε,2 (1,t)and Theorem 2. Suppose that Assumptions 1 5 hold for the models (θ,x, Y ) k=1,2 and that Assumptions 1 5 hold for the approximate models (θ ε,x ε, Y ε ) k=1,2. Suppose (θ ε,x ε ) (θ,x ) as ε. Then, as ε, for k = 1,2, (i) Y ε Y ; (ii) ε,k (f, t) k (f, t); (iii) π ε,k (f, t) π k (f, t); (iv) (q ε,1 (f 1, t), q ε,2 (f 2,t)) (q 1 (f 1, t), q 2 (f 2,t))for all bounded continuous functions, f, f 1, and f 2, and t>. Proof. To simplify notation, we exclude the superscript,. Since (θ ε,x ε ) (θ,x), continuous mapping theorem implies that the stochastic intensities of Y ε converges weakly to that of Y. This implies Y ε Y. Note that Eq. (3) implies that Q k {E Q k[l k (X k,y k ) Y k = } = f or k = 1, 2. To show parts (ii) (iv), since (by Remark 1) under the reference measure Q ε, (θ ε,x ε ) and Y ε are independent, it suffices to show ((θ ε,x ε ), Y ε,l ε ) ((θ,x), Y,L) under the reference measures. The Radon Nikodym derivative dp ε /dq ε is L ε (t) = n j=1 { exp log λ j (θ ε (s ), X ε (s ), s ) dy ε,j (s) [λ j (θ ε (s), X ε (s), s) 1 ds }.

9 M.A. Kouritzin, Y. Zeng / Nonlinear Analysis 6 (25) Noting (θ ε,x ε, Y ε ) (θ,x, Y) under the reference measures, one finds that continuous mapping theorem and Kurtz and Protter s Theorem [5 imply that log λ j (θ ε (s ), X ε s ), s ) dy ε,j (s) [λ j (θ ε (s), X ε (s), s) 1 ds log λ j (θ(s ), X(s ), s ) dy k (s), [λ j (θ(s), X(s), s) 1 ds, and ((θ ε,x ε ), Y ε,l ε ) ((θ,x), Y,L) under the reference measures (condition C2.2(i) ofkurtz and Protter [5 holds in our case, see Example 3.3 there,). Hence, Theorem 1 and Corollaries 1 and 2 imply (ii) (iv), respectively. References [1 S. Ethier, T. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, [2 E. Goggin, Convergence in distribution ofconditional expectations, Ann. Probab. 22 (1994) [3 R.E. Kass, A.E. Raftery, Bayes factors and model uncertainty, J. Am. Stat. Assoc. 9 (1995) [4 M. Kouritzin, Y. Zeng, Bayesian model selection via filtering for a class ofmicro-movement models ofasset price. International Journal oftheoretical and Applied Finance, submitted. [5 T. Kurtz, P. Protter, Weak limit theorems for stochastic integrals and stochastic differential equations, Ann. Probab. 19 (1991) [6 Y. Zeng, A partially-observed model for micro-movement of asset prices with bayes estimation via filtering, Math. Financ. 13 (23) [7 I. Crimaldi, L. Pratelli, Convergence results for conditional expectations, to appear in Bernoulli.