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Convergence of Markovian price processes in a financial market transaction model

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Abstract

This paper studies a financial market transaction model and convergence of Markovian price processes generated by an \(\alpha\)-double auction in Xu et al. (Expert Syst Appl 41(16):7032–7045, 2014) and extends their results for a fixed \(\alpha\) in [0, 1] to the case where \(\alpha\) is governed by a time non-homogeneous Markov chain over a set of finite states defined by \(R\equiv \{\alpha _1, \alpha _2, \ldots , \alpha _r\}\), \(0\le \alpha _1<\alpha _2<\cdots <\alpha _r\le 1\). A convergence result similar to that in Xu et al. (2014) holds, with the fixed \(\alpha\) replaced with the average \(\alpha ^*=\frac{1}{r}\sum _{\theta =1}^r \alpha _\theta\). We also identify the conditions under which a price process generated by such a Markovian \(\alpha\)-double auction converges in probability to a Walrasian equilibrium of the underlying financial market transaction model. A number of simulations are conducted and these simulations are consistent with the theoretical results of the paper.

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Notes

  1. This is the same as the k-double auction but k has been reserved to denote iterations in this paper.

  2. (1) \(a_k>0\) and \(b_k>0\); (2) \(\sum _{k=0}^{\infty } a_{k}=+\!\infty\) and \(\sum _{k=0}^{\infty }b_{k}=+\!\infty\); (3) \(\sum _{k=0}^{\infty } a_{k}^{2}<+\!\infty\) and \(\sum _{k=0}^{\infty } b_{k}^{2}<+\!\infty\). See Nedić and Bertsekas (2001) and Ram et al. (2009).

  3. Bubbles and crashes in economics and finance are the price movements of an asset that are not supported by its fundamentals. Changes in fundamentals that result in higher or lower prices are not seen as bubbles or crashes. But sentiments that over or under-react to the news in fundamentals can lead to a formation of a bubble and a crash.

  4. In practice, an execution of a trade is complicated. Given a quote (bid, ask) \(=(200, 201)\) for a stock in a dealer’s market, the ask 201 is the best price available from the dealer to sell when a buyer places a market order to buy and the bid 200 is the best offer available from the dealer to buy when a seller places a market order to sell. The actual posted price may often lie within (200, 201). In a “bull” market for a stock where its price goes up more often than down, the actual executed price is likely closer to the ask price of the dealer.

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Acknowledgments

The authors thank two anonymous referees for many helpful comments that have greatly improved the paper.

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Correspondence to Jinpeng Ma.

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This study was funded by Major Research Plan of National Science Foundation of China (91430105).

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The authors declare that they have no conflict of interest.

Additional information

This paper supersedes the paper entitled “Markovian \(\alpha\)-Double Auctions”.

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Appendix: Proofs of Lemma 7.1 and Theorem 7.1

Appendix: Proofs of Lemma 7.1 and Theorem 7.1

1.1 Proof of Lemma 7.1

Proof

The proof is similar to that of Lemma 4.2 in Ram et al. (2009). In fact, from the iteration (5.1)–(5.2), the “non-expansion” property of the Euclidean projection and the definition of subgradient, we have

$$\begin{aligned}&||x_{k+1}-y||^2\nonumber \\&\quad =\left| \left| P_{Y}\left( \alpha _{w_k^{\prime \prime }} \psi _{k+1}+\left( 1-\alpha _{w_k^{\prime \prime }}\right) \varphi _{k+1}\right) -y\right| \right| ^2\nonumber \\&\quad =\left| \left| P_{Y}\left( x_k-\alpha _{w_k^{\prime \prime }} a_k \bigtriangledown f_{w_k}(x_k)-\alpha _{w_k^{\prime \prime }} a_k \epsilon _{w_k,k}-(1-\alpha _{w_k^{\prime \prime }})b_k \bigtriangledown g_{w_k^{\prime }}(x_k)-\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right) -y\right| \right| ^2\nonumber \\&\quad \le \left| \left| (x_{k}-y)-\left( \alpha _{w_k^{\prime \prime }} a_k \bigtriangledown f_{w_k}(x_k)+\alpha _{w_k^{\prime \prime }} a_k \epsilon _{w_k,k}+(1-\alpha _{w_k^{\prime \prime }})b_k \bigtriangledown g_{w_k^{\prime }}(x_k)+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right) \right| \right| ^2\nonumber \\&\quad \le ||x_{k}-y||^2-2\alpha _{w_k^{\prime \prime }} a_{k}(x_{k}-y)^T\bigtriangledown f_{w_k}(x_k) -2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_{k}(x_{k}-y)^T\bigtriangledown g_{w_k^{\prime }}(x_k) \nonumber \\&\qquad +\left| \left| \alpha _{w_k^{\prime \prime }} a_k \bigtriangledown f_{w_k}(x_k)+\alpha _{w_k^{\prime \prime }} a_k \epsilon _{w_k,k}+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k \bigtriangledown g_{w_k^{\prime }}(x_k)+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right| \right| ^2\nonumber \\&\qquad -\left( 2\alpha _{w_k^{\prime \prime }} a_k (x_k-y)^T\epsilon _{w_k,k}+2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k(x_k-y)^T\delta _{w_k^{\prime },k}\right) \nonumber \\&\quad \le ||x_{k}-y||^2-2\alpha _{w_k^{\prime \prime }} a_{k}\left( f_{w_k}(x_k)-f_{w_k}(y)\right) -2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_{k}\left( g_{w_k^{\prime }}(x_k)-g_{w_k^{\prime }}(y)\right) \nonumber \\&\qquad +\left| \left| \alpha _{w_k^{\prime \prime }} a_k \bigtriangledown f_{w_k}(x_k)+\alpha _{w_k^{\prime \prime }}a_k \epsilon _{w_k,k}+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k \bigtriangledown g_{w_k^{\prime }}(x_k)+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right| \right| ^2\nonumber \\&\qquad -\left( 2\alpha _{w_k^{\prime \prime }} a_k (x_k-y)^T\epsilon _{w_k,k}+2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k(x_k-y)^T\delta _{w_k^{\prime },k}\right) . \end{aligned}$$
(10.1)

Taking conditional expectations with respect to \(G_{d(k)}\), we get

$$\begin{aligned}&E\left[ ||x_{k+1}-y||^2\left| G_{d(k)}\right. \right] \nonumber \\&\quad \le E\left[ ||x_{k}-y||^2\left| G_{d(k)}\right. \right] -2 a_{k}E\left[ \alpha _{w_k^{\prime \prime }}\left( f_{w_k}(x_k)-f_{w_k}(y)\right) \left| G_{d(k)}\right. \right] \nonumber \\&\qquad -2 b_{k}E\left[ \left( 1-\alpha _{w_k^{\prime \prime }}\right) \left( g_{w_k^{\prime }}(x_k)-g_{w_k^{\prime }}(y)\right) \left| G_{d(k)}\right. \right] \nonumber \\&\qquad +E\left[ \left| \left| \alpha _{w_k^{\prime \prime }} a_k \bigtriangledown f_{w_k}(x_k)+\alpha _{w_k^{\prime \prime }} a_k \epsilon _{w_k,k}+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k \bigtriangledown g_{w_k^{\prime }}(x_k)+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right| \right| ^2\left| G_{d(k)}\right. \right] \nonumber \\&\qquad -E\left[ \left( 2\alpha _{w_k^{\prime \prime }} a_k (x_k-y)^T\epsilon _{w_k,k}+2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k(x_k-y)^T\delta _{w_k^{\prime },k}\right) \left| G_{d(k)}\right. \right] \nonumber \\&\quad \le E\left[ ||x_{k}-y||^2\left| G_{d(k)}\right. \right] -2 a_{k}E\left[ \alpha _{w_k^{\prime \prime }}\left( f_{w_k}(x_k)-f_{w_k}(x_{d(k)})\right) \left| G_{d(k)}\right. \right] \nonumber \\&\qquad -2 a_{k}E\left[ \alpha _{w_k^{\prime \prime }}\left( f_{w_k}(x_{d(k)})-f_{w_k}(y)\right) \left| G_{d(k)}\right. \right] -2 b_{k}E\left[ \left( 1-\alpha _{w_k^{\prime \prime }}\right) \left( g_{w_k^{\prime }}(x_k)-g_{w_k^{\prime }}(x_{d(k)})\right) \left| G_{d(k)}\right. \right] \nonumber \\&\qquad -2 b_{k}E\left[ \left( 1-\alpha _{w_k^{\prime \prime }}\right) \left( g_{w_k^{\prime }}(x_{d(k)})-g_{w_k^{\prime }}(y)\right) \left| G_{d(k)}\right. \right] \nonumber \\&\qquad +E\left[ \left| \left| \alpha _{w_k^{\prime \prime }} a_k \bigtriangledown f_{w_k}(x_k)+\alpha _{w_k^{\prime \prime }} a_k \epsilon _{w_k,k}+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k \bigtriangledown g_{w_k^{\prime }}(x_k)+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right| \right| ^2\left| G_{d(k)}\right. \right] \nonumber \\&\qquad -E\left[ \left( 2\alpha _{w_k^{\prime \prime }} a_k (x_k-y)^T\epsilon _{w_k,k}+2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k(x_k-y)^T\delta _{w_k^{\prime },k}\right) \left| G_{d(k)}\right. \right] . \end{aligned}$$
(10.2)

Now, we will estimate the second term to the seventh term in the second “\(\le\)” of (10.2) respectively. For the second term , by the definition of subgradient, subgradient boundedness (5.4) and the fact that \(\alpha _{w_k^{\prime \prime }} \in [0, 1]\), we have

$$\begin{aligned}&E\left[ \alpha _{w_k^{\prime \prime }}\left( f_{w_k}(x_k)-f_{w_k}(x_{d(k)})\right) | G_{d(k)}\right] \nonumber \\&\quad \ge E\left[ \alpha _{w_k^{\prime \prime }}\bigtriangledown f_{w_k}(x_{d(k)})^T(x_k-x_{d(k)})\left| G_{d(k)}\right. \right] \nonumber \\&\quad \ge -C E[||x_{d(k)}-x_k|| ~| G_{d(k)}]\nonumber \\&\quad \ge -C\sum _{\ell =d(k)}^{k-1}E[||x_{\ell +1}-x_{\ell }|| ~| G_{d(k)}]\nonumber \\&\quad \ge -C\sum _{\ell =d(k)}^{k-1}E\left[ \left| \left| \alpha _{w_\ell ^{\prime \prime }} a_\ell \bigtriangledown f_{w_\ell }(x_\ell )+\alpha _{w_\ell ^{\prime \prime }} a_\ell \epsilon _{w_\ell ,\ell }+\left( 1-\alpha _{w_\ell ^{\prime \prime }}\right) b_\ell \bigtriangledown g_{w_\ell ^{\prime }}(x_\ell )+\left( 1-\alpha _{w_\ell ^{\prime \prime }}\right) b_\ell \delta _{w_\ell ^{\prime },\ell }\right| \right| \left| G_{d(k)}\right. \right] \nonumber \\&\quad \ge -C\sum _{\ell =d(k)}^{k-1} \left( a_{\ell } (C+\nu _{\ell })+b_{\ell } (D+\sigma _{\ell })\right) . \end{aligned}$$
(10.3)

Similarly, we can obtain

$$\begin{aligned} E\left[ \left( 1-\alpha _{w_k^{\prime \prime }}\right) (g_{w_k^{\prime }}(x_k)-g_{w_k^{\prime }}(x_{d(k)})| G_{d(k)})\right]\ge & {} -D\sum _{\ell =d(k)}^{k-1} \left( a_{\ell } (C+\nu _{\ell })+b_{\ell } (D+\sigma _{\ell })\right) . \end{aligned}$$
(10.4)

As for the third term in the second “\(\le\)” of (10.2), noting that \(G_{d(k)}\) denotes the entire history of the method up to time \(d(k)-1\), and the probability transition matrices for the Markov chains \(\{w_k\}\), \(\{w_k^{\prime \prime }\}\) from time \(d(k)-1\) to k are \(\Phi _I(k, d(k)-1)\) and \(\Phi _R(k, d(k)-1)\) respectively, we have

$$\begin{aligned}&E\left[ \alpha _{w_k^{\prime \prime }}\left( f_{w_k}(x_{d(k)})-f_{w_k}(y)\right) | G_{d(k)}\right] \nonumber \\&\quad =\sum _{t=1}^{r}\sum _{i=1}^{m}[\Phi _{R}(k,d(k)-1)]_{w^{\prime \prime }_{d(k)},t}[\Phi _{I}(k,d(k)-1)]_{w_{d(k)},i}\alpha _t\left( f_i(x_{d(k)})-f_i(y)\right) \nonumber \\&\quad \ge \sum _{t=1}^{r}[\Phi _{R}(k,d(k)-1)]_{w^{\prime \prime }_{d(k)},t} \alpha _t \sum _{i=1}^{m}\frac{1}{m}\left( f_i(x_{d(k)})-f_i(y)\right) \nonumber \\&\qquad -\sum _{t=1}^{r}[\Phi _{R}(k,d(k)-1)]_{w^{\prime \prime }_{d(k)},t} \alpha _t\sum _{i=1}^{m}|\Phi _{I}(k,d(k)-1)]_{w_{d(k)},i}-\frac{1}{m}| |f_i(x_{d(k)})-f_i(y)| \nonumber \\&\quad \ge \sum _{t=1}^{r}[\Phi _{R}(k,d(k)-1)]_{w^{\prime \prime }_{d(k)},t} \alpha _t \sum _{i=1}^{m}\frac{1}{m}\left( f_i(x_{d(k)})-f_i(y)\right) \nonumber \\&\qquad -r\sum _{i=1}^{m}|\Phi _{I}(k,d(k)-1)]_{w_{d(k)},i}-\frac{1}{m}| |f_i(x_{d(k)})-f_i(y)| \nonumber \\&\quad \ge \sum _{t=1}^{r}\left( \frac{1}{r} +\left( [\Phi _{R}(k,d(k)-1)]_{w^{\prime \prime }_{d(k)},t}- \frac{1}{r}\right) \right) \frac{\alpha _t}{m}\left( f(x_{d(k)})-f(y)\right) \nonumber \\&\qquad -r b_I \beta _I^{k+1-d(k)}m C||x_{d(k)}-y|| \nonumber \\&\quad \ge \frac{\alpha ^*}{m}\left( f(x_{d(k)})-f(y)\right) -r C \left( b_R\beta _R^{k+1-d(k)} + m b_I \beta _I^{k+1-d(k)}\right) ||x_{d(k)}-y||, \end{aligned}$$
(10.5)

where in the second “\(\ge\)”, we have used the fact that \([\Phi _{R}(k,d(k)-1)]_{w^{\prime \prime }_{d(k)},t}\le 1\) and \(\alpha _t\in [0,1]\), in the third “\(\ge\)”, we have used (3.1), and in the last “\(\ge\)”, we have used (3.3). Similarly, it holds

$$\begin{aligned}&E\left[ \left( 1-\alpha _{w_k^{\prime \prime }}\right) \left( g_{w_k^{\prime }}(x_{d(k)})-g_{w_k^{\prime }}(y)\right) | G_{d(k)}\right] \nonumber \\&\quad \ge \frac{1-\alpha ^*}{n}\left( g(x_{d(k)})-g(y)\right) -rD\left( b_R \beta _R^{k+1-d(k)} +n b_J\beta _J^{k+1-d(k)}\right) ||x_{d(k)}-y||. \end{aligned}$$
(10.6)

For the sixth term in the second “\(\le\)” of (10.2), from Assumption 7.1, the boundedness (5.4)–(5.5) and the fact that \(\alpha _{w_k^{\prime \prime }}\in [0,1]\), it follows

$$\begin{aligned}&E\left[ \left( \alpha _{w_k^{\prime \prime }} a_k \bigtriangledown f_{w_k}(x_k)+\alpha _{w_k^{\prime \prime }} a_k \epsilon _{w_k,k}+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k \bigtriangledown g_{w_k^{\prime }}(x_k)+\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right) ^2\left| G_{d(k)}\right. \right] \nonumber \\&\quad \le \left( a_k C+a_k \nu _k+b_k D +b_k \sigma _k\right) ^2. \end{aligned}$$
(10.7)

As for the last term in the second “\(\le\)” of (10.2), since \(G_{d(k)} \subset G_k\), it holds

$$\begin{aligned}&E\left[ \left( 2\alpha _{w_k^{\prime \prime }} a_k (x_k-y)^T\epsilon _{w_k,k}+2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k(x_k-y)^T\delta _{w_k^{\prime },k}\right) \left| G_{d(k)}\right. \right] \nonumber \\&\quad =E\left[ E\left[ \left( 2\alpha a_k (x_k-y)^T\epsilon _{w_k,k}+2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k(x_k-y)^T\delta _{w_k^{\prime },k}\right) |G_{k}\right] \left| G_{d(k)}\right. \right] \nonumber \\&\quad =E\left[ (x_k-y)^TE\left[ \left( 2\alpha _{w_k^{\prime \prime }} a_k \epsilon _{w_k,k}+2\left( 1-\alpha _{w_k^{\prime \prime }}\right) b_k\delta _{w_k^{\prime },k}\right) |G_{k}\right] \left| G_{d(k)}\right. \right] \nonumber \\&\quad \ge -E\left[ ||x_k-y|| \left( ||E[2a_k \epsilon _{w_k,k}|G_{k}]||+||E[2b_k\delta _{w_k^{\prime },k}|G_{k}]||\right) \left| G_{d(k)}\right. \right] \nonumber \\&\quad \ge \left( -2 a_k \mu _k -2 b_k \tau _k\right) E\left[ ||x_k-y|| \left| G_{d(k)}\right. \right] . \end{aligned}$$
(10.8)

Substituting the preceding estimates (10.3)–(10.8) into (10.2) yields the desired estimate (7.1). \(\square\)

1.2 Proof of Theorem 7.1

Proof

Since Y is compact, fg is convex functions, it follows that \(Y^{*}(\alpha ^*,\lambda )\) is nonempty, closed and convex. By Lemma 7.1, we obtain that for any \(y^*\in Y^*(\alpha ^*,\lambda )\)

$$\begin{aligned}&E[\text {dist}(x_{k+1}, Y^*(\alpha ^*,\lambda ))^2|G_{d(k)}] \nonumber \\&\quad \le E[\text {dist}(x_{k}, Y^*(\alpha ^*,\lambda ))^2|G_{d(k)}]- \frac{2\alpha ^*a_{k}}{m}\left( f(x_{d(k)})-f(y^{*})\right) -\frac{2(1-\alpha ^*) b_{k}}{n}\left( g(x_{d(k)})-g(y^{*})\right) \nonumber \\&\qquad +\left( 2a_k r C\left( b_R \beta _R^{k+1-d(k)} + m b_I \beta _I^{k+1-d(k)}\right) +2b_k r D\left( b_R \beta _R^{k+1-d(k)} + n b_J \beta _J^{k+1-d(k)}\right) \right) \max _{x,y\in Y}||x-y||\nonumber \\&\qquad +2\left( a_{k}C + b_{k}D \right) \sum _{\ell =d(k)}^{k-1} \left( a_{\ell } (C+\nu _{\ell })+b_{\ell } (D+\sigma _{\ell })\right) +\left( a_k C+a_k \nu _k+b_k D +b_k \sigma _k\right) ^2\nonumber \\&\qquad +\left( 2a_k\mu _k+2b_k \tau _k\right) E\left[ \text {dist}\left( x_{k}, Y^*(\alpha ^*,\lambda )\right) \left| G_{d(k)}\right. \right] . \end{aligned}$$
(10.9)

Denote

$$\begin{aligned} \eta _k:= & {} \left( 2a_k r C\left( b_R \beta _R^{k+1-d(k)} + m b_I \beta _I^{k+1-d(k)}\right) +2b_k r D\left( b_R \beta _R^{k+1-d(k)} + n b_J \beta _J^{k+1-d(k)}\right) \right) \max _{x,y\in Y}||x-y||\nonumber \\&+2\left( a_{k}C + b_{k}D \right) \sum _{\ell =d(k)}^{k-1} \left( a_{\ell } (C+\nu _{\ell })+b_{\ell } (D+\sigma _{\ell })\right) +\left( a_k C+a_k \nu _k+b_k D +b_k \sigma _k\right) ^2\nonumber \\&+\left( 2a_k\mu _k+2b_k \tau _k\right) E[\text {dist}(x_{k}, Y^*(\alpha ^*,\lambda ))]. \end{aligned}$$
(10.10)

Taking expectations to (10.9) yields

$$\begin{aligned}&E\left[ \text {dist}\left( x_{k+1}, Y^*\left( \alpha ^*,\lambda \right) \right) ^2\right] \\&\quad \le E\left[ \text {dist}\left( x_{k}, Y^*\left( \alpha ^*,\lambda \right) \right) ^2\right] -\frac{2\alpha ^* a_k}{m}\left( E\left[ f(x_{d(k)})\right] -f(y^*)\right) \\&\qquad -\frac{2\left( 1-\alpha ^*\right) b_k}{n}\left( E\left[ g(x_{d(k)})\right] -g(y^*)\right) +\eta _k\\&\quad \le E\left[ \text {dist}\left( x_{k}, Y^*\left( \alpha ^*,\lambda \right) \right) ^2\right] -\frac{2 a_k}{m}\left( E\left[ \left( \alpha ^* f+\lambda \left( 1-\alpha ^*\right) g\right) (x_{d(k)})\right] \right. \\&\left. \qquad -\left( \alpha ^* f+\lambda (1-\alpha ^*)g\right) (y^*)\right) +2(1-\alpha ^*)\left( \frac{b_k}{n}-\lambda \frac{a_k}{m}\right) \left( E\left[ g\left( x_{d(k)}\right) \right] -g(y^*)\right) +\eta _k\\&\quad \le E\left[ \text {dist}\left( x_{k}, Y^*(\alpha ^*,\lambda )\right) ^2\right] -\frac{2 a_k}{m}\left( E\left[ \left( \alpha ^* f+\lambda (1-\alpha ^*) g\right) (x_{d(k)})\right] \right. \\&\left. \qquad -\left( \alpha ^* f+\lambda (1-\alpha ^*)g\right) (y^*)\right) +4M\left| \frac{b_k}{n}-\lambda \frac{a_k}{m}\right| +\eta _k. \end{aligned}$$

Following the same routine as in the proof of Theorem 4.3 in Ram et al. (2009), we easily know that for some non-negative integer sequence \(\{d(k)\}\), it holds

$$\begin{aligned} \sum _{k=2}^{\infty }\eta _k < \infty . \end{aligned}$$

In addition, in view of (7.2), the inequality

$$\begin{aligned} E\left[ \left( \alpha ^* f+\lambda (1-\alpha ^*) g\right) (x_{d(k)})\right] \ge \left( \alpha ^* f+\lambda (1-\alpha ^*)g\right) (y^*) \end{aligned}$$

and Lemma 1 in Bertsekas and Tsitsiklis (2000), we conclude that \(E[\text {dist}(x_{k}, Y^*(\alpha ^*,\lambda ))^2]\) converges to a non-negative scalar and

$$\begin{aligned} \sum _{k=2}^{\infty }\frac{2 a_k}{m}\left( E\left[ \left( \alpha ^* f+\lambda (1-\alpha ^*) g\right) (x_{d(k)})\right] -\left( \alpha ^* f+\lambda (1-\alpha ^*)g\right) (y^*)\right) <\infty , \end{aligned}$$
(10.11)

which, together with the fact \(\sum _{k=2}^{\infty }a_{k}=\infty\) for \(\frac{2}{3}< p\le 1\), yields

$$\begin{aligned} \liminf _{k\rightarrow \infty } E\left[ \left( \alpha ^* f+\lambda (1-\alpha ^*) g)(x_{k}\right) \right] =\left( \alpha ^* f+\lambda (1-\alpha ^*)g\right) (y^*). \end{aligned}$$
(10.12)

Since fg are continuous and Y is compact, from Fatou’s lemma it follows

$$\begin{aligned}&E\left[ \liminf _{k\rightarrow \infty } \left( \alpha ^* f +\lambda (1-\alpha ^*)g\right) (x_{k})\right] \le \liminf _{k\rightarrow \infty } E\left[ \left( \alpha ^* f+\lambda (1-\alpha ^*) g\right) (x_{k})\right] \nonumber \\&\quad =\left( \alpha ^* f+\lambda (1-\alpha ^*)g\right) (y^*), \end{aligned}$$
(10.13)

which implies that

$$\begin{aligned} \liminf _{k\rightarrow \infty } \left( \alpha ^* f +\lambda (1-\alpha ^*)g\right) (x_{k})=\left( \alpha ^* f+\lambda (1-\alpha ^*)g\right) (y^*) \end{aligned}$$

with probability 1, i.e. the first result of (7.4) holds. Using again the continuity of fg and the compactness of Y, we know that

$$\begin{aligned} \liminf _{k\rightarrow \infty }\text {dist}\left( x_k,Y^*(\alpha ^*,\lambda )\right) =0 \end{aligned}$$

with probability 1, i.e. the second result of (7.4) holds.

Now we aim to prove (7.5). Since \(\liminf _{k\rightarrow \infty }\text {dist}\left( x_k,Y^*(\alpha ^*,\lambda )\right) =0\) with probability 1, there exists a subsequence of \(\{\text {dist}\left( x_k,Y^*(\alpha ^*,\lambda )\right) ^2\}\), which we denote by \(\{\text {dist}\left( x_{k_\ell },Y^*(\alpha ^*,\lambda )\right) ^2\}\), such that \(\lim _{k_\ell \rightarrow \infty }\text {dist}\left( x_{k_\ell },Y^*(\alpha ^*,\lambda )\right) ^2=0\) with probability 1. For the set Y is bounded, we know the sequence \(\{\text {dist}\left( x_{k_\ell },Y^*(\alpha ^*,\lambda )\right) ^2\}\) is bounded. By the dominated convergence theorem, we have

$$\begin{aligned} \lim _{k_\ell \rightarrow \infty }E\left[ \text {dist}\left( x_{k_\ell },Y^*(\alpha ^*,\lambda )\right) ^2\right] =E\left[ \lim _{k_\ell \rightarrow \infty }\text {dist}\left( x_{k_\ell },Y^*(\alpha ^*,\lambda )\right) ^2\right] =0. \end{aligned}$$

Since we already obtain that \(E[\text {dist}(x_{k}, Y^*(\alpha ^*,\lambda ))^2]\) converges to a non-negative scalar, then it has to converge to 0, i.e.

$$\begin{aligned} \lim _{k\rightarrow \infty }E\left[ \text {dist}\left( x_k,Y^*(\alpha ^*,\lambda )^2\right. \right] =0, \end{aligned}$$

which completes the proof. \(\square\)

1.3 Experiments for Sect. 8.5

The weight \(\alpha\) follows a Markov chain in a state \(R=\{\alpha _1, \alpha _2,\alpha _3\}\) and in all cases, the transition matrix for the Markov chain for \(\alpha\) is given by, \(k=0, 1,2,\ldots\),

$$\begin{aligned} P_{R}(k)=\left( \begin{array}{ccc} 2/3&{}\quad 1/3 &{}\quad 0 \\ 1/3&{}\quad 1/3 &{}\quad 1/3 \\ 0 &{}\quad 1/3 &{}\quad 2/3 \\ \end{array}\right) , \end{aligned}$$

which equals \(P_I\) in Fig. 2. We let \(\lambda =1\). The two step sizes are given by

$$\begin{aligned} a_k=\frac{1}{k+1}\quad \text{ and }\quad b_k=\frac{\lambda n}{m}a_k \end{aligned}$$

so that the \(\lambda\) condition (7.2) is satisfied. In the situations with noises, we add the Gauss noises with mean and variance (0, 25), as before.

Experiment 1 \(m=5, n=5\). The two transition matrices \(P_I(k)\) and \(P_J(k)\) are given by, \(k=0,1,2,\ldots ,\)

$$\begin{aligned} P_I(k)= & {} \left( \begin{array}{ccccc} 4/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 1/5&{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 4/5 \\ \end{array}\right) ,\\ P_J(k)= & {} \left( \begin{array}{ccccc} 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 \\ 1/5&{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5\\ 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5\\ \end{array}\right) . \end{aligned}$$

We let \(R=\{0.25, 0.5, 0.75\}\). Thus, \(\alpha ^*=0.5\) and \(Y^*=Y^*(\alpha ^*, \lambda )=\{1\}\), since \(\lambda =1\). Figure 8 reports the result, which shows that the price process \(\{x_k\}\) converges to the equilibrium price 1, with or without noises. Next we increase the number of buyers from \(n=5\) to \(n=10\) while m remains at 5.

Fig. 8
figure 8

Average \(\{\hbox {x}_{\mathrm{k}}\}\) of 100 samples for \(\hbox {m}=5\) and \(\hbox {n}=5\). \(\hbox {x}_{0}\) randomly generated in [1, 5] and iterations from 1 to 20,000. Note: equilibrium price = 1 when \(\alpha ^*=0.5\)

Fig. 9
figure 9

Average \(\{\hbox {x}_{\mathrm{k}}\}\) of 100 samples for \(\hbox {m}=5\) and \(\hbox {n}=10\). \(\hbox {x}_{0}\) randomly generated in [1, 5] and iterations from 1 to 20,000. Note: equilibrium price = 1.8 when \(\alpha^ *\) is 0.5. For \(\alpha^ *=2/3\), the process converges to 1.27

Experiment 2 \(m=5, n=10\). The two transition matrices \(P_I(k)\) and \(P_J(k)\) are given by, \(k=0,1,2,\ldots\),

$$\begin{aligned} P_I(k)= & {} \left( \begin{array}{ccccc} 4/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 1/5&{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 4/5 \\ \end{array}\right) ,\\ P_J(k)= & {} \left( \begin{array}{ccccccc } 8/10&{}\quad 1/10 &{}\quad &{}\quad &{}\quad 1/10 \\ 1/10&{}\quad 8/10 &{}\quad \ddots &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad 1/10 \\ 1/10 &{}\quad &{}\quad &{}\quad 1/10&{}\quad 8/10\\ \end{array}\right) _{10\times 10}. \end{aligned}$$

We select two different states for \(\alpha\)s, \(R=\{0.25, 0.5, 0.75\}\) and \(R1=\{0.25, 0.75, 1\}\), with different average weights \(\alpha ^*=\frac{1}{2}\) and \(\alpha _1^*=\frac{2}{3}\). Thus, \(Y^*=Y^*(\alpha ^*, \lambda )=\{1.8028\}\) and \(Y^*(\alpha _1^*, \lambda )=\{1.2748\}\). Clearly, \(Y^*\ne Y^*(\alpha _1^*, \lambda )\). Figure 9 presents the results, which confirm what has been shown in Theorem 7.1. Note that the fundamentals remain the same. But the price process converges to the price 1.2748, purely due to a change in the average weight from \(\alpha ^*\) to \(\alpha ^*_1\). The interesting part is that what matters is the average \(\alpha ^*\). This implies that if we change R1 to \(R1^\prime =\{0.1, 0.9, 1\}\), then our experiments will also converge to the same price 1.2748. Next we increase the number of buyers further from \(n=10\) to \(n=20\) while m stays put at 5.

Experiment 3 \(m=5, n=20\). The two transition matrices \(P_I(k)\) and \(P_J(k)\) are given by, \(k=0,1,2,\ldots ,\)

$$\begin{aligned} P_I(k)= & {} \left( \begin{array}{ccccc} 4/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 0 \\ 1/5&{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 \\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 4/5 \\ \end{array}\right) ,\\ P_J(k)= & {} \left( \begin{array}{cccccc } 18/20&{}\quad 1/20 &{}\quad &{}\quad &{}\quad 1/20 \\ 1/20&{}\quad 18/20 &{}\quad \ddots &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad \ddots &{}\quad \ddots &{}\quad \\ &{}\quad &{}\quad \ddots &{}\quad \ddots &{}\quad 1/20 \\ 1/20 &{}\quad &{}\quad &{}\quad 1/20&{}\quad 18/20\\ \end{array}\right) _{20\times 20}. \end{aligned}$$

We also consider two different states \(R=\{0.25, 0.5, 0.75\}\) and \(R2=\{0.4, 0.8, 1\}\). Then \(\alpha ^*=0.5\) and \(\alpha ^*_2=2.2/3\). So, \(Y^*=\{3.3912\}\) and \(Y^*(\alpha ^*_2, \lambda )=\{2.0449\}\). Figure 10 presents the experimental results. Once again, with a higher average weight \(\alpha ^*_2\) than \(\alpha ^*\), the price process converges to a price that is substantially lower than the equilibrium price of the original economy. Next we keep \(n=5\) as in Experiment 1 while increase the number of sellers from \(m=5\) to \(m=10\). One can expect that the equilibrium price is lower than 1 because there are more sellers.

Fig. 10
figure 10

Average \(\{\hbox {x}_{\mathrm{k}}\}\) of 100 samples for \(\hbox {m}=5\) and \(\hbox {n}=20\). \(\hbox {x}_{0}\) randomly generated in [1, 5] and iterations from 1 to 20,000. Note: equilibrium = 3.39 for \(\alpha^ *=0.5\). For \(\alpha^ *=2.2/3\), the process converges to 2.04

Fig. 11
figure 11

Average \(\{\hbox {x}_{\mathrm{k}}\}\) of 100 samples for \(\hbox {m}=10\) and \(\hbox {n}=5\). \(\hbox {x}_{0}\) randomly generated in [1, 5] and iterations from 1 to 20,000. Note: equilibrium = 0.55 when \(\alpha^ *\) equals 0.5 For \(\alpha^ *=1/3\), the process converges to 0.78

Experiment 4 \(m=10, n=5\). The two transition matrices \(P_I(k)\) and \(P_J(k)\) are given by, \(k=0,1,2,\ldots\),

$$\begin{aligned} P_I(k)= & {} \left( \begin{array}{ccccccc } 9/10&{}\quad 1/10 &{}\quad &{}\quad &{}\quad \\ 1/10&{}\quad 8/10 &{}\quad \ddots &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad \ddots &{}\quad 1/10&{}\quad \\ &{}\quad &{}\quad 1/10&{}\quad 8/10 &{}\quad 1/10\\ &{}\quad &{}\quad &{}\quad 1/10&{}\quad 9/10\\ \end{array}\right) _{10\times 10},\\ P_J(k)= & {} \left( \begin{array}{ccccc} 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 \\ 1/5&{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 \\ 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 \\ \end{array}\right) . \end{aligned}$$

We set \(R=\{0.25, 0.5, 0.75\}\) and \(R3=\{0, 0.25, 0.75\}\) in our experiments. Thus, \(\alpha ^*=0.5\) and \(\alpha ^*_3=\frac{1}{3}\). So, \(Y^*=\{0.5547\}\) and \(Y^*(\alpha ^*_3, \lambda )=\{0.7845\}\). The experimental results are reported in Fig. 11, which confirms our theoretical result in Theorem 7.1. Note that when \(\alpha ^*\) is lower from \(\frac{1}{2}\) to \(\frac{1}{3}\), the price process converges to a higher price 0.7845 than the equilibrium price 0.5547 of the original economy, higher by more than 41 %. Next we increase the number of sellers from \(m=10\) to \(m=20\). The equilibrium price in \(Y^*\) will be even lower, as expected.

Experiment 5 \(m=20, n=5\). The two transition matrices \(P_I(k)\) and \(P_J(k)\) are given by, \(k=0,1,2,\ldots\),

$$\begin{aligned} P_I(k)= & {} \left( \begin{array}{ccccccc } 19/20&{}\quad 1/20 &{}\quad &{}\quad &{}\quad \\ 1/20&{}\quad 18/20 &{}\quad \ddots &{}\quad &{}\quad \\ &{}\quad \ddots &{}\quad \ddots &{}\quad 1/20&{}\quad \\ &{}\quad &{}\quad 1/20&{}\quad 18/20 &{}\quad 1/20\\ &{}\quad &{}\quad &{}\quad 1/20&{}\quad 19/20\\ \end{array}\right) _{20\times 20},\\ P_J(k)= & {} \left( \begin{array}{ccccc} 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 \\ 1/5&{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 &{}\quad 1/5 \\ 1/5 &{}\quad 0 &{}\quad 0 &{}\quad 1/5 &{}\quad 3/5 \\ \end{array}\right) . \end{aligned}$$

We set \(R=\{0.25, 0.5, 0.75\}\) and \(R4=\{0, 0.2, 0.4\}\) in our experiments. Thus, \(\alpha ^*=0.5\) and \(\alpha ^*_4=0.2\). So, \(Y^*=\{0.2949\}\) and \(Y^*(\alpha ^*_4, \lambda )=\{0.5898\}\). The experimental results are reported in Fig. 12. When the average weight \(\alpha ^*\) moves lower from 0.5 to 0.2, the price process converges to a higher price 0.5898, which is twice as much as the equilibrium price of the original economy because \(\sqrt{\frac{\lambda (1-\alpha _4^*)}{\alpha ^*_4}}=2\). In summary, we have done 18 experiments each of which has shown how a change in \(\alpha\) may affect the convergence of the price process \(\{x_k\}\) of an \(\alpha\)-double auction. These experiments provide solid evidence a double auction implemented in a real exchange market may indeed contribute to the excess volatility.

Fig. 12
figure 12

Average \(\{\hbox {x}_{\mathrm{k}}\}\) of 100 samples for \(\hbox {m}=20\) and \(\hbox {n}=5\). \(\hbox {x}_{0}\) randomly generated in [1, 5] and iterations from 1 to 20,000. Note: equilibrium price = 0.29 for \(\alpha^ *=0.5\). For \(\alpha^ *=1/5\), the process converges to 0.59

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Xu, X., Ma, J. & Xie, X. Convergence of Markovian price processes in a financial market transaction model. Oper Res Int J 17, 239–273 (2017). https://doi.org/10.1007/s12351-015-0224-7

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