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Market-reaction-adjusted optimal central bank intervention policy in a forex market with jumps

Abstract

Impulse control with random reaction periods (ICRRP) is used to derive a country’s optimal foreign exchange (forex) rate intervention policy when the forex market reacts to the interventions. This paper extends the previous work on ICRRP by incorporating a multi-dimensional jump diffusion process to model the state dynamics, and hence, enhance the viability of the extant model for applications. Furthermore, we employ a novel minimum cost operator that simplifies the computations of the optimal solutions. Finally, we demonstrate the efficacy of our framework by finding a market-reaction-adjusted optimal central bank intervention (CBI) policy for a country. Our numerical results suggests that market reactions and the jumps in the forex market are complements when the reactions increase the forex rate volatility; otherwise, they are substitutes.

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Notes

  1. 1.

    Empirical evidence on market reactions to central bank interventions can be found in Beine et al. (2002), Beine et al. (2003), Bonser-Neal and Tanner (1996), Caporale and Doroodian (2001), Dominguez (1998), Hung (1997), Mundaca (2001), Mundaca (2011) and Wilfling (2009).

  2. 2.

    Note that each column \(\gamma ^{j}\) of the \(k\times l\) matrix \(\gamma =[\gamma _{ij}]\) depends on z only through the jth coordinate \(z_{j}\), i.e.

    $$\begin{aligned} \gamma ^{(j)}(t,z)=\gamma ^{(j)}(t,z_{j}), \ j=1,\ldots , l;\quad z=(z_{1},\ldots ,z_{l})^{T}\in \mathbb {R}^{l}. \end{aligned}$$

    Thus the integral on the right-hand side of (1) is just a shorthand matrix notation, i.e.,

    $$\begin{aligned}&\int _{\mathbb {R}^l} \gamma (X(t-),z)\widetilde{N}(dt,dz)\\&\quad =\left( \sum _{j=1}^{l}\int _{\mathbb {R}} \gamma _{1j}(X(t-),z_{j})\widetilde{N}_{j}(dt,dz_{j}),\ldots \,,\sum _{j=1}^{l}\int _{\mathbb {R}} \gamma _{kj}(X(t-),z_{j})\widetilde{N}_{j}(dt,dz_{j})\right) ^{T}. \end{aligned}$$

    In component form (1) becomes: \( dX_{d}(t)=\mu _{d}(X(t))dt+\sum _{j=1}^{m} \sigma _{dj}(X(t))dB_{j}(t)+\sum _{j=1}^{l}\int _{\mathbb {R}} \gamma _{dj}(X(t-),z_{j})\widetilde{N}_{j}(dt,dz_{j});\qquad 1\le d\le k. \)

  3. 3.

    The gradient operator \(\nabla \phi (x)\) denotes \(\left( \frac{\partial \phi }{\partial x_1},\frac{\partial \phi }{\partial x_2},\ldots ,\frac{\partial \phi }{\partial x_k}\right) \), and the dot product is defined by \(x\cdot y=\sum _{i=1}^{k}x_i y_i\) for \(x=(x_1,x_2,\ldots ,x_k)\) and \(y=(y_1,y_2,\ldots ,y_k)\).

  4. 4.

    The number of domestic currency units per unit of foreign currency or a given basket of foreign currencies.

  5. 5.

    Note that, from \(\tau _j\) to \(\tau _j+t\), the process \(X^{(u)}\) follows \(X^{i_j}\).

References

  1. Ahn, C. M., Cho, D. C., & Park, K. (2007). The pricing of foreign currency options under jump-diffusion processes. Journal of Futures Markets, 27(7), 669–695.

    Article  Google Scholar 

  2. Ahn, C. M., & Thompson, H. E. (1992). The impact of jump risks on nominal interest rates and foreign exchange rates. Review of Quantitative Finance and Accounting, 2(1), 17–31.

    Article  Google Scholar 

  3. Aït-Sahalia, Y., & Jacod, J. (2011). Testing whether jumps have finite or infinite activity. Annals of Statistics, 39(3), 1689–1719.

    Article  Google Scholar 

  4. Akgiray, V., & Booth, G. (1988). Mixed diffusion-jump process modeling of exchange rate movements. The Review of Economics and Statistics, 70(4), 631–637.

    Article  Google Scholar 

  5. Ball, C. A., & Roma, A. (1993). A jump diffusion model for the European Monetary System. Journal of International Money and Finance, 12(5), 475–492.

    Article  Google Scholar 

  6. Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic modelling. Scandinavian Journal of Statistics, 24(1), 1–13.

    Article  Google Scholar 

  7. Bar-Ilan, A., Perry, D., & Stadje, W. (2004). A generalized impulse control model of cash management. Journal of Economic Dynamics and Control, 28(6), 1013–1033.

    Article  Google Scholar 

  8. Bates, D. S. (1988). The crash premium: Option pricing under asymmetric processes, with applications to options on deutschmark futures. Rodney L. White Center for financial research working paper no. 36–88, University of Pennsylvania.

  9. Bates, D. S. (1996). Jumps and stochastic volatility: Exchange rate processes implicit in deutsche mark options. Review of Financial Studies, 9(1), 69–107.

    Article  Google Scholar 

  10. Beine, M., Bénassy-Quére, A., & Lecourt, C. (2002). Central bank intervention and foreign exchange rates: New evidence from FIGARCH estimations. Journal of International Money and Finance, 21(1), 115–144.

    Article  Google Scholar 

  11. Beine, M., Laurent, S., & Lecourt, C. (2003). Official central bank intervention and exchange rate volatility: Evidence from a regime-switching analysis. European Economic Review, 47(5), 891–911.

    Article  Google Scholar 

  12. Bensoussan, A., Long, H., Perera, S., & Sethi, S. (2012). Impulse control with random reaction periods: A central bank intervention problem. Operations Research Letters, 40(6), 425–430.

    Article  Google Scholar 

  13. Bonser-Neal, C., & Tanner, G. (1996). Central bank intervention and the volatility of foreign exchange rates: Evidence from the options market. Journal of International Money and Finance, 15(6), 853–878.

    Article  Google Scholar 

  14. Boyce, W. E., & Di Prima, R. C. (1997). Elementary differential equations. New York: Wiley.

    Google Scholar 

  15. Buncak, T. (2013). Jump processes in exchange rates modeling. Masaryk Institute of Advanced Studies, MPRA Paper No: Czech Technical University in Prague 49882.

  16. Cadenillas, A., & Huamán-Aguilar, R. (2015). Explicit formula for the optimal government debt ceiling. Annals of Operations Research. doi:10.1007/s10479-015-2052-9.

  17. Cadenillas, A., & Zapatero, F. (1999). Optimal central bank intervention in the foreign exchange market. Journal of Economic Theory, 87(1), 218–242.

    Article  Google Scholar 

  18. Caporale, T., & Doroodian, K. (2001). Central bank intervention and foreign exchange volatility. International Advances in Economic Research Journal, 7(4), 385–392.

    Article  Google Scholar 

  19. Carr, P., Geman, H., Madan, D. B., & Yor, M. (2002). The fine structure of asset returns: An empirical investigation. Journal of Business Research, 75(2), 305–332.

    Google Scholar 

  20. Carr, P., & Wu, L. (2003). The finite moment log stable process and option pricing. The Journal of Finance, 58(2), 753–777.

    Article  Google Scholar 

  21. Chiang, M.-H., Li, C.-Y., & Chen, S.-N. (2016). Pricing currency options under double exponential jump diffusion in a Markov-modulated HJM economy. Review of Quantitative Finance and Accounting, 46(3), 459–482.

    Article  Google Scholar 

  22. Constantinides, G. M. (1976). Stochastic cash management with fixed and proportional transaction costs. Management Science, 22(12), 1320–1331.

    Article  Google Scholar 

  23. Constantinides, G. M., & Richard, S. F. (1978). Existence of optimal simple policies for discounted-cost inventory and cash management in continuous time. Operations Research, 26(4), 620–636.

    Article  Google Scholar 

  24. Cont, R., & Tankov, P. (2004). Financial modelling with jump processes. London: Chapman and Hall/CRC.

    Google Scholar 

  25. Davis, M. H. A., Guo, X., & Wu, G. (2010). Impulse Control of multidimensional jump diffusions. SIAM Journal on Control and Optimization, 48(8), 5276–5293.

    Article  Google Scholar 

  26. De Jong, F., Drost, F. C., & Werker, B. J. M. (2001). A jump-diffusion model for exchange rates in a target zone. Statistica Neerlandica, 55(3), 270–300.

    Article  Google Scholar 

  27. Doffou, A., & Hilliard, J. E. (2001). Pricing currency options under stochastic interest rates and jump-diffusion processes. Journal of Financial Resource, 24(4), 565–586.

    Article  Google Scholar 

  28. Dominguez, K. (1998). Central bank intervention and exchange rate volatility. Journal of International Money and Finance, 17, 161–190.

    Article  Google Scholar 

  29. Dumas, B., Jennergren, L. P., & Näslund, B. (1995). Realignment risk and currency option pricing in target zones. European Economic Review, 39(8), 1523–1544.

    Article  Google Scholar 

  30. Guo, J.-H., & Hung, M.-W. (2007). Pricing American options on foreign currency with stochastic volatility, jumps, and stochastic interest rates. Journal of Futures Markets, 27(9), 867–891.

    Article  Google Scholar 

  31. Huamán-Aguilar, R., & Cadenillas, A. (2015). Government debt control: Optimal currency portfolio and payments. Operations Research, 63(5), 1044–1057.

    Article  Google Scholar 

  32. Hung, J. H. (1997). Intervention strategies and exchange rate volatility: A noise trading perspective. Journal of International Money and Finance, 16(5), 779–793.

    Article  Google Scholar 

  33. Jiang, G. J. (1998). Jump diffusion model of exchange rate dynamics—Estimation via indirect inference. Organizations and Management, University of Groningen: Published by Graduate School/Research Institute Systems.

  34. Jorion, P. (1988). On jump processes in the foreign exchange and stock markets. Review of Financial Studies, 1(4), 427–445.

    Article  Google Scholar 

  35. Kou, S. (2002). A jump-diffusion model for option pricing. Management Science, 48(8), 1086–1101.

    Article  Google Scholar 

  36. Madan, D. B., & Seneta, E. (1990). The Variance Gamma (V.G.) model for share market returns. Journal of Business, 63(4), 511–524.

    Article  Google Scholar 

  37. Madan, D. B., Carr, P., & Chang, E. C. (1998). The Variance Gamma process and option pricing. The European Financial Review, 2(1), 79–105.

    Article  Google Scholar 

  38. Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3(1–2), 125–144.

    Article  Google Scholar 

  39. Mundaca, G. (2001). Central bank interventions and exchange rate band regimes. Journal of International Money and Finance, 20(5), 677–700.

    Article  Google Scholar 

  40. Mundaca, G. (2011). How does public information on central bank intervention strategies affect exchange rate volatility? The case of Peru. World Bank’s policy research working paper series 5579.

  41. Mundaca, G., & Øksendal, B. (1998). Optimal stochastic intervention control with application to the exchange rate. Journal of Mathematical Economics, 29(2), 225–243.

    Article  Google Scholar 

  42. Nascimento, J., & Powell, W. (2010). Dynamic programming models and algorithms for the mutual fund cash balance problem. Management Science, 56(5), 801–815.

    Article  Google Scholar 

  43. Neave, E. H. (1970). The stochastic cash balance problem with fixed costs for increases and decreases. Management Science, 16(7), 472–490.

    Article  Google Scholar 

  44. Nieuwland, F. G., Verschoor, W. F., & Wolff, C. C. (1994). Stochastic trends and jumps in EMS exchange rates. Journal of International Money and Finance, 13(6), 699–727.

    Article  Google Scholar 

  45. Nirei, M., & Sushko, V. (2011). Jumps in foreign exchange rates and stochastic unwinding of carry trades. International Review of Economics & Finance, 20(1), 110–127.

    Article  Google Scholar 

  46. Øksendal, B., & Sulem, A. (2007). Applied stochastic control of jump diffusions. Berlin: Springer.

    Book  Google Scholar 

  47. Park, K., Ahn, C. M., & Fujihara, R. (1993). Optimal hedged portfolios: The case of jump-diffusion risks. International Journal of Monetary Economics and Finance, 12(5), 493–510.

    Google Scholar 

  48. Schoutens, W. (2003). Lévy processes in finance: Pricing financial derivatives. Chichester, UK: Wiley.

    Book  Google Scholar 

  49. Schoutens, W., & Teugels, J. L. (1998). Lévy processes, polynomials and martingales. Communications in Statistics-Stochastic Models, 14(1–2), 335–349.

    Article  Google Scholar 

  50. Svensson, L. (1992). The foreign exchange risk premium in a target zone with devaluation risk. Journal of International Economics, 33(1–2), 21–40.

    Article  Google Scholar 

  51. White, H. (1982). Maximum likelihood estimation of misspecified models. Econometrica, 50(1), 1–25.

    Article  Google Scholar 

  52. Wilfling, B. (2009). Volatility regime-switching in European exchange rates prior to monetary unification. Journal of International Money and Finance, 28(2), 240–270.

    Article  Google Scholar 

  53. Yu, J. (2007). Closed form likelihood approximation and estimation of jump-diffusions with application to the realignment risk of the Chinese Yuan. Journal of Econometrics, 141(2), 1245–1280.

    Article  Google Scholar 

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Acknowledgments

The authors thank the guest editors Jun-ya Gotoh and Stan Uryasev for their contribution, and the anonymous reviewers for their valuable comments and suggestions. Moreover, the authors are grateful to Alain Bensoussan for his thoughtful suggestions that resulted in the improvement of this paper. We also would like to thank participants of the session of OM-Finance interface at INFORMS Annual Meeting 2011 in Charlotte, session of Financial Services Section (Best Student Research Paper Competition) at INFORMS Annual Meeting 2012 in Phoenix, OM seminar of the Jindal School of Management at the University of Texas Dallas, Brown Bag Lunch Seminar of the Swiss Finance Institute, and Advances of Computational Economics & Finance Seminar of the Institute of Operations Research at the University of Zurich for their helpful comments.

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Correspondence to Sandun Perera.

Appendix: Proof of the verification theorem

Appendix: Proof of the verification theorem

Let \(\phi \) be a solution of the QIVI; then, \(\phi \in C^{1}\). However, using (i)–(iii) and the Approximation Theorem of Øksendal and Sulem (2007), we can and will assume that \(\phi \in C^{2}\). Now, let \(u= (\tau _{1},\tau _{2},\ldots \,,\tau _{j},\ldots \,;(i_{1},\zeta _{1}),(i_{2},\zeta _{2}),\ldots \,,(i_{j},\zeta _{j}),\ldots )\in \mathcal {U}\) with \(\tau _{0}=0\), and define \(\phi _{d}\left( X_{x}^{(u)}(s)\right) =e^{-rs}\phi \left( X_{x}^{(u)}(s)\right) \) for \(s \ge 0\). Then, since \(T^{i_j}_{j}\) is independent of \(X_{x}^{(u)}\) for \(j \ge 1\) and \(i_j \in \{1,2,\ldots ,n\}\), we have

$$\begin{aligned} E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j}+T^{i_{j}}_{j})\right) \bigg \vert T^{i_{j}}_{j}=t\right] =E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j}+t)\right) \right] , \end{aligned}$$
(24)

where \(t \ge 0\).

Next, we derive an expression for the right-hand side of (24). For this, consider \(g(s,X_{x}^{(u)}(s)):=e^{-rs}\phi \left( X_{x}^{(u)}(s)\right) =\phi _{d}\left( X_{x}^{(u)}(s)\right) \) and observe that g(sy) is \(C^1\) w.r.t. \(s \ge 0\) (i.e., for a given \(y \in \mathbb {R}^{k}\), \(g(\cdot ,y)\) is a continuous real-valued function on \([0,\infty )\) with continuous first order derivatives) and \(C^2\) w.r.t. \(y \in \mathbb {R}^{k}\) (i.e., for a given \(s\ge 0\), \(g(s,\cdot )\) is a continuous real-valued function on \(\mathbb {R}^{k}\) with continuous second order derivatives). Therefore, applying the multi-dimensional Itô ’s formula (cf. Øksendal and Sulem (2007)) for g from \(\tau _{j}\) to \(\tau _{j}+t\), we obtainFootnote 5

$$\begin{aligned}&\phi _{d}\left( X_{x}^{(u)}(\tau _{j}+t)\right) =\phi _{d}\left( X_{x}^{(u)}(\tau _{j})\right) -r\int _{\tau _{j}}^{\tau _{j}+t}\phi _{d}\left( X_{x}^{(u)}(s)\right) ds\\&\quad + \int _{\tau _{j}}^{\tau _{j}+t} \sum _{p=1}^{k}\frac{\partial \phi _{d}}{\partial x_{p}}\left( X_{x}^{(u)}(s)\right) \left[ \mu ^{i_{j}}_{p}\left( X_{x}^{(u)}(s)\right) ds+\sigma ^{i_{j}}_{p}\left( X_{x}^{(u)}(s)\right) dB(s)\right] \\&\quad +\frac{1}{2} \int _{\tau _{j}}^{\tau _{j}+t} \sum _{p,q=1}^{k}(\sigma ^{i_{j}} (\sigma ^{i_{j}})^{T})_{pq}\left( X_{x}^{(u)}(s)\right) \frac{\partial ^{2} \phi _{d}}{\partial x_{p} \partial x_{q}}\left( X_{x}^{(u)}(s)\right) ds\\&\quad +\int _{\tau _{j}}^{\tau _{j}+t} \sum _{p=1}^{l} \int _{\mathbb {R}} \{ \phi _{d}\left( X_{x}^{(u)}(s-)+(\gamma ^{i_{j}})^{(p)}(X_{x}^{(u)}(s-),z_{p})\right) -\phi _{d}\left( X_{x}^{(u)}(s-)\right) \\&\qquad \qquad \quad - \sum _{q=1}^{k} (\gamma ^{i_{j}})_{q}^{(p)}\left( X_{x}^{(u)}(s-),z_{p}\right) \frac{\partial \phi _{d}}{\partial x_{q}}\left( X_{x}^{(u)}(s-)\right) \} \nu _{p}(dz_{p})ds\\&\quad + \int _{\tau _{j}}^{\tau _{j}+t}\sum _{p=1}^{l}\int _{\mathbb {R}} \left\{ \phi _{d}\left( X_{x}^{(u)}(s-)+(\gamma ^{i_{j}})^{(p)}(X_{x}^{(u)}(s-),z_{p})\right) \right. \\&\qquad \qquad \quad \left. -\,\phi _{d}\left( X_{x}^{(u)}(s-)\right) \right\} {\widetilde{N}_{p}}(ds,dz_{p}). \end{aligned}$$

Rearranging the terms in the above equation then yields

$$\begin{aligned} \phi _{d}\left( X_{x}^{(u)}(\tau _{j}+t)\right)= & {} \phi _{d}\left( X_{x}^{(u)}(\tau _{j})\right) + \int _{\tau _{j}}^{\tau _{j}+t} \sum _{p=1}^{k}\frac{\partial \phi _{d}}{\partial x_{p}}\left( X_{x}^{(u)}(t)\right) \sigma ^{i_{j}}_{p}\left( X_{x}^{(u)}(s)\right) dB(s)\nonumber \\&+\int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X_{x}^{(u)}(s)\right) ds\nonumber \\&+ \int _{\tau _{j}}^{\tau _{j}+t}\sum _{p=1}^{l}\int _{\mathbb {R}} \left\{ \phi _{d}\left( X_{x}^{(u)}(s-)+(\gamma ^{i_{j}})^{(p)}(X_{x}^{(u)}(s-),z_{p})\right) \right. \nonumber \\&\qquad \qquad \quad \left. -\,\phi _{d}\left( X_{x}^{(u)}(s-)\right) \right\} {\widetilde{N}_{p}}(ds,dz_{p}), \end{aligned}$$
(25)

where, for \(x \in \mathbb {R}^{k}\),

$$\begin{aligned} A_{i_{j}}\phi _{d}(x):= & {} \displaystyle \sum _{p=1}^{k} \mu ^{i_{j}}_{p}(x)\frac{\partial \phi _{d}}{\partial x_{p}}(x)+\frac{1}{2} \displaystyle \sum _{p,q=1}^{k}(\sigma ^{i_{j}} (\sigma ^{i_{j}})^{T})_{pq}(x) \frac{\partial ^{2} \phi _{d}}{\partial x_{p} \partial x_{q}}(x)-r\phi _{d}(x)\\&+\displaystyle \sum _{p=1}^{l} \int _{\mathbb {R}} \{ \phi _{d}(x\!+\!(\gamma ^{i_{j}})^{(p)} (x,z_{j}))\!-\!\phi _{d}(x)\!-\!\nabla \phi _{d}(x)\cdot (\gamma ^{i_{j}})^{(p)}(x,z_{p})\}\nu _{p}(dz_{p}). \end{aligned}$$

Thus, since

$$\begin{aligned} E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t} \sum _{p=1}^{k}\frac{\partial \phi _{d}}{\partial x_{p}}\left( X^{(u)}(t)\right) \sigma ^{i_{j}}_{p}\left( X^{(u)}(s)\right) dB(s)\right] =0 \end{aligned}$$

and

$$\begin{aligned}&E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}\sum _{p=1}^{l}\int _{\mathbb {R}} \left\{ \phi _{d}( X^{(u)}(s-)+(\gamma ^{i_{j}})^{(p)}(X^{(u)}(s),z_{p}))\right. \right. \\&\left. \left. \qquad -\,\phi _{d}\left( X^{(u)}(s-)\right) \right\} {\widetilde{N}_{p}}(ds,dz_{p}) \right] =0, \end{aligned}$$

taking the expectation of both sides of (25), we have

$$\begin{aligned} E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j}+t)\right) \right] =E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j})\right) \right] +E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\right] . \end{aligned}$$
(26)

Now, note that, from \(\tau _j+t\) to \(\tau _{j+1}\), the process \(X^{(u)}\) follows the original process X. Therefore, using an argument similar to that leading up to (26), it can be easily shown that

$$\begin{aligned} E^{x}\left[ \phi _{d}\left( \check{X}^{(u)}(\tau _{j+1}-)\right) \right] =E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j}+t)\right) \right] +E^{x}\left[ \int _{\tau _{j}+t}^{\tau _{j+1}}A\phi _{d}\left( X^{(u)}(s)\right) ds\right] , \end{aligned}$$
(27)

where \(\check{X}_{x}^{(u)}(\tau _{j+1}-)=X_{x}^{(u)}(\tau _{j+1}-)+\triangle _{N}X_{x}(\tau _{j+1})\).

It follows from (26) and (27) that

$$\begin{aligned} E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j})\right) -\phi _{d}\left( \check{X}^{(u)}(\tau _{j+1}-)\right) \right]&= -E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\right] \nonumber \\&\quad \,\,-E^{x}\left[ \int _{\tau _{j}+t}^{\tau _{j+1}}A\phi _{d}\left( X^{(u)}(s)\right) ds\right] . \end{aligned}$$
(28)

Next, observe that

$$\begin{aligned}&\int _{0}^{\infty }E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j})\right) -\phi _{d}\left( \check{X}^{(u)}(\tau _{j+1}-)\right) \right] dF_{i_{j}}(t)\\&\quad =E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j})\right) -\phi _{d}\left( \check{X}^{(u)}(\tau _{j+1}-)\right) \right] \int _{0}^{\infty }dF_{i_{j}}(t)\\&\quad =E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j})\right) -\phi _{d}\left( \check{X}^{(u)}(\tau _{j+1}-)\right) \right] , \end{aligned}$$

where the first equality is due the independence. Therefore, integrating (28) with respect to the probability measure induced by \(T^{i_{j}}\) on \([0,\infty )\) first and then summing from \(j=1\) to \(j=m\), we obtain

$$\begin{aligned}&\sum _{j=1}^{m}E^{x}\left[ \phi _{d}( X^{(u)}(\tau _{j}))-\phi _{d}\left( \check{X}^{(u)}(\tau _{j+1}-)\right) \right] \nonumber \\&\quad =-\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t)\nonumber \\&\qquad -\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}+t}^{\tau _{j+1}}A\phi _{d}\left( X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t). \end{aligned}$$
(29)

Moreover, using an argument similar to that leading up to (26) again, it can be easily observed that

$$\begin{aligned} E^{x}\left[ \phi _{d}\left( \check{X}^{(u)}(\tau _{1}-)\right) \right]&=E^{x}\left[ \phi _{d}\left( X_x^{(u)}(0)\right) \right] + E^{x}\left[ \int _{0}^{\tau _{1}}A\phi _{d}\left( X^{(u)}(s)\right) ds\right] \nonumber \\&=\phi \left( x\right) + E^{x}\left[ \int _{0}^{\tau _{1}}A\phi _{d}\left( X^{(u)}(s)\right) ds\right] . \end{aligned}$$
(30)

Combining (29) and (30) yield

$$\begin{aligned}&\phi \left( x\right) +\sum _{j=1}^{m}E^{x}\left[ \phi _{d}( X^{(u)}(\tau _{j}))-\phi _{d}\left( \check{X}^{(u)}(\tau _{j+1}-)\right) \right] -E^{x}\left[ \phi _{d}\left( \check{X}^{(u)}(\tau _{1}-)\right) \right] \\&\quad =-\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t)-E^{x}\left[ \int _{0}^{\tau _{1}}A\phi _{d}\left( X^{(u)}(s)\right) ds\right] \\&\qquad \quad -\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}+t}^{\tau _{j+1}}A\phi _{d}\left( X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t). \end{aligned}$$

Equivalently, we have

$$\begin{aligned}&\phi (x)+\sum _{j=1}^{m} E^{x}\left[ \phi _{d}(X^{(u)}(\tau _{j}))-\phi _{d}(\check{X}^{(u)}(\tau _{j}-)) \right] -E^{x}\left[ \phi _{d}(\check{X}^{(u)}(\tau _{m+1}-))\right] \nonumber \\&\quad =\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}\left\{ -A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) \right\} ds\right] dF_{i_{j}}(t)\nonumber \\&\qquad +E^{x}\left[ \int _{0}^{\tau _{1}}\left\{ -A\phi _{d}\left( X^{(u)}(s)\right) \right\} ds\right] \nonumber \\&\qquad +\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}+t}^{\tau _{j+1}}\left\{ -A\phi _{d}\left( X^{(u)}(s)\right) \right\} ds\right] dF_{i_{j}}(t). \end{aligned}$$
(31)

Now, since \(\phi \) satisfies the QIVI, we have \(f \ge -A\phi \). Therefore, \(f_{d} \ge -A\phi _{d}\) where \(f_{d}( X^{(u)}(s))=e^{-rs}f(X^{(u)}(s))\). Then, it follows from (31) that

$$\begin{aligned}&\phi (x)+\sum _{j=1}^{m} E^{x}\left[ \phi _{d}(X^{(u)}(\tau _{j}))-\phi _{d}(\check{X}^{(u)}(\tau _{j}-)) \right] -E^{x}\left[ \phi _{d}(\check{X}^{(u)}(\tau _{m+1}-))\right] \nonumber \\&\quad \le \sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}\left\{ -A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) \right\} ds\right] dF_{i_{j}}(t)+E^{x}\left[ \int _{0}^{\tau _{1}}f_{d}\left( X^{(u)}(s)\right) ds\right] \nonumber \\&\qquad +\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}+t}^{\tau _{j+1}}f_{d}\left( X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t). \end{aligned}$$
(32)

Next, since \(T^{i_j}\) is independent of \(X_{x}^{i_j}\) for \(i_j \in \{1,2,\ldots ,n\}\), by the definition of \(\mathcal {M}_{r}\), we have

$$\begin{aligned} \mathcal {M}_{r}\phi (\check{X}_{x}^{(u)}(\tau _{j}-))\le & {} K_{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})\\&+\int _{0}^{\infty }E^{\Gamma _{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})}\left[ \int _{0}^{t}e^{-rs}f\left( X^{i_{j}}(s)\right) ds+e^{-rt}\phi \left( X^{i_{j}}(t)\right) \right] dF_{i_{j}}(t) \end{aligned}$$

But \(X_{x}^{(u)}(\tau _{j})=\Gamma _{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})\), therefore

$$\begin{aligned} \mathcal {M}_{r}\phi (\check{X}_{x}^{(u)}(\tau _{j}\!-\!))\!\le & {} \! K_{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})\!+\!\int _{0}^{\infty }E^{X_{x}^{(u)}(\tau _{j})}\left[ \int _{0}^{t}e^{-rs}f\left( X^{i_{j}}(s)\!\right) ds\right] dF_{i_{j}}(t)\nonumber \\&+\int _{0}^{\infty }E^{X_{x}^{(u)}(\tau _{j})}\left[ e^{-rt}\phi \left( X^{i_{j}}(t)\right) \right] dF_{i_{j}}(t). \end{aligned}$$
(33)

Now, using the strong Markov property and the fact that \(X_{x}^{(u)}(s)=X^{i_{j}}_{X_{x}^{(u)}(\tau _{j})}(s)\) for \(\tau _{j}\le s \le \tau _{j}+t\), we observe that

$$\begin{aligned}&\int _{0}^{\infty }E^{X_{x}^{(u)}(\tau _{j})}\left[ \int _{0}^{t}e^{-rs}f\left( X^{i_{j}}(s)\right) ds\right] dF_{i_{j}}(t)\\&\quad = \int _{0}^{\infty }\left\{ \int _{0}^{t}e^{-rs}E^{X_{x}^{(u)}(\tau _{j})}\left[ f\left( X^{i_{j}}(s)\right) \right] ds\right\} dF_{i_{j}}(t) \\&\quad = \int _{0}^{\infty }\left\{ \int _{0}^{t}e^{-rs}E^{x}\left[ f\left( X^{(u)}(\tau _{j}+s)\right) \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] ds\right\} dF_{i_{j}}(t) \\&\quad = \int _{0}^{\infty }E^{x}\left[ \int _{0}^{t}e^{-rs}f\left( X^{(u)}(\tau _{j}+s)\right) ds\, \bigg \vert \, \mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t) \\&\quad = \int _{0}^{\infty } e^{r \tau _{j}}E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}e^{-rs}f\left( X^{(u)}(s)\right) ds \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t) \end{aligned}$$

and

$$\begin{aligned}&\int _{0}^{\infty }E^{X_{x}^{(u)}(\tau _{j})}\left[ e^{-rt}\phi \left( X^{i_{j}}(t)\right) \right] dF_{i_{j}}(t)\\&\quad = \int _{0}^{\infty }E\left[ e^{-rt}\phi \left( X^{i_{j}}_{X^{(u)}_{x}(\tau _{j})}(\tau _{j}+t)\right) \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t)\\&\quad = \int _{0}^{\infty }E^{x}\left[ e^{-rt}\phi \left( X^{(u)}(\tau _{j}+t)\right) \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t). \end{aligned}$$

Then, from (33), we have

$$\begin{aligned} \mathcal {M}_{r}\phi (\check{X}_{x}^{(u)}(\tau _{j}-))\le & {} K_{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})\nonumber \\&+\int _{0}^{\infty }e^{r\tau _{j}} E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}e^{-rs}f\left( X^{(u)}(s)\right) ds\,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t)\\&+\int _{0}^{\infty }E^{x}\left[ e^{-rt}\phi \left( X^{(u)}(\tau _{j}+t)\right) \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t). \end{aligned}$$

Multiplying the above equation by \(e^{-r \tau _{j}}\) yields

$$\begin{aligned} \mathcal {M}_{r}\phi _{d}(\check{X}_{x}^{(u)}(\tau _{j}-))\le & {} e^{-r \tau _{j}}K_{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})\nonumber \\&+\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}f_{d}\left( X^{(u)}(s)\right) ds \,\bigg \vert \, \mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t)\nonumber \\&+\int _{0}^{\infty }E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j}+t)\right) \,\bigg \vert \, \mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t). \end{aligned}$$
(34)

Next, taking the conditional expectation of both sides of (25) first and then using an argument similar to that leading up to (26) again, we obtain

$$\begin{aligned}&E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j}+t)\right) \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] \nonumber \\&\quad =E^{x}\left[ \phi _{d}\left( X_{x}^{(u)}(\tau _{j})\right) \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] + E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] .\nonumber \\&\quad =\phi _{d}\left( X_{x}^{(u)}(\tau _{j})\right) + E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] , \end{aligned}$$
(35)

where the second equality is due to the \(\mathcal {F}_{\tau _{j}}\)-measurability of \(\phi _{d}\left( X_{x}^{(u)}(\tau _{j})\right) \).

Integrating (35) with respect to the probability measure induced by \(T^{i_{j}}\) on \([0,\infty )\), we have

$$\begin{aligned}&\int _{0}^{\infty }E^{x}\left[ \phi _{d}\left( X^{(u)}(\tau _{j}+t)\right) \,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t)\nonumber \\&\quad =\phi _{d}\left( X_{x}^{(u)}(\tau _{j})\right) +\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t). \end{aligned}$$
(36)

Substituting (36) into (34) and subtracting \(\phi _{d}(\check{X}_{x}^{(u)}(\tau _{j}-))\) from both sides then yield

$$\begin{aligned} \mathcal {M}_{r}\phi _{d}(\check{X}_{x}^{(u)}(\tau _{j}-))-\phi _{d}(\check{X}_{x}^{(u)}(\tau _{j}-))\le & {} e^{-r \tau _{j}} K_{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})\\&+\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}f_{d}\left( X^{(u)}(s)\right) ds\,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t)\\&+\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\,\bigg \vert \,\mathcal {F}_{\tau _{j}}\right] dF_{i_{j}}(t)\\&+\,\phi _{d}\left( X_{x}^{(u)}(\tau _{j})\right) -\phi _{d}(\check{X}_{x}^{(u)}(\tau _{j}-)). \end{aligned}$$

Now, taking the expectation of both sides of the above inequality first and then summing the resulting inequality from \(j=1\) to \(j=m\), we obtain

$$\begin{aligned}&\sum _{j=1}^{m}E^{x}\left[ \mathcal {M}_{r}\phi _{d}(\check{X}_{x}^{(u)}(\tau _{j}-))-\phi _{d}(\check{X}_{x}^{(u)}(\tau _{j}-))\right] \nonumber \\&\quad \le E^{x}\left[ \sum _{j=1}^{m}e^{-r \tau _{j}}K_{i_{j}}(\check{X}_{x}^{(u)}(\tau _{j}-),\zeta _{j})\right] \nonumber \\&\qquad +\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}f_{d}\left( X^{(u)}(t)\right) dt\right] dF_{i_{j}}(t)\nonumber \\&\qquad +\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(t)\right) dt\right] dF_{i_{j}}(t)\nonumber \\&\qquad +\sum _{j=1}^{m} E^{x}\left[ \phi _{d}(X^{(u)}(\tau _{j}))-\phi _{d}(\check{X}^{(u)}(\tau _{j}-)) \right] . \end{aligned}$$
(37)

Finally, it follows from (32) and (37) that

$$\begin{aligned}&\phi (x)+\sum _{j=1}^{m} E^{x}\left[ \mathcal {M}_{r}\phi _{d}(\check{X}^{(u)}(\tau _{j}-))-\phi _{d}(\check{X}^{(u)}(\tau _{j}-))\right] \nonumber \\&\quad \le E^{x}\left[ \phi _{d}(\check{X}^{(u)}(\tau _{m+1}-))\right] -\sum _{j=1}^{m} E^{x}\left[ \phi _{d}(X^{(u)}(\tau _{j}))-\phi _{d}(\check{X}^{(u)}(\tau _{j}-)) \right] \nonumber \\&\quad \quad -\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t)+E^{x}\left[ \int _{0}^{\tau _{1}}f_{d}\left( X^{(u)}(s)\right) ds\right] \nonumber \\&\quad \quad +\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}+t}^{\tau _{j+1}}f\left( X^{(u)}(t)\right) dt\right] dF_{i_{j}}(t)\nonumber \\&\quad \quad +E^{x}\left[ \sum _{j=1}^{m}e^{-r\tau _{j}}K_{i_{j}}(\check{X}^{(u)}(\tau _{j}-),\zeta _{j})\right] \!+\! \sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}f\left( \!X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t)\nonumber \\&\quad \quad +\sum _{j=1}^{m}\int _{0}^{\infty }E^{x}\left[ \int _{\tau _{j}}^{\tau _{j}+t}A_{i_{j}}\phi _{d}\left( X^{(u)}(s)\right) ds\right] dF_{i_{j}}(t)\nonumber \\&\qquad +\sum _{j=1}^{m} E^{x}\left[ \phi _{d}(X^{(u)}(\tau _{j}))-\phi _{d}(\check{X}^{(u)}(\tau _{j}-)) \right] \nonumber \\&\quad = E^{x}\left[ \int _{0}^{\tau _{m+1}}f_{d}(X^{(u)}(s))ds\right] +E^{x}\left[ \phi _{d}(\check{X}^{(u)}(\tau _{m+1}-))\right] \nonumber \\&\qquad +E^{x}\left[ \sum _{j=1}^{m}e^{-r\tau _{j}}K_{i_{j}}(\check{X}^{(u)}(\tau _{j}-),\zeta _{j})\right] \nonumber \\&\quad = E^{x}\left[ \int _{0}^{\tau _{m+1}}e^{-rs}f(X^{(u)}(s))ds+e^{-r\tau _{m+1}}\phi (\check{X}^{(u)}(\tau _{m+1}-))\right. \nonumber \\&\qquad +\left. \sum _{j=1}^{m}e^{-r\tau _{j}}K_{i_{j}}(\check{X}^{(u)}(\tau _{j}-),\zeta _{j}) \right] .\qquad \end{aligned}$$
(38)

Since the second term on the left-hand side of (38) is non-negative, letting \(m\rightarrow \infty \) in (38) and using conditions (iv)-(v), we have

$$\begin{aligned} \phi (x)\le E^{x}\left[ \int _{0}^{\infty }e^{-rt}f(X^{(u)}(t))dt+\sum _{j=1}^{\infty }e^{-r\tau _{j}}K_{i_{j}}(\check{X}^{(u)}(\tau _{j}-),\zeta _{j}) \right] =J^{(u)}(x). \end{aligned}$$
(39)

Hence, we have that \(\phi (x)\le \varPhi (x)=\inf \{ J^{(u)}(x); \ u \in \mathcal{U}\}\).

Moreover, if the QIVI-control \(\hat{u}\) corresponding to \(\phi \) is admissible, then we can apply the above argument to \(\hat{u}= (\hat{\tau }_{1},\hat{\tau }_{2},\ldots \,,\hat{\tau }_{j},\ldots \,;(\hat{i}_{1},\hat{\zeta }_{1}),(\hat{i}_{2},\hat{\zeta }_{2}),\ldots \,,(\hat{i}_{j},\hat{\zeta }_{j}),\ldots )\). Now, since \(A\phi +f=0\) on the boundary of \(\mathbb {C}\), we obtain the equality in (32) and by the choices of \(\zeta _{i}=\hat{\zeta }_{i}\) and \(i_{j}=\hat{i}_{j}\), we also have the equality in (34) and (37). Then, since condition (i) implies that the second term on the left-hand side of (38) goes to zero as \(m \rightarrow \infty \), we achieve the desired equality in (39). Therefore, we have \(\phi (x)=J^{(\hat{u})}(x)\). Hence, \(\phi (x)=\varPhi (x)\) and \(u^{*}=\hat{u}\). \(\square \)

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Perera, S., Buckley, W. & Long, H. Market-reaction-adjusted optimal central bank intervention policy in a forex market with jumps. Ann Oper Res 262, 213–238 (2018). https://doi.org/10.1007/s10479-016-2297-y

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Keywords

  • Optimal central-bank/government intervention policy
  • Financial market reactions
  • Jump diffusions
  • Stochastic control