Stability for gains from large investors’ strategies in \(M_{1}/J_{1}\) topologies. (English) Zbl 1459.60121

Summary: We prove continuity of a controlled SDE solution in Skorokhod’s \(M_{1}\) and \(J_{1}\) topologies and also uniformly, in probability, as a nonlinear functional of the control strategy. The functional comes from a finance problem to model price impact of a large investor in an illiquid market. We show that \(M_{1}\)-continuity is the key to ensure that proceeds and wealth processes from (self-financing) càdlàg trading strategies are determined as the continuous extensions for those from continuous strategies. We demonstrate by examples how continuity properties are useful to solve different stochastic control problems on optimal liquidation and to identify asymptotically realizable proceeds.


60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60F17 Functional limit theorems; invariance principles
91G10 Portfolio theory
Full Text: DOI arXiv Euclid


[1] Alfonsi, A., Fruth, A. and Schied, A. (2010). Optimal execution strategies in limit order books with general shape functions. Quant. Finance10 143–157. · Zbl 1185.91199
[2] Bank, P. and Baum, D. (2004). Hedging and portfolio optimization in financial markets with a large trader. Math. Finance14 1–18. · Zbl 1119.91040
[3] Becherer, D., Bilarev, T. and Frentrup, P. (2015). Multiplicative limit order markets with transient impact and zero spread. Available at arXiv:1501.01892v1. · Zbl 1403.35330
[4] Becherer, D., Bilarev, T. and Frentrup, P. (2018). Optimal asset liquidation with multiplicative transient price impact. Appl. Math. Optim. To appear. DOI:10.1007/s00245-017-9418-0. · Zbl 1403.35330
[5] Becherer, D., Bilarev, T. and Frentrup, P. (2018). Optimal liquidation under stochastic liquidity. Finance Stoch.22 39–68. · Zbl 1391.91164
[6] Billingsley, P. (1999). Convergence of Probability Measures. Wiley Series in Probability and Statistics. New York: Wiley. · Zbl 0944.60003
[7] Blümmel, T. and Rheinländer, T. (2017). Financial markets with a large trader. Ann. Appl. Probab.27 3735–3786. · Zbl 1408.91191
[8] Borodin, A.N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae. Probability and Its Applications. Basel: Birkhäuser. · Zbl 1012.60003
[9] Bouchard, B., Loeper, G. and Zou, Y. (2016). Almost-sure hedging with permanent price impact. Finance Stoch.20 741–771. · Zbl 1369.91172
[10] Carr, P., Geman, H., Madan, D.B. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. J. Bus.75 305–332.
[11] Çetin, U., Jarrow, R.A. and Protter, P.E. (2004). Liquidity risk and arbitrage pricing theory. Finance Stoch.8 311–341. · Zbl 1064.60083
[12] Çetin, U., Soner, H.M. and Touzi, N. (2010). Option hedging for small investors under liquidity costs. Finance Stoch.14 317–341. · Zbl 1226.91072
[13] Chan, L.K.C. and Lakonishok, J. (1995). The behavior of stock prices around institutional trades. J. Finance50 1147–1174.
[14] Cont, R. and De Larrard, A. (2013). Price dynamics in a Markovian limit order market. SIAM J. Financial Math.4 1–25. · Zbl 1288.91092
[15] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Amsterdam: North-Holland. · Zbl 0494.60002
[16] Esche, F. and Schweizer, M. (2005). Minimal entropy preserves the Lévy property: How and why. Stochastic Process. Appl.115 299–327. · Zbl 1075.60049
[17] Friz, P. and Chevyrev, I. (2018). Canonical RDEs and general semimartingales as rough paths. Ann. Probab. To appear. · Zbl 1475.60203
[18] Guo, X. and Zervos, M. (2015). Optimal execution with multiplicative price impact. SIAM J. Financial Math.6 281–306. · Zbl 1310.93083
[19] Henderson, V. and Hobson, D. (2011). Optimal liquidation of derivative portfolios. Math. Finance21 365–382. · Zbl 1215.91073
[20] Hindy, A., Huang, C.-F. and Kreps, D. (1992). On intertemporal preferences in continuous time: The case of certainty. J. Math. Econom.21 401–440. · Zbl 0765.90023
[21] Jacod, J. and Shiryaev, A.N. (2003). Limit Theorems for Stochastic Processes. Berlin: Springer. · Zbl 1018.60002
[22] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd ed. Probability and Its Applications. New York: Springer. · Zbl 0996.60001
[23] Kallsen, J. and Shiryaev, A.N. (2002). The cumulant process and Esscher’s change of measure. Finance Stoch.6 397–428. · Zbl 1035.60042
[24] Kardaras, C. (2013). On the closure in the Emery topology of semimartingale wealth-process sets. Ann. Appl. Probab.23 1355–1376. · Zbl 1408.91196
[25] Klein, O., Maug, E. and Schneider, C. (2017). Trading strategies of corporate insiders. J. Financ. Mark.34 48–68.
[26] Kurtz, T.G., Pardoux, É. and Protter, P. (1995). Stratonovich stochastic differential equations driven by general semimartingales. Ann. Inst. Henri Poincaré Probab. Stat.31 351–377. · Zbl 0823.60046
[27] Kurtz, T.G. and Protter, P.E. (1996). Weak convergence of stochastic integrals and differential equations. In Probabilistic Models for Nonlinear Partial Differential Equations (D. Talay and L. Tubaro, eds.). Lecture Notes in Math.1627 1–41. Berlin: Springer. · Zbl 0862.60041
[28] Løkka, A. (2014). Optimal liquidation in a limit order book for a risk-averse investor. Math. Finance24 696–727. · Zbl 1314.91250
[29] Lorenz, C. and Schied, A. (2013). Drift dependence of optimal trade execution strategies under transient price impact. Finance Stoch.17 743–770. · Zbl 1278.91065
[30] Marcus, S.I. (1981). Modeling and approximation of stochastic differential equations driven by semimartingales. Stochastics4 223–245. · Zbl 0456.60064
[31] Obizhaeva, A. and Wang, J. (2013). Optimal trading strategy and supply/demand dynamics. J. Financ. Mark.16 1–32.
[32] Pang, G., Talreja, R. and Whitt, W. (2007). Martingale proofs of many-server heavy-traffic limits for Markovian queues. Probab. Surv.4 193–267. · Zbl 1189.60067
[33] Pang, G. and Whitt, W. (2010). Continuity of a queueing integral representation in the \(M_{1}\) topology. Ann. Appl. Probab.20 214–237. · Zbl 1186.60098
[34] Predoiu, S., Shaikhet, G. and Shreve, S. (2011). Optimal execution in a general one-sided limit-order book. SIAM J. Financial Math.2 183–212. · Zbl 1222.91062
[35] Protter, P.E. (2005). Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability21. Berlin: Springer. Second edition. Version 2.1, corrected third printing.
[36] Roch, A.F. and Soner, H.M. (2013). Resilient price impact of trading and the cost of illiquidity. Int. J. Theor. Appl. Finance16 1350037 (27 pages). · Zbl 1295.91047
[37] Roch, A.F. (2011). Liquidity risk, price impacts and the replication problem. Finance Stoch.15 399–419. · Zbl 1303.91096
[38] Skorokhod, A.V. (1956). Limit theorems for stochastic processes. Theory Probab. Appl.1 261–290.
[39] Whitt, W. (2002). Stochastic-Process Limits: An Introduction to Stochastic-Process Limits and Their Application to Queues. Springer Series in Operations Research. New York: Springer. · Zbl 0993.60001
[40] Wong, E. and Zakai, M. (1965). On the convergence of ordinary integrals to stochastic integrals. Ann. Math. Stat.36 1560–1564. · Zbl 0138.11201
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.