Optimal price management in retail energy markets: an impulse control problem with asymptotic estimates

We consider a retailer who buys energy in the wholesale market and resells it to final consumers. The retailer has to decide when to intervene to change the price he asks to his customers, in order to maximize his income. We model the problem as an infinite-horizon stochastic impulse control problem. We characterize an optimal price strategy and provide analytical existence results for the equations involved. We then investigate the dependence on the intervention cost. In particular, we prove that the measure of the continuation region is asymptotic to the fourth root of the cost. Finally, we provide some numerical results and consider a suitable extension of the model.

[1]  Luis H. R. Alvarez,et al.  A Class of Solvable Impulse Control Problems , 2004 .

[2]  Luciano Campi,et al.  Nonzero-Sum Stochastic Differential Games with Impulse Controls: A Verification Theorem with Applications , 2016, Math. Oper. Res..

[3]  B. Øksendal,et al.  Applied Stochastic Control of Jump Diffusions , 2004, Universitext.

[4]  Peter P. Belobaba,et al.  Survey Paper - Airline Yield Management An Overview of Seat Inventory Control , 1987, Transp. Sci..

[5]  Xin Guo,et al.  Smooth Fit Principle for Impulse Control of Multidimensional Diffusion Processes , 2009, SIAM J. Control. Optim..

[6]  A. Shiryaev,et al.  Optimization of the flow of dividends , 1995 .

[7]  F. Zapatero,et al.  Optimal Central Bank Intervention in the Foreign Exchange Market , 1999 .

[8]  Feng Zhang,et al.  Stochastic differential games involving impulse controls , 2011 .

[9]  Ralf Korn,et al.  Portfolio optimisation with strictly positive transaction costs and impulse control , 1998, Finance Stochastics.

[10]  F. Zapatero,et al.  OPTIMAL DIVIDEND POLICY WITH MEAN‐REVERTING CASH RESERVOIR , 2007 .

[11]  B. Øksendal,et al.  NON-ROBUSTNESS OF SOME IMPULSE CONTROL PROBLEMS WITH RESPECT TO INTERVENTION COSTS , 2002 .

[12]  Michael Pinedo,et al.  Optimal Control of a Mean-Reverting Inventory , 2010, Oper. Res..

[13]  S. Federico,et al.  Irreversible investment with fixed adjustment costs: a stochastic impulse control approach , 2018, Mathematics and Financial Economics.

[14]  B. Øksendal.  Stochastic Control Problems where Small Intervention Costs Have Big Effects , 1999 .

[15]  Alain Bensoussan,et al.  Inventory control with an order-time constraint: optimality, uniqueness and significance , 2010, Ann. Oper. Res..

[16]  Haoyang Cao,et al.  Nonzero-sum stochastic games with impulse controls , 2019, 1901.08085.

[17]  Bernt Øksendal,et al.  Optimal stochastic intervention control with application to the exchange rate , 1998 .

[18]  Luis H. R. Alvarez,et al.  On the Optimal Stochastic Impulse Control of Linear Diffusions , 2008, SIAM J. Control. Optim..

[19]  Xin Guo,et al.  Impulse Control of Multidimensional Jump Diffusions , 2009, SIAM J. Control. Optim..

[20]  Pinar Keskinocak,et al.  Dynamic pricing in the presence of inventory considerations: research overview, current practices, and future directions , 2003, IEEE Engineering Management Review.

[21]  Yngve Willassen,et al.  The Stochastic Rotation Problem - A Generalization of faustmann's Formula to Stochastic Forest Growth , 1998 .

[22]  J. Michael Harrison,et al.  Impulse Control of Brownian Motion , 1983, Math. Oper. Res..

[23]  Jeffrey I. McGill,et al.  Revenue Management: Research Overview and Prospects , 1999, Transp. Sci..

[24]  Ralf Korn,et al.  Some applications of impulse control in mathematical finance , 1999, Math. Methods Oper. Res..

[25]  Lei Zhang,et al.  CLASSICAL AND IMPULSE STOCHASTIC CONTROL FOR THE OPTIMIZATION OF THE DIVIDEND AND RISK POLICIES OF AN INSURANCE FIRM , 2006 .

[26]  M. Jeanblanc-Picqué,et al.  Impulse Control Method and Exchange Rate , 1993 .

[27]  F. Benth,et al.  Stochastic modeling of financial electricity contracts , 2008 .

[28]  Giorgio Ferrari,et al.  On a strategic model of pollution control , 2017, Ann. Oper. Res..

[29]  Kumar Muthuraman,et al.  Impulse Control of Interest Rates , 2013, Oper. Res..

[30]  Luis H. R. Alvarez,et al.  Stochastic Forest Stand Value and Optimal Timber Harvesting , 2003, SIAM J. Control. Optim..