An Optimal Dividend Problem with Capital Injections over a Finite Horizon

Ferrari G, Schuhmann P (2019)
SIAM Journal on Control and Optimization 57(4): 2686-2719.

Zeitschriftenaufsatz | Veröffentlicht| Englisch
 
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Abstract / Bemerkung
In this paper, we propose and solve an optimal dividend problem with capital injections over a finite time horizon. The surplus dynamics obeys a linearly controlled drifted Brownian motion that is reflected at the origin: dividends give rise to time-dependent instantaneous marginal profits, whereas capital injections are subject to time-dependent instantaneous marginal costs. The aim is to maximize the sum of a liquidation value at terminal time and of the total expected profits from dividends, net of the total expected costs for capital injections. Inspired by the study of El Karoui and Karatzas [Integration of the optimal risk in a stopping problem with absorption, in Seminaire de Probabilites XXIII, Springer, Berlin, 1989, pp. 405-420] on reflected follower problems, we relate the optimal dividend problem with capital injections to an optimal stopping problem for a drifted Brownian motion that is absorbed at the origin. We show that whenever the optimal stopping rule is triggered by a time-dependent boundary, the value function of the optimal stopping problem gives the derivative of the value function of the optimal dividend problem. Moreover, the optimal dividend strategy is also triggered by the moving boundary of the associated stopping problem. The properties of this boundary are then investigated in a case study in which instantaneous marginal profits and costs from dividends and capital injections are constants discounted at a constant rate.
Stichworte
optimal dividend problem; capital injections; singular stochastic; control; optimal stopping; free boundary
Erscheinungsjahr
2019
Zeitschriftentitel
SIAM Journal on Control and Optimization
Band
57
Ausgabe
4
Seite(n)
2686-2719
ISSN
0363-0129
eISSN
1095-7138
Page URI
https://pub.uni-bielefeld.de/record/2937742

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Ferrari G, Schuhmann P. An Optimal Dividend Problem with Capital Injections over a Finite Horizon. SIAM Journal on Control and Optimization. 2019;57(4):2686-2719.
Ferrari, G., & Schuhmann, P. (2019). An Optimal Dividend Problem with Capital Injections over a Finite Horizon. SIAM Journal on Control and Optimization, 57(4), 2686-2719. doi:10.1137/18M1184588
Ferrari, G., and Schuhmann, P. (2019). An Optimal Dividend Problem with Capital Injections over a Finite Horizon. SIAM Journal on Control and Optimization 57, 2686-2719.
Ferrari, G., & Schuhmann, P., 2019. An Optimal Dividend Problem with Capital Injections over a Finite Horizon. SIAM Journal on Control and Optimization, 57(4), p 2686-2719.
G. Ferrari and P. Schuhmann, “An Optimal Dividend Problem with Capital Injections over a Finite Horizon”, SIAM Journal on Control and Optimization, vol. 57, 2019, pp. 2686-2719.
Ferrari, G., Schuhmann, P.: An Optimal Dividend Problem with Capital Injections over a Finite Horizon. SIAM Journal on Control and Optimization. 57, 2686-2719 (2019).
Ferrari, Giorgio, and Schuhmann, Patrick. “An Optimal Dividend Problem with Capital Injections over a Finite Horizon”. SIAM Journal on Control and Optimization 57.4 (2019): 2686-2719.