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A control problem with fuel constraint and Dawson-Watanabe superprocesses. (English) Zbl 1288.60100

The author is concerned with the problem of minimizing the functional \[ \mathrm{E}_{0,z}\Biggl[\int^T_0|\dot x(t)|^p \eta(Z_\tau)\,dt+ \int^T_0 |x(t)|^p A(dt)\Biggr] \] over adapted and absolutely continuous strategies \(x(t)\) satisfying \(x(0)= x_0\) and \(x(T)= 0\). Here \(p\in[2, \infty]\), \(\eta\) is a strictly positive function, and \(A\) a nonnegative additive functional of the time-inhomogeneous Markov process \(Z\) with \(Z_0= z\) a.s. \([\mathrm{P}_{0,z}]\). This control problem is related to the monotone follower problems with fuel constraint introduced by V. E. Benes, L. A. Shepp and H. S. Witsenhausen [Stochastics 4, 39–83 (1980; Zbl 0451.93068)]. It also occurs in mathematical finance when looking into strategies minimizing the cost of liquidating a given amount of stock within a certain interval of time, see [R. Almgren, SIAM J. Financ. Math. 3, No. 1, 163–181 (2012; Zbl 1256.49031)]. The author solves his problem using the log-Laplace transforms of \(J\)-functionals (as introduced by E. B. Dynkin [Probab. Theory Relat. Fields 90, No. 1, 1–36 (1991; Zbl 0727.60095)]) of superprocesses with not necessarily homogeneous branching parameters.
The solution is related to the solution of quasilinear parabolic PDEs of the form \[ v_t- (p- 1) \eta^{1/(p-1)} v^{1+ 1/(1-p)}+ a+ L_t v= 0,\;v(T, z)= \infty, \] where \(L_t\) is the generator of \(Z\). This is the type of equations solved by E. B. Dynkin [Ann. Probab. 20, No. 2, 942–962 (1992; Zbl 0756.60074)] by means of superprocesses.
Clearly, the direct probabilistic approach followed here is more elegant than the classical, often cumbersome approach via the HJB equation. As a byproduct, the author obtains sharp bounds on the blow-up behavior of the log-Laplace functionals, which are of interest in themselves.

MSC:

60J68 Superprocesses
93E20 Optimal stochastic control
60H20 Stochastic integral equations
60H30 Applications of stochastic analysis (to PDEs, etc.)
91G80 Financial applications of other theories
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[1] Almgren, R. (2012). Optimal trading with stochastic liquidity and volatility. SIAM J. Financial Math. 3 163-181. · Zbl 1256.49031 · doi:10.1137/090763470
[2] Almgren, R. and Chriss, N. (2000). Optimal execution of portfolio transactions. Journal of Risk 3 5-39.
[3] Beneš, V. E., Shepp, L. A. and Witsenhausen, H. S. (1980). Some solvable stochastic control problems. Stochastics 4 39-83. · Zbl 0451.93068 · doi:10.1080/17442508008833156
[4] Björk, T. and Murgoci, A. (2010). A general theory of Markovian time inconsistent stochastic control problems. · Zbl 1297.49038
[5] Dawson, D. A. and Perkins, E. A. (1991). Historical processes. Mem. Amer. Math. Soc. 93 iv+179. · Zbl 0754.60062 · doi:10.1090/memo/0454
[6] Dellacherie, C. and Meyer, P.-A. (1982). Probabilities and Potential. B. Theory of Martingales. North-Holland Mathematics Studies 72 . North-Holland, Amsterdam. · Zbl 0494.60002
[7] Duffie, D., Filipović, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Probab. 13 984-1053. · Zbl 1048.60059 · doi:10.1214/aoap/1060202833
[8] Dynkin, E. B. (1991a). A probabilistic approach to one class of nonlinear differential equations. Probab. Theory Related Fields 89 89-115. · Zbl 0722.60062 · doi:10.1007/BF01225827
[9] Dynkin, E. B. (1991b). Path processes and historical superprocesses. Probab. Theory Related Fields 90 1-36. · Zbl 0727.60095 · doi:10.1007/BF01321132
[10] Dynkin, E. B. (1992). Superdiffusions and parabolic nonlinear differential equations. Ann. Probab. 20 942-962. · Zbl 0756.60074 · doi:10.1214/aop/1176989812
[11] Dynkin, E. B. (1994). An Introduction to Branching Measure-valued Processes. CRM Monograph Series 6 . Amer. Math. Soc., Providence, RI. · Zbl 0824.60001
[12] Dynkin, E. B. (2004). Superdiffusions and Positive Solutions of Nonlinear Partial Differential Equations. University Lecture Series 34 . Amer. Math. Soc., Providence, RI. · Zbl 1079.60006
[13] Ekeland, I. and Lazrak, A. (2006). Being serious about non-commitment: Subgame perfect equilibrium in continuous time. Preprint. Available at .
[14] Engländer, J. and Pinsky, R. G. (1999). On the construction and support properties of measure-valued diffusions on \(D\subseteq\mathbf{R}^d\) with spatially dependent branching. Ann. Probab. 27 684-730. · Zbl 0979.60078 · doi:10.1214/aop/1022677383
[15] Fleischmann, K. and Mueller, C. (1997). A super-Brownian motion with a locally infinite catalytic mass. Probab. Theory Related Fields 107 325-357. · Zbl 0924.60076 · doi:10.1007/s004400050088
[16] Forsyth, P., Kennedy, J., Tse, T. S. and Windclif, H. (2012). Optimal trade execution: A mean-quadratic-variation approach. Journal of Economic Dynamics and Control 36 1971-1991. · Zbl 1347.91228
[17] Gatheral, J. and Schied, A. (2013). Dynamical models of market impact and algorithms for order execution. In Handbook on Systemic Risk (J.-P. Fouque and J. Langsam, eds.) 579-602. Cambridge Univ. Press, Cambridge.
[18] Karatzas, I. (1985). Probabilistic aspects of finite-fuel stochastic control. Proc. Natl. Acad. Sci. USA 82 5579-5581. · Zbl 0572.93078 · doi:10.1073/pnas.82.17.5579
[19] Perkins, E. (2002). Dawson-Watanabe superprocesses and measure-valued diffusions. In Lectures on Probability Theory and Statistics ( Saint-Flour , 1999). Lecture Notes in Math. 1781 125-324. Springer, Berlin. · Zbl 1020.60075
[20] Schied, A. (1996). Sample path large deviations for super-Brownian motion. Probab. Theory Related Fields 104 319-347. · Zbl 0851.60083 · doi:10.1007/BF01213684
[21] Schied, A. (1999). Existence and regularity for a class of infinite-measure \((\xi,\psi,K)\)-superprocesses. J. Theoret. Probab. 12 1011-1035. · Zbl 0963.60046 · doi:10.1023/A:1021645204173
[22] Schöneborn, T. (2008). Trade execution in illiquid markets. Optimal stochastic control and multi-agent equilibria. Ph.D. thesis, TU Berlin.
[23] Tse, T. S., Forsyth, P. A., Kennedy, J. S. and Windclif, H. (2011). Comparison between the mean variance optimal and the mean quadratic variation optimal trading strategies. · Zbl 1396.91705
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