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.


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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