Optimal impulse control on an unbounded domain with nonlinear cost ...

Optimal impulse control on an unbounded domain with nonlinear cost functions Stefano Baccarin∗ and Simona Sanfelici† Keywords: impulse control, stochastic cash management, quasi-variational inequalities, finite element approximation. Mathematics Subject Classification (2000): 49J40, 60G40, 65N30. Abstract In this paper we consider the optimal impulse control of a system which evolves randomly in accordance with a homogeneous diffusion process in 0) or selling (when x(t) < 0) short-term securities, but each time he converts cash into securities he must bear transaction costs. The presence of a fixed component in the transaction costs makes a continuous control unprofitable and the financial manager intervenes only at isolated points of time and for discrete operations. From a theoretical point of view our problem is one of impulse control where a policy is made up of a sequence ti of stopping times and corresponding random jumps ξi enforced upon the system. It is well known that applying the optimality principle of Dynamic Programming we can associate an impulse control problem with a quasi-variational inequality1 . If one succeeds in finding a regular solution of this inequality it can be shown that this solution is the value function and from this solution it is possible to derive the optimal policy (see, for example, Richard (1977), Harrison, Sellke and Taylor (1983), Eastham and Hastings (1988), Korn(1997,1998)). In this paper using a variational approach and the functional analysis techniques developed in the monographs of Bensoussan and Lions (1982,1984) we go beyond “verification theorems”. Under general assumptions we show that the value function of our problem is always a weak solution of the associated quasi-variational inequality in a suitable Sobolev space. From this solution we can obtain the optimal control whose existence is therefore always ensured, even under very general conditions. The analytical solution in the form of a control band policy can be obtained only in some special cases involving, for instance, linear or quadratic holdingpenalty costs (see Harrison, Sellke and Taylor (1983), Baccarin (2002)). For more general specifications, a numerical approach to the problem is compulsory. We perform a Finite Element (FE) approximation by means of continuous, piecewise linear functions defined on a suitable truncated domain [−r, r] and vanishing on the boundary. The convergence of the solution ur of the truncated problem to the value function as r → +∞ can be shown. The FE approximation urh is obtained by means of the “Bensoussan-Lions discrete iterative process”, whose convergence analysis on bounded domains can be found in Loinger (1980), Cortey-Dumont (1980) and Boulbrachene (1998). The paper is organized as follows. In §2 we give a precise formulation of our impulse control problem and we recall some results of Bensoussan and Lions (1984) concerning a variational inequality in a weighted Sobolev space. In §3 we prove some properties of the non local operator associated with the transaction cost structure. Furthermore we consider a weak formulation of the quasi variational inequality in a weighted Sobolev space and we show the existence of 1 A formal argument which shows how to derive the quasi-variational inequality from Belmann’s optimality principle can be found, for the cash management problem, in Constantinides and Richard (1978) or in Baccarin (2002).

2

a minimum solution for this inequality. §4 is devoted to the characterization of the value function of the problem as the minimum solution of the quasi variational inequality. Moreover using the value function we prove the existence of an optimal impulse policy dividing < in two regions: a continuation set where the system evolves freely and a transaction region where the system is controlled. Finally, in §5 we show the FE approach for computing the value function and some numerical simulations, with different data specifications, are performed and analyzed in §6.

2

Problem formulation and preliminary results

We consider a filtered probability space S = (Ω, F, P, Ft ) and a one-dimensional standard Wiener process Wt adapted to Ft . We are given the following functions ( σ(x) > λ > 0 , σ(x) ∈ W 1,∞ (R) (1) b(x) ∈ W 1,∞ (R) . The dynamics of the cash fund x(t), in absence of control, is described by the following Itˆo stochastic differential equation ( dx(t) = b(x)dt + σ(x)dWt x(0) = x . An impulse policy V = (t1 , ξ1 ; t2 , ξ2 ; ...; ti , ξi ; .....) is an increasing sequence (i ≥ 1) of stopping times 0 ≤ t1 ... ≤ ti ≤ ti+1 .....(with respect to Ft ) and corresponding random variables ξi , which represent the jumps enforced upon the system. An impulse control is said to be admissible if it satisfies the two feasibility conditions ½ ti → ∞ a.s. when i → ∞ (2) ξi is Fti measurable ∀i ≥ 1 where Fti is the minimum σ-algebra of events prior to ti 2 . We will denote by A the set of admissible policies. When policy V is used, the controlled process y(t) is generated by the following set of stochastic differential equations with random initial conditions3  when ti ≤ t < ti+1 (i ≥ 0)  dy(t) = b(y)dt + σ(y)dWt (3) y(ti ) = y(t− ) + ξ ∀i ≥1 i i  y(t0 ) = x 2 t → ∞ implies that only a finite number of actions can be taken in any bounded interval; i ξi is Fti measurable means that the ξi decision depends only on the information available at ti . 3 Since σ(x) and b(x) are Lipschitz continuous this system has a strong solution: the probability space S = (Ω, F, P, Ft ) is fixed and it will not depend on the control V .

3

− 4 where t0 ≡ 0 and y(t− i ) = lim y(t) if ti−1 < ti , or y(ti ) = y(ti−1 ) if ti−1 = ti . t↑ti

By yx (t) we will denote the controlled process y(t) starting in x = y(0). We make the following assumptions on the holding/penalty costs f (x) and on the variable part c(ξ) of the transaction costs C(ξ) : | ¯ ¯ ¯ ¯ ¯ ¯ ¯

s

f (x) is measurable; 0 ≤ f (x) ≤ f0 (1 + |x| ), s > 0 c(ξ) : < →