Maximum Process Problems in Optimal Control Theory

Transcription

1 J. Appl. Math. Stochastic Anal. Vol. 25, No., 25, (77-88) Research Report No. 423, 2, Dept. Theoret. Statist. Aarhus (2 pp) Maximum Process Problems in Optimal Control Theory GORAN PESKIR 3 Given a standard Brownian motion (B t ) t and the equation of motion: p dx t = v t dt + 2 db t we set S t = max st X s and consider the optimal control problem: sup E S c v where c > and the supremum is taken over all admissible controls v satisfying v t 2 [ ; ] for all t up to = inf f t > j X t =2 (`; `) g with < < and ` < < ` given and fixed. The following control v 3 is proved to be optimal: Pull as hard as possible that is v 3 = t if X t < g3(s t ), and push as hard as possible that is v 3 = t if X t > g3(s t ), where s 7! g3(s) is a switching curve that is determined explicitly (as the unique solution to a nonlinear differential equation). The solution found demonstrates that the problem formulations based on a maximum functional can be successfully included in optimal control theory (calculus of variations) in addition to the classic problem formulations due to Lagrange, Mayer and Bolza.. Introduction Stochastic control theory deals with three basic problem formulations which were inherited from classical calculus of variations (cf. [4] pp ). Given the equation of motion: (.) dx t = (X t ; u t ) dt + (X t ; u t ) db t where (B t ) t is standard Brownian motion, consider the optimal control problem: (.2) inf E u xz D L(X t ; u t ) dt + M (X D ) where the infimum is taken over all admissible controls u = (u t ) t applied before the exit time D = inf f t > j X t =2 C g for some open set C = D c and the process (X t ) t starts at x under P x. If M and L 6=, the problem (.2) is said to be Lagrange formulated. If L and M 6=, the problem (.2) is said to be Mayer formulated. If both L 6= and M 6=, the problem (.2) is said to be Bolza formulated. The Lagrange formulation goes back to the 8th century, the Mayer formulation originated in the 9th century, and the Bolza formulation [2] was introduced in 93. We refer to [] (pp ) with the references for a historical account of the Lagrange, Mayer and Bolza problems. 3 Centre for Mathematical Physics and Stochastics (funded by the Danish National Research Foundation). Mathematics Subject Classification 2. Primary 49J55, 93E2, 6J65. Secondary 6J6, 34A34, 35G5. Key words and phrases: Optimal control theory, maximum process, the Hamilton-Jacobi-Bellman equation, bang-bang control, switching curve, diffusion, Brownian motion, nonlinear differential equation. goran@imf.au.dk

2 Although the three problem formulations are formally known to be equivalent (see e.g. [] pp or [4] pp ), this fact is rarely proved to be essential when solving a concrete problem. Setting Z t = L(X t ; u t ) or Z t = M (X t ), and focusing upon the sample path t 7! Z t for t 2 [; D ], we see that the three problem formulations measure the performance associated with a control u by means of the following two functionals: (.3) Z D Z t dt and Z D where the first one represents the surface area below (or above) the sample path, and the second one represents the sample-path s terminal value. In addition to these two functionals, it is suggested by elementary geometric considerations that the maximal value of the sample path: (.4) max Z t t D provides yet another performance measure which, to a certain extent, is more sensitive than the previous two ones. Clearly, a sample path can have a small integral but still a large maximum, while a large maximum cannot be detected by the terminal value either. The main purpose of the present paper is to show that the problem formulations based on a maximum functional can be successfully added to optimal control theory (calculus of variations). This is done by formulating a specific problem of this type (Section 2) and solving it in a closed form (Section 3). The result suggests a number of new avenues for further research upon extending the Bolza formulation (.2) to optimize the following expression: (.5) E x max t D K(X t ; u t ) +Z D L(X t ; u t ) dt + M (X D ) where some of the maps K ; L and M may also be identically zero. Optimal stopping problems for the maximum process have been studied by a number of authors in the 99 s (see e.g. [6, 3,, 8, 5]) and the subject seems to be well-understood now. 2. Formulation of the problem Consider a process X = (X t ) t (2.) dx t = v t dt + solving the stochastic differential equation: p 2 db t where B = (B t ) t is standard Brownian motion, and associate with X the maximum process: (2.2) S t = max X r rt so that X = x and S = s under P x;s where x s. Introduce the exit time: _ s (2.3) = inf f t > j X t =2 (`; `) g where ` < < ` are given and fixed, and let so be c > in the sequel. 2

3 The optimal control problem to be examined in this paper is formulated as follows: (2.4) J(x; s) := sup v E x;s S c where the supremum is taken over all admissible controls v satisfying v t 2 [ ; ] for all t with some < < given and fixed. By admissible we mean that the s.d.e. (2.) can be solved in Itô s sense (either strongly or weakly). Since v t is required to be uniformly bounded, it is well-known that a weak solution (unique in law) always exists under a measurability condition (see e.g. [9] p. 55) where v t may depend on the entire sample path r 7! X r up to time t. Moreover, if v t = v(x t ) for some (bounded) measurable function v, then a strong solution (pathwise unique) also exists (see e.g. [9] pp. 79-8). The optimal control problem (2.4) has some interesting interpretations. The equation (2.) may be viewed as describing the motion of a Brownian particle (subject to a fluctuating force B _ t ) that is under influence of a (slowly varying) external force v t (see [7] pp ). The objective in (2.4) is therefore to determine an optimum of the external force one needs to exert upon the particle so to make its maximal height at the time of exit as large as possible in the course of time needed for the same exit to happen as short as possible. Clearly, the interpretation and objective just described carry over to many other problems where the equation (2.) plays a role. It appears intuitively clear that the optimal control should be of the following bang-bang type: At each time either push or pull as hard as possible so to reach either ` or ` as soon as possible. The solution of the problem presented in the next section confirms this guess and makes the statement precise in analytic terms. It is also apparent that at each time t we need to keep track of both X t and S t so that the problem is inherently two-dimensional. 3. Solution of the problem. In the setting of the previous section consider the optimal control problem (2.4). Note that X t = (X t ; S t ) is a two-dimensional process with the state space S = f (x; s) 2 IR 2 j x s g that changes (increases) in the second coordinate only after hitting the diagonal x = s in IR 2. Off the diagonal, the process X = ( Xt ) t changes only in the first coordinate and thus may be identified with X. Moreover, if v t = v(x t ) for some (bounded) measurable function v in (2.), then X is a Markov process. The later feedback controls are sufficient to be considered under fairly general hypotheses (see e.g. [4] pp ), and this fact will also be proved below. The infinitesimal generator of X may be therefore formally described as follows: (3.) IL X = IL X in x < = at x = s where IL X is the infinitesimal generator of X. This means that the infinitesimal generator of X is acting on a space of C 2 functions f on S satisfying lim x"s (@f =@s)(x; s) =. The formal description (3.)+(3.2) appears in [8] (pp ) where the latter fact is also verified. The condition of normal reflection (3.2) was used for the first time in [3] in the case of Bessel processes (it was also noted in [6] (p. 8) in the case of a Bessel process of dimension one). 3

4 2. Assuming for a moment that the supremum in (2.4) is attained at some feedback control, and making use of the formal description of the infinitesimal generator (3.)+(3.2), we are naturally led to formulate the following HJB system: (3.3) sup IL vx (J) c = in C for x < s (x; s) = (normal reflection) x=s (3.5) J(x; s) = s (instantaneous stopping) x=`+ (3.6) J(x; s) = s (instantaneous stopping) x=` for ` < s < ` where the infinitesimal generator of X given v is expressed by (3.7) IL v X 2 and we set C = f (x; s) 2 IR 2 j ` < x s < ` g. Our main effort in the sequel will be directed to solving the system (3.3)-(3.6) in a closed form. More explicitly, the HJB equation (3.3) with J = J(x; s) reads as follows: (3.8) sup v v J x + J xx c = so that we may expect a bang-bang solution v 3 depending on the sign of J x : If J x < then v 3 t =, and if J x > then v 3 t =. The equation J x = defines an optimal switching curve s 7! g 3 (s), and the main task in the sequel will be to determine it explicitly. Further heuristic considerations based on the bang-bang principle just stated (when close to ` apply so to exit at `, and when close to ` apply so to exit at ` ) suggest to partition C into the following three subsets (modulo two curves x = g 3 (s) and s = s 3 to be found): (3.9) C = f (x; s) 2 IR 2 j ` < x < g 3 (s) ; s 3 < s < ` g (3.) C 2 = f (x; s) 2 IR 2 j g 3 (s) < x s ; s 3 < s < ` g (3.) C 3 = f (x; s) 2 IR 2 j ` < x s < s 3 g where s 3 is a unique point in (`; `) satisfying: (3.2) g 3 (s 3 ) = `. In addition to (3.) we also set: (3.3) g 3 (`) = x 3 to denote another point in (`; `) playing a role. We refer to Figure below to obtain a better geometric understanding of (3.9)-(3.3). 4

5 3. We construct a solution to the system (3.3)-(3.6) in three steps. In the first two steps we determine C [ C 2 together with a boundary curve s 7! g 3 (s) separating C from C 2. Step. Consider the HJB equation: (3.4) J x + J xx c = in C to be found. The general solution of (3.4) is given by (3.5) J(x; s) = a (s) e x + c x + b (s) where a (s) and b (s) are some undetermined functions of s. Using (3.5) we can eliminate b (s) from (3.5) and this yields: (3.6) J(x; s) = a (s) e x e ` + c (x `) + s. Solving J x (x; s) = for x gives x 3 = g 3 (s) as a candidate for the switching curve, and also that a (s) can be expressed in terms of g 3 (s) as follows: (3.7) a (s) = c Inserting this back into (3.6) gives: (3.8) J () (x; s) = c ( ) 2 e g3(s) e g3(s). e x e ` as a candidate for the value function (2.4) when (x; s) 2 C. Step 2. Consider the HJB equation: (3.9) J x + J xx c = in C 2 to be found. The general solution of (3.9) is given by (3.2) J(x; s) = a (s) e x + c x + b (s) + c (x `) + s where a (s) and b (s) are some undetermined functions of s. Solving J x (x; s) = for x gives x 3 = g 3 (s) as a candidate for the switching curve, and also that a (s) can be expressed in terms of g 3 (s) as follows: (3.2) a (s) = c Inserting this back into (3.2) gives: (3.22) J (2) (x; s) = c e g3(s). ( ) 2 e (g3(s)x) + c x + b (s). The two functions (3.8) and (3.22) must coincide at the switching curve, i.e. 5

6 (3.23) J () (x; s) x=g3(s)+ = J (2) (x; s) x=g3(s) giving a closed expression for b (s), which after inserted back into (3.22) yields: (3.24) J (2) (x; s) = c ( ) 2 e (g3(s)x) c ( ) 2 e (g3(s)`) c g3(s) x + c g3(s) ` + s + c c ( ) 2 ( ) 2 as a candidate for the value function (2.4) when (x; s) 2 C 2. The condition (3.6) with x = s = ` can now be used to determine a unique x3 2 (`; `) satisfying (3.3). It follows from the expression (3.24) above that x3 solves (and is uniquely determined by) the following transcendental equation: (3.25) ( ) 2 e (x3`) ( ) 2 e (x3`) + ` x3 ` =. It may be noted that this equation, and therefore x3 as well, does not depend on c. Finally, applying the condition (3.4) to the expression (3.24) we obtain a differential equation for the switching curve s 7! g3(s) that takes the following form: (3.26) g (s) = c e (g(s)s) c e (g(s)`) for all s 2 (s; `) where s < ` happens to be a singularity point at which dg=ds = +. The equation (3.26) is solved backwards under the initial condition (3.3) where x3 2 (`; `) is found by solving (3.25). The switching curve s 7! g3(s) is a unique solution of (3.26) obtained in this way. It can also be verified that (3.2) holds for some s3 2 (s; `). 4. It turns out that the solution of (3.26) satisfying (3.3) can hit the diagonal in IR 2 at a point s 3 2 (`; `) taken to be closest to x3, if c c3 for some large c3 to be determined below. When this happens, the construction of the solution becomes more complicated, and the solution of (3.26) for s 2 (s; s 3 ) is of no use. We thus first treat the simpler case when the solution of (3.26) stays below the diagonal ( the case of small c ), and this case is then followed by the more complicated case when the solution of (3.26) hits the diagonal ( the case of large c ). Step 3 ( The case of small c ). Having characterized the curve s 7! g 3 (s) for [s 3 ; `] we have obtained a candidate (3.8)+(3.24) for the value function (2.4) when (x; s) 2 C [ C 2. It remains to determine J(x; s) for (x; s) 2 C 3, and this is what we do in the final step. As clearly the control should be applied, consider the HJB equation: (3.27) J x + J xx c = in C 3 given and fixed. The general solution of (3.27) is given by (3.28) J(x; s) = a 2 (s) e x + c x + b 2 (s) 6

7 where a2(s) and b2(s) are some undetermined functions of s. Using the condition (3.5) we can eliminate b2(s) from (3.28) and this yields: (3.29) J (3) (x; s) = a 2(s) e x e ` + c (x `) + s. Applying the condition (3.4) to the expression (3.29) we find that a2(s) (3.3) a 2 (s) = e s e ` for ` < s < s 3. This equation can be solved explicitly and this gives: (3.3) a2(s) = e ` s + log e s e ` + d should solve: where d is an undetermined constant. To determine d we may use the fact that the maps (3.24) and (3.29) must coincide at (s 3 ; s 3 ), i.e. (3.32) J (2) (x; s) x=s=s3 = J (3) (x; s) the latter being known explicitly due to (3.2). With this d (3.33) a2(s) = e ` (s 3 s) log x=s=s3 we can then rewrite (3.3) as follows: c. e s e ` e s3 e ` Inserting this expression back into (3.29) we obtain a candidate for the value function (2.4) when (x; s) 2 C3, thus completing the construction of a solution to the system (3.3)-(3.6). Steps 3-5 ( The case of large c ). In this case the solution s 7! g 3 (s) of (3.26) satisfying (3.3) hits the diagonal in IR 2 at some s 2 (`; `). The set C 3 from (3.9) naturally splits into the following three subsets (modulo two curves): (3.34) C ; = f (x; s) 2 IR 2 j ` < x < g 3 (s) ; s 3 < s < ` g (3.35) C ;2 = f (x; s) 2 IR 2 j ` < x < s ; s 3 < s < s 3 g (3.36) C ;3 = f (x; s) 2 IR 2 j ` < x < h 3 (s) ; s 3 < s < s 3 g where s C 2 3, the map s 7! h 3(s), and s 3 in this context will soon be defined. Similarly, the set from (3.) naturally splits into the following two subsets (modulo one curve): (3.37) C 2; = f (x; s) 2 IR 2 j g 3 (s) < x s ; s 3 < s < ` g (3.38) C 2;2 = f (x; s) 2 IR 2 j h 3 (s) < x s ; s 3 < s < s 3 g. The set C 3 from (3.) remains the same, however, with a new value for s 3 to be introduced. We refer to Figure 2 below to obtain a better geometric understanding of (3.34)-(3.38). It is clear from the construction above that in C ; the value function (2.4) is given by (3.8), and that in C 2; the value function (2.4) is given by (3.24), where s 7! g 3 (s) is the solution of (3.26) satisfying (3.3). For the points s < s 3 (close to s 3 ) we can no longer make use 7

8 of the solution g 3 (s), and the expression (3.8), as clearly the condition (3.4) fails to hold. We need, moreover, to apply the control instead of, and thus Step 3 presented above must be modified. The HJB equation (3.27) is considered with in place of, and this again using (3.5) leads to the expression (3.29) with in place of : (3.39) J (;2) (x; s) = a 3(s) e x e ` + c (x `) + s as well as the equation (3.3) and its solution (3.3). To determine d (3.4) J (;2) (x; s) = J (i;) (x; s) x=s=s 3 x=s=s 3 in (3.3) we may use that for i either or 2, where we set J (;) = J () and J (2;) = J (2) with J () from (3.8) and J (2) from (3.24). The latter expression in (3.4) is known explicitly due to the fact that. With this d we can then rewrite an analogue of (3.33) as follows: g 3 (s 3 ) = s 3 (3.4) a3(s) = e ` e s (s 3 s) log e ` c e s 3 e ` e s Inserting this expression back into (3.39) we obtain a candidate for the value function (2.4) when (x; s) 2 C;2. Moreover, the equation J (;2) x = determines an optimal switching curve: (3.42) h 3 (s) = log c a 3(s) and the points s 3 2 (`; s 3 ) and s 3 2 (`; s 3 ) appearing in (3.35)+(3.36) and (3.38) are determined by the following identities: (3.43) h 3 (s 3 ) = s 3 (3.44) h 3 (s 3 ) = `. Note that (3.44) is reminiscent of (3.2) above, and so is h 3 (s) of g 3 (s) above. However, the two functions are different for s < s 3. It is clear that the value function (2.4) is also given by the formula (3.39) for (x; s) 2 C;3. To determine the value function (2.4) in C2;2, where clearly the control is to be applied, we can use the result of Step 2 above which leads to the following analogue of (3.24) above: (3.45) J (2;2) (x; s) = c () 2 e (h3(s)x) c () 2 e (h3(s)`) c h 3 (s) x + c h 3 (s) ` + s + c () 2 3. c () 2 for (x; s) 2 C2;2. Finally, in C3 we should also apply the control, and thus the result of Step 3 ( the case of small c ) above can be used. Due to (3.44) the expressions (3.29)+(3.33) carry over unchanged to the present case. It must be kept in mind, however, that s 3 satisfies (3.44) and not (3.2). This completes the construction of a solution to the system (3.3)-(3.6). 8

9 5. In this way we have obtained a candidate for the value function (2.4) when (x; s) 2 C in both cases of small and large c. The preceding considerations can now be summarized as follows. Theorem 3. In the setting of (2.)-(2.3) consider the optimal control problem (2.4) where the supremum is taken over all admissible controls as explained above.. In the case of small c in the sense that the solution of (3.26) stays below the diagonal in IR 2, the value function J = J(x; s) solves the system (3.3)-(3.6) and is explicitly given by (3.8) in C from (3.9), by (3.24) in C 2 from (3.), and by (3.29)+(3.33) in C 3 from (3.) with s 3 from (3.2), where s 7! g 3 (s) is the unique solution of the differential equation (3.26) for s 2 (s 3 ; `) satisfying g 3 (`) = x 3, and x 3 2 (`; `) is the unique solution of the transcendental equation (3.25). (The values of J at the two boundary curves in C not mentioned are determined by continuity.) 2. In the case of large c in the sense that the solution of (3.26) hits the diagonal in IR 2, the value function J = J(x; s) solves the system (3.3)-(3.6) and is explicitly given by (3.8) in C ; from (3.34), by (3.24) in C 2; from (3.37), by (3.39)+(3.4) in C ;2 [ C ;3 from (3.35)+(3.36), by (3.45) in C 2;2 from (3.38), and by (3.29)+(3.33) in C 3 from (3.) with s 3 from (3.44), where s 7! g 3 (s) is the unique solution of the differential equation (3.26) for s 2 (s ; `) satisfying 3 g 3 (`) = x 3 with x 3 2 (`; `) as above, and s 7! h 3 (s) is given by (3.42)+(3.4). (The values of J at the four boundary curves in C not mentioned are determined by continuity.) 3. The optimal control v 3 in the problem (2.4) is described as follows: Pull as hard as possible that is vt 3 = if X t < g 3 (S t ), and push as hard as possible that is vt 3 = if X t > g 3 (S t ), both as long as t where is given in (2.3). This description holds in the case of large c by modifying the solution g 3 (s) of (3.26) for s < s as follows: g 3 3 (s) := s + if s 2 (s ; ) 3 s 3 and g 3 (s) := h 3 (s) for s 2 (s 3 ; s ). In both cases of small and large c we formally set 3 g 3 (s) := ` for s 2 (`; s 3 ). (The control vt 3 for X t = g 3 (S t ) is arbitrary and has no effect on optimality of the result.) To verify the statements denote the candidate function by J(x; s) for (x; s) 2 C. Then, by construction, it is clear that (x; s) 7! J(x; s) is C 2 outside the set f (g 3 (s); s) j s 3 < s < ` g in C, and x 7! J(x; s) is C at g(s) when s is fixed. (Actually, the latter mapping is also C 2 as is easily seen from (3.4) and (3.9) above.) Thus Itô s formula can be applied to the process J(X t ; S t ) with any admissible v, and under P x;s in this way we get: (3.46) J(X t ; S t ) = J(x; s)+ = J(x; s) + Z t Z t J x (X r ; S r )dx r + Z t p 2 J x (X r ; S r ) db r + Z t J s (X r ; S r )ds r + 2 Z t ((IL v X )(J))(X r; S r ) dr J xx (X r ; S r )d X; X r where the integral with respect to ds r is zero, since the increment S r equals zero off the diagonal in IR 2, and at the diagonal we have (3.4). Since X equals either ` or `, we see by (3.5)+(3.6) that J(X ; S ) = S, and thus (3.46) implies the following identity: (3.47) S = J(x; s) + M + Z ((IL v X)(J))(X r ; S r ) dr 9

10 R t p where M t = 2 Jx (X r ; S r ) db r is a continuous (local) martingale. Since (x; s) 7! J x (x; s) is uniformly bounded in C, it follows by the optional sampling theorem that E x;s (M ) =. (Note that there is no restriction to consider only those controls v for which E x;s () < so that E x;s ( p ) < as well.) On the other hand, by (3.3) we know that (IL v X(J))(x; s) c for all (x; s) 2 C, with equality if v = v 3, and thus (3.47) yields: (3.48) J(x; s) = E x;s S Z ((IL v X)(J))(X r ; S r ) dr Ex;s S c with equality if v = v 3. This establishes that the candidate function equals the value function, and that the optimal control equals v 3, thus completing the proof. REFERENCES [] BLISS, G. A. (946). Lectures on the Calculus of Variations. Univ. Chicago Press. [2] BOLZA, O. (93). Über den Anormalen Fall beim Lagrangeschen und Mayerschen Problem mit gemischten Bedingungen und variablen Endpunkten. Math. Ann. 74 (43-446). [3] DUBINS, L. E. SHEPP, L. A. and SHIRYAEV, A. N. (993). Optimal stopping rules and maximal inequalities for Bessel processes. Theory Probab. Appl. 38 (226-26). [4] FLEMING, W. H. and RISHEL, R. W. (975). Deterministic and Stochastic Optimal Control. Springer-Verlag. [5] GRAVERSEN, S. E. PESKIR, G. and SHIRYAEV, A. N. (999). Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Research Report No. 45, 999, Dept. Theoret. Statist. Aarhus ( pp). Theory Probab. Appl. 45, 2 (25-36). [6] JACKA, S. D. (99). Optimal stopping and best constants for Doob-like inequalities I: The case p =. Ann. Probab. 9 (798-82). [7] NELSON, E. (967). Dynamical Theories of Brownian Motion. Princeton Univ. Press. [8] PESKIR, G. (997). Optimal stopping of the maximum process: The maximality principle. Research Report No. 377, 997, Dept. Theoret. Statist. Aarhus (3 pp). Ann. Probab. 26, 998 (64-64). [9] ROGERS, L. C. G. and WILLIAMS, D. (987). Diffusions, Markov Processes, and Martingales. Volume 2: Itô Calculus. John Wiley and Sons. [] SHEPP, L. A. and SHIRYAEV, A. N. (993). The Russian option: Reduced regret. Ann. Appl. Probab. 3 (63-64). Goran Peskir Department of Mathematical Sciences University of Aarhus, Denmark Ny Munkegade, DK-8 Aarhus home.imf.au.dk/goran goran@imf.au.dk

11 x * C C 2 s g * (s) s x s * C 3 Figure. The case of small c in the problem (2.4). The state space (triangle) of the process (X t; S t) from (2.)+(2.2) splits into three regions. In the region C the optimal control 3 vt equals, and in the region C 2 [ C 3 the optimal control 3 vt equals. The switching curve s 7! g3(s) is determined as the unique solution of the differential equation (3.26) satisfying g3(`) = x3 where x3 2 (`; `) is the unique solution of the transcedental equation (3.25). In this specific case we took ` = =, ` = = and c = 2. It turns out that x3 =, s3 = :574 8 and s = : (The point s is a singularity point at which dg3=ds = + and g3 takes a finite value.)

12 x * C 2, C, s g * (s) s x s h * (s) s * C,3 s * s * C,2 C 2,2 C 3 Figure 2. The case of large c in the problem (2.4). The state space (triangle) of the process (X t; S t) from (2.)+(2.2) splits into six regions. In the region C ; [ C ;2 [ C ;3 the optimal control v 3 t equals, and in the region C 2; [ C 2;2 [ C 3 the optimal control v 3 t equals. The switching curve s 7! g3(s) is determined as the unique solution of the differential equation (3.26) satisfying g3(`) = x3 where x3 2 (`; `) is the unique solution of the transcedental equation (3.25). The switching curve s 7! h3(s) differs from the solution s 7! g3(s) of (3.26) and is determined to meet the condition (3.4) as explained in the text. In this specific case we took ` = =, ` = = and c = 3:8. It turns out that x3 =, s 3 = :492 2, s 3 = :82273 and s3 = : For comparison we state that the smaller (of two) ~s3 satisfying g3(~s3 ) = ~s3 equals :8247 4, that ~s3 satisfying g3(~s3) = ` equals :9658 5, and that s = : (The point s is a singularity point at which dg3=ds = + and g3 takes a finite value.) 2