数理解析研究所講究録
第 1557 巻 2007 年 132-139

132

OPTIMAL STOPPING OF A RISK PROCESS WHEN CLAIMSARE COVERED IMMEDIATELYBOGDAN K. MUCIEK AND KRZYSZTOF J. SZAJOWSKI

ABSTR.1CT. The optimal stopping problem for the risk process with interests
rates and when claims are covered immediately is considered. An insurancecompany receives premiums and pays out claims which have occured according
to a renewal process and which have been recognized by them. The capital of, the size of claims increase at
the company is invested at interest rate$\beta\in\Re+according$
process.
The
immediate payment of claims
toinflation
rate
decreases the company investment by rate . The aim to find the stopping
time which maximizes the capital of the company. The improvement to the
known models by taking into account different scheme of claims payment andthe possibility of rejection of the request by the insurance company is made.
It leads to essentially new risk process and the solution of optimal stopping$\alpha\in\Re+$$\alpha_{1}$$is$problem is different.

1. INTRODUCTION
The following problem in collective risk theory (see Rolski et al. (1998)) is considered. An insurance company, endowed with an initial capital $a>0$ , receives premiums and pays out claims that occur according to a renewal process $\{N(t), t\geq 0\}$ ,where $N(t)$ is the number of losses up till time . The initial capital of the insurance.
company and received premiums are invested at a constant rate of return
denotes the time of the i-th loss, then random variables
Let $T_{0}=0$ and let) with cumulative$(_{i}=T_{i}-T_{i-1}$ are independent and identically distributed (i.i.
be a sequence
distribution function (cdf) $F$ , such that $F(O)=0$ . Let, with cdf $H$ with $H(O)=0$ . Theof i.i. . random variables independent ofrepresents values of successive claims. Usually the costs of damsequenoe
ages elimination increase. It is modelled by the rate $\beta\in\Re+$ . If a claim appears$t$$\alpha\in\Re+$$T_{i}$$d.$$X_{1},$ $X_{2},$$d$$\ldots$$\{\zeta_{i}\}$$\{X_{i}\}_{i=1}^{\infty}$

2000 Mathematics Subject Classification. Primary $60G40;60K99$ ; Secondary $90A46$ .
Key words and phrases. Risk reserve process, optimal stopping, dynamic programming, interest rates.
This paper is a preliminary version, and the final form will be published elsewhere.
The research was supported by KBN grant no 2 P03A 02122 (350228).
JEL classification: C61.
Subject Category and Insurance Branch Category: IE43 IM51 IMIO.

133
B.K. MUCIEK AND K.J. SZAJOWSKI, then the company have to pay. This amount of money decreases the company investment by rate and, as a consequence of that, at the endof investment period the claim at.
decreases the capital by
Although it may seem somewhat surprising at first glance, claims of size zero arise
in some insurance contexts (see Panjer and Willmot (1992)). If a company records

at moment$X_{n}e^{\beta T_{n}}$$T_{n}$$\alpha_{1}$$t$$X_{n}e^{\beta T_{n}}e^{\alpha_{1}(t-T_{n})}$$T_{n}$

all claims as they are presented to the company and some claims are resisted, refused or a complete recovery of losses is made from another insurer, the net costof the claim is zero. This effect is modelled by additional sequence of i.i. random, independent of claims size process and the process of moments ofvariables
claims. It is assumed that $P\{\epsilon_{n}=1\}=p$ and $P\{\epsilon_{n}=0\}=1-p$ . The investigatedprocess of capital assets of the insurance company is$d$$\{\epsilon_{i}\}_{i=1}^{\infty}$(1)$U_{t}=ae^{\alpha t}+ \int_{0}^{t}ce^{\alpha(t-s)}ds-\sum_{n=0}^{N(t)}\epsilon_{n}X_{n}e^{\beta T_{n}}e^{\alpha_{1}(t-T_{n})}$,where $a>0$ is the initial capital, $c>0$ is a constant rate of income from theinsurance premiums, $X_{0}=0$ and $N(O)=0$ . The form of capital assets (1) reduces
to(2)$U_{t}$

where
return at time$\beta_{1}=\beta-\alpha_{1}$

(3)

$t$

$=$

$ae^{\alpha t}+ce^{\alpha t} \frac{1-e^{-\alpha t}}{\alpha}-e^{\alpha_{1}}{}^{t}\sum_{n=0}^{N(t)}\epsilon_{n}X_{n}e^{\beta_{1}T_{\mathfrak{n}}}$. Let $g(u,t)=g_{1}(u)I_{\{t\geq 0\}}$ , where$g_{1}$

is a utility function. The

is $\{Z(t), t\geq 0\}$ and it is given by$Z(t)=g(U_{t}, t_{0}-t) \prod_{j=0}^{N(t)}I_{\{U_{T_{j}}>0\}}=g(U_{t})I_{\{U_{l}>0,\epsilon\leq t\}}$The optimal stopping problem for the process $Z(t)$ is investigated. The model with$\alpha=\beta=0$ have been considered by Ferenstein and Sieroci\’{n}ski (1997). Jensen(1997) investigated a similar model with a claim process modulated by periodic
Markovian processes but without care for time value of money, formulated in (2).
When the claims are paid from the capital of the company, it can be assumed thatwhich
. Muciek (2002) investigated the model given by (1) withdescribed the case when the claims were paid at the end of the investing period.
The improvement introduced here, which takev into account the consequence of theimmediate payment of claims, change the considered risk process essentially. The
model admitted will have an impact on the form of the strong generator for theprocess
as well as on the form of the dynamic programming equations, which
are the tools for describing the solution of the optimal stopping problem for (3).
The considered process $Z(t)$ is the piecewise-deterministic process. The methodsof solving the optimal stopping problem for such processes can by found in papers by
Boshuizen and Gouweleeuw (1993) and Jensen (1997); Jensen and Hsu (1993) and$\alpha_{1}=0$$\alpha=\alpha_{1}$$(^{\underline{\tau)}})$

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ON OPTIMAL STOPPING OF A RISK PROCESS

the monography by Davis (1993). Muciek (2002) has solved the optimal stopping
problem for process (3) withwhich is not direct consequence of the optimal
stopping problem solution for model (1) with.
The organization of the paper is following. In the next section the optimal
stopping problem for the process (3) is formulated. The case of the optimal stopping
up to the fixed number of claims is the subject of investigation in the section 3.
The solution of the problem for the infinite number of claims is given in the section
4.
$\alpha_{1}=0$$\alpha_{1}\neq 0$

2. THE OPTIMIZATION PROBLEM
In this section we define an optimization problem for the model introduced in
the previous section. This optimization problem will be solved in the next section.
Let $\mathcal{F}(t)=\sigma(U_{8}, s\leq t)=\sigma(X_{1}, \epsilon_{1},T_{1}, \ldots, X_{N(t)}, \epsilon_{N(t)},T_{N(t)})$ be thegenerated by all the events up to time $t\geq 0$ . Let
be the set of all stopping
times with respect to the family. Furthermore, for fixed $K$ and for$n=0,1,$$k<K$ letdenote the subset of , such that
if and only
if $T_{n}\leq\tau\leq T_{K}$ a.s.
. The essence of the considerations in the next chapter will beLetto find the optimal stopping timesuch that $EZ(\tau_{K}^{*})=\sup\{EZ(\tau) : \tau\in \mathcal{T}_{0,K}\}$ .
, we first consider optimal stopping
In order to find the optimal stopping time
, such thattimes$\sigma- field$$\mathcal{T}$$\{\mathcal{F}(t), t\geq 0\}$$\mathcal{T}$$\mathcal{T}_{n,K}$$\ldots,$$\tau\in \mathcal{T}_{n,K}$$\mathcal{F}_{n}=\mathcal{F}(T_{n})$$\tau_{K}^{*},$$\tau_{K}^{*}$$\tau_{\mathfrak{n},K}^{*}$(4)$E(Z(\tau_{n,K}^{*})|\mathcal{F}_{n})=ess\sup\{E(Z(\tau)|\mathcal{F}_{n}) :\tau\in \mathcal{T}_{n,K}\}$and using backward induction as in dynamic programming, we will obtain$\tau_{K}^{*}=$$\tau O_{K}$After finding the optimal stopping time for fixed $K$ we will deal with unlimited
number of claims and the aim will be to find the optimal stopping time , such$\tau_{K}^{*}$$\tau^{*}$

that
(5)

$EZ(\tau^{*})=\sup\{EZ(\tau):\tau\in T\}$

can be defined as the limit of the finite horizon
is fulfilled. It will be shown that
optimal stopping times. Such a stopping time in an insurance company management
can be used as the best moment to recalculate premium rate.
$\tau^{*}$

3. CASE WITH FIXED NUMBER OF CLAIMSIn this section we find the form of optimal stopping time in the finite horizon, where $K$ is finite
case, which means the optimal stopping time in the classand fixed (the number of claims is fixed, but the time of the Kth claim, ie. time
horizon, remains non-deterministic). This is a technical assumption which allows$\mathcal{T}_{0,K}$

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B.K. MUCIEK AND K.J. SZAJOWSKI

calculations for finite number of claims and generalize this result to the infinite
number of claims. First we present dynamic programming equations satisfying$n=K,$ $K-1,$. Then in Corollary,
and optimal mean values of returnand3.3 we find optimal stopping times
related to them.
The following representation lemma (see for example Davis (1976)) plays the
crucial role in consequent considerations:$\Gamma_{n,K}=ess\sup\{E(Z(\tau)|\mathcal{F}_{n}):\tau\in \mathcal{T}_{n,K}\}$$1$$\ldots,$$\tau_{K}^{*}$$\tau_{n,K}^{*}$, there vists a positive,
Lemma 3.1. If, such that $\tau\wedge T_{n+1}=(T_{n}+\xi)\wedge T_{n+1}a.s$ .
$\tau\in \mathcal{T}_{n,K}$$\mathcal{F}_{n}$-measurable random variable$\xi$. Then $\Gamma_{K,K}=Z(T_{K})=g(U_{T_{K}},t_{0}-T_{K})\mu_{K}$ .
Let $\mu 0=1$ andNote that the sum of claims ffom (2) can be expressed as$\mu_{n}=\prod_{j=1}^{n}I_{\{U_{T_{f}}>0\}}$(6)$\sum_{n=0}^{N(t)}\epsilon_{n}X_{n}e^{\beta_{1}T_{n}}=(ae^{\alpha t}+\frac{c}{\alpha}(e^{\alpha t}-1)-U_{t})e^{-\alpha_{1}t}$

Let us define for $\xi>0$ such that there is no jump between(7)$d(t,\xi, U_{t})$$=$and $t+\xi$$t$$U_{t+\xi}-U_{t}=e^{\alpha t}(a+ \frac{c}{\alpha})(e^{\alpha\xi}-e^{\alpha_{1}\xi})$$+ \frac{c}{\alpha}(e^{\alpha_{1}\xi}-1)+(e^{\alpha_{1}\xi}-1)U_{t}$,then we have(8)$\mu_{K}=\mu_{K-1}I_{\{U\tau_{K-1}+d(T_{K-1},\zeta_{K},U\tau_{K-1})-\epsilon_{K}X_{K}e^{\beta\langle\tau_{K-1}+c_{K)}}>0\}}$Similarly as in Muciek (2002), Theorem 1, from (6) and from (7) we get thefollowing dynamic programming equations:(i): For $n=K-1,$ $K-2,$$\Gamma_{n,K}$

$=$

$0$

$\ldots,$,

ess $sup\{\mu_{n}\overline{F}(\xi)g(U_{T_{n}}+d(T_{n},\xi, U_{T_{n}}), t_{0}-T_{n}-\xi)$$+E(I_{\{\xi\geq\zeta_{n+1}\}}\Gamma_{n+1,K}|\mathcal{F}_{n})$:$\xi\geq 0$

is$\mathcal{F}_{n}$-measurable}

a.s.,where $\overline{F}=1-F$ is the survival function.
$\Gamma_{n,K}=\mu_{n}\gamma_{K-n}(U_{T_{n}},T_{n})$a.s., where the(ii): For $n=K,$ $K-1,$
sequence of functions $\{\gamma j(u,t),u\in \mathbb{R}, t\geq 0\}$ , using (7), (6) and (8) isdefined as follows$\ldots,$$\gamma_{0}(u,t)=g(u,t_{0}-t)$

$0,$

,

$\gamma_{j}(u,t)=\sup_{r\geq 0}[\overline{F}(r)g(u+d(t, r,u), t_{0}-t-r)$$+p \int_{0}^{r}dF(s)\int_{0}^{e^{-\beta(t+\cdot)}(u+d(t,\epsilon,u))}\gamma_{j-1}(u+d(t,s,u)-xe^{\beta(t+e)},t+s)dH(x)$$+(1-p) \int_{0}^{r}\gamma_{j-1}(u+d(t, s, u),t+s)H(e^{-\beta(t+\epsilon)}(u+d(t, s,u)))dF(s)]$$j=1,2,$$\ldots$

136
ON OPTIMAL STOPPING OF A RISK PROCESSThe above equations differ from the ones in Theorem 1 in Muciek (2002) as aresult of a different form of the capital assets process .
The next step is to find the optimal stopping time . To do this we should ana. Let $B=B[(-\infty, +\infty)\cross$lyze the properties ofthe sequence of functions$[0, +\infty)]$ be the space of all bounded and continuous functions with the norm$t)=\delta_{1}(u,$$|| \delta||=\sup_{u,t}|\delta(u, t)|$ and let $B^{0}=$ {and:}.
. For eachare included in
One should notioe that the functions$t,r\geq 0$ let
and any$U_{t}$$\tau_{K}^{*}$$\{\gamma_{n}, n\geq 0\}$$\delta(u,$$\delta$$B^{0}$$\{\gamma_{n}, n\geq 0\}$$\delta\in B^{0}$$\phi_{\delta}(r,u, t)$$\delta_{1}\in B$$t)I_{\{t\leq t_{0}\}}$$u\in \mathbb{R},$$=$$\overline{F}(r)g(u+d(t,r, u),t_{0}-t-r)+$$+(1-p) \int_{0}^{r}\delta(u+d(t, s,u), t+s)H(e^{-\beta(t+s)}(u+d(t, s, u)))dF(s)$$+p \int_{0}^{r}dF(s)\int_{0}^{e^{-\beta(l+*)}(u+d(t,s,u))}\delta(u+d(t, s,u)-xe^{\beta(t+\epsilon)},t+s)dH(x)$From the properties of the cumulative distribution function $F$ we know thathas at most a countable number of points of discontinuity according tobeing continuous andand is continuous according to $(u, t)$ in the case of$t\neq t_{0}-r$ . Therefore, for further considerations we assume that the function
is bounded and continuous.
let
For each$\phi_{\delta}(r, u,t)$$g_{1}(\cdot)$$r$$g_{1}(\cdot)$$\delta\in B^{0}$(9)$( \Phi\delta)(u,t)=\sup_{r\geq 0}\{\phi_{\delta}(r,u, t)\}$Lemma 3.2. For each$\delta\in B^{0}$.

we have$( \Phi\delta)(u,t)=0\leq r\leq t_{0}-t\max\{\phi_{\delta}(r,u, t)\}\in B^{0}$and there enists a function$r_{\delta}(u, t)$such that$(\Phi\delta)(u, t)=\phi_{\delta}(r_{\delta}(u, t),$ $u,$ $t$).

In subsequent considerations more properties of will be presented.
may be expressed as follows
and $u\in R,$ $t\geq 0,$
For $i=1,2,$$\Phi$$\gamma_{i}(u, t)$$\ldots$$\gamma_{i}(u,t)=\{\begin{array}{ll}(\Phi\gamma_{i-1})(u,t) if u\geq 0 and t\leq t_{0},0 otherwise,\end{array}$and $hom$ Lemma 3.2 there exist functions$r_{\gamma:-1}$

such that$\gamma_{\text{\’{i}}}(u,t)=\{\begin{array}{ll}\phi_{\gamma_{i-1}}(r_{\gamma:-1}, u, t) if u\geq 0 and t\leq t_{0},0 otherwise.\end{array}$, we need to define the
To specify the form of the optimal stopping times
and $\sigma_{n,K}=K\wedge\inf\{i\geq n:R;<$
following random variables$\tau_{n,K}^{*}$$R_{i}^{*}=r_{\gamma\kappa-i+1}(U_{T\iota}, T_{i})$$\zeta_{i+1}\}$.

Finally in Corollary 3.3 we present the fom of the optimal stopping time.

.

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B.K. MUCIEK AND K.J. SZAJOWSKI

Corollary 3.3. Let$\tau_{n,K}^{*}=T_{\sigma_{n,K}}+R_{\sigma_{\mathfrak{n}.K}}^{*}$

and$\tau K=\tau 0_{K}$,then for all $0\leq n\leq K$ the following hold$\Gamma_{n,K}=E(Z(\tau_{n,K}^{*})|\mathcal{F}_{n})a.s$

which means$\tau_{n,K}^{*}$

and$\tau_{K}^{*}$.

and$\Gamma_{0,K}=E(Z(\tau_{K}^{*}))$ $=\gamma_{K}(a, 0)$

are optimal stopping times in the classes$\mathcal{T}_{n,K}$,

and$\mathcal{T}_{0,K}$

respectively.

4. CASE WITH AN INFINITE NUMBER OF CLAIMS,
While is the set of all stopping times with respect to the family
we would like to maximize the mean retum (3), i.e. to find the optimal stopping
time “, such that$\{\mathcal{F}(t),t\geq 0\}$$\mathcal{T}$$\tau$(10)$EZ(\tau^{*})=\sup\{EZ(\tau) :is fulfilled. It will be shown that$\tau^{*}$\tau\in \mathcal{T}\}$

can be defined as the limit of the finite horizon

optimal stopping times.
Let be an optimal stopping time taken from the set of stopping times which oc:
cur not earlier than at , the time of n-th claim, ie.
$\tau_{n}^{*}$$EZ(\tau_{n}^{*}|\mathcal{F}_{n})=\sup\{EZ(\tau|\mathcal{F}_{n})$$T_{n}$$\tau\in \mathcal{T}\cap\{\tau :\tau\geq T_{n}\}\}$.

The solution of this case will be based on the iteration of the operator definedby (9). By an assumption that the interarrival time is greater than with non-zerois a contraction and it has a fixedprobability it can proved that the operator$\Phi$$t_{0}$$\Phi$

point.
The essence of this section is contained in the following theorem. The proof
is based on the proof of the existence of optimal stopping times for semi-Markov
processes presented by Boshuizen and Gouweleeuw (1993).

Theorem 4.1. Assuming the utility function
and $F$ has a density function we have$g_{1}$

is differentiable and nondecreasing$f$(i) for $n=0,1,$ , the limit
stopping time in$\tilde{\tau}_{n}$$\ldots$$\mathcal{T}\cap\{\tau :\tau\geq T_{n}\}(\tau_{n,K}^{*}$

number of claimg defined by (4)),(ii)$E[Z(\tilde{\tau}_{n})|\mathcal{F}_{n}]=\mu_{n}\gamma(U_{T_{n}},T_{n})$is an optimalenists andis a solution of the case with finite$:= \lim_{Karrow\infty}\tau_{n,K}^{*}$$a.s$$\tilde{\tau}_{n}$.

may be proved in a similar way as in Boshuizen and
The optimality ofGouweleeuw (1993). The details may be found in Muciek (2002). Let us denote$\eta_{t}=(t, U_{t},Y_{t}, V_{t})$ , where $Y_{t}=t-T_{N(t)},$ $V_{t}=\mu_{N(t)},$ $t\geq 0$ .
For such that$\tilde{\tau}_{n}$$\tilde{g}$138
ON OPTIMAL STOPPING OF A RISK PROCESS

the strong generator
Skorokhod (1975))$Z(t)=\tilde{g}(\eta_{t})$$(A\tilde{g})(t,u,y,v)$

$=$

$A$

of

$\eta_{t}$on$\tilde{g}$has the form (see Gihman and$\{[(\alpha a+c)e^{\alpha t}-(\alpha_{1}a+c\frac{\alpha_{1}}{\alpha})e^{\alpha t}+(\frac{\alpha_{1}}{\alpha}c+\alpha_{1}u)]g_{1}’(u)$$- \frac{f(y)}{\overline{F}(y)}[g_{1}(u)\overline{H}(ue^{-(\alpha_{1}y+\beta(t-y))})$$-(\alpha_{1}y+\beta(t-y))$$- p\int_{0}^{ue}$$g_{1}(u-xe^{\alpha_{1}y+\beta(t-y)})dH(x)$$+pg_{1}(u)H(ue^{-(\alpha_{1}\nu+\beta(t-y))})]\}v$,

where $t<t_{0},$ $y\geq 0$ and $v\in\{0,1\}$ . Thus we get$(A\tilde{g})(\eta_{*})$

$=$

$+$

$\{[(\alpha a+c)e^{\alpha s}-(\alpha_{1}a+c\frac{\alpha_{1}}{\alpha})e^{\alpha s}+(\frac{\alpha_{1}}{\alpha}c+\alpha_{1}U_{s})]g_{1}’(U_{s})$$\frac{f(s-T_{N(s)})}{\overline{F}(s-T_{N(\cdot)})}[p\int_{0}^{U_{\theta}\epsilon^{-(\alpha_{1}(\cdot-?_{N(s)})+\beta T_{N(\epsilon)})}}g_{1}(U_{l}-xe^{a_{1}(s-T_{N(\cdot)})+\beta T_{N(\cdot)}})dH(x)$$-pg_{1}(U_{*})H(U_{\epsilon}e^{-(\alpha_{1}(s-T_{N(\cdot)})+\beta T_{N(\epsilon)})})$$-\overline{H}(U.e^{-(\alpha_{1}(*-T_{N(\iota)})+\beta T_{N(\cdot)})})g_{1}(U_{\epsilon})]\}\mu_{N(\epsilon)}$.

It should also be marked that the limit of optimal stopping times as $Karrow\infty$
coincide with the overall optimal stopping time.

5. FINAL REMARKSThe presented results generalize known solutions of the optimal stopping prob(1997), Muciek (2002)) tolems for the risk process (see Ferenstein andmore realistic models of risk reserve processes. For $\alpha=\beta=0$ the solution presented by Ferenstein and Sieroci\’{n}ski (1997) are obtained. The model considered by
Muciek (2002) is not direct consequence of model (1). It implies that the solutionof the optimal stopping problem for the risk reserve process investigated in Muciek(2002) is not simple conclusion from the formulae of Corollary 3.3 and Section 4.
There are also similar optimal stopping problems considered by Yasuda (1984)
and Sch\"ottl (1998). The solution of the problem (5) is not direct consequence ofthe results from neither Yasuda (1984) nor Sch\"ottl (1998).
$Sieroci\text{\’{n}}’ski$

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Davis, M., 1976. The representation of martingales of jump processes. SIAM J.
Control Optim. 14, 623-638.

Davis, M., 1993. Markov Models and Optimization. Chapman&Hall, London.

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B.K. MUCIEK AND K J. SZAJOWSKI

Ferenstein, E., Sieroci\’{n}ski, A., 1997. Optimal stopping of a risk process. Applicationes Mathematicae 24(3), 335-342.
Gikhman, I., Skorokhod, A., 1975. The theory of stochastic processes II. Vol. 218
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Muciek, B., 2002. Optimal stopping of a sk process: model with interest rates. J.
Appl. Probab. 39, 261-270.
Panjer, H., Willmot, G., 1992. Insurance Risk Models. Society of Actuaries, 475
North Martingale Road, Suite 800, Schaumburg, IL 60173-2226, U.S.A.
Rolski, T., Schmidli, H., Schimdt, V., Teugels, J., 1998. Stochastic Processes for
Insurance and Finance. John Wiley&Sons, Chichester.
Sch\"ottl, A., 1998. Optimal stopping of a risk reserve process with interest and cost
rates. J. Appl. Probab. 35, 115-123.
Yasuda, M., 1984. On the optimal stopping problem of markov chains by variational
inequalities. J. College of Arts and Sciences B-17, 11-16.
$ri$

INSTITUTE OF MATHEMATICS AND COMPUTER SCIENCE, WROCLAW UNIVERSITY OF $TECHNOL-$
OGY. WYBRZEZE $WYSPIA\acute{N}$ SKIEGO 27, WROCLAW. POLAND

Current address: AIG Credit SA, ul. Strzegomska $42c,$ $53- 611$ Wroclaw, Poland
E-mail address: Bogdan.MuclekOaigcrodit. pl$URL$ : http: $//www$ . im. pwr. wroc. $p1/$ -muciek(Corresponding author) INSTITUTE OF MATHEMATICS AND COMPUTER SCIENCE, WROCLAW $UNI-$
VERSITY OF TECHNOLOGY, WYBRZEZE WYSPIANSKIECO 27,
POLAND$WROCLA\backslash V$

E-mail address: Krzysztof. Szaj owskiOpwr. wroc. pl$URL$ : http: $//neyman$ . im. pwr. wroc. $p1/\sim$ szaj ow.