Optimal Strategies for Utility from Terminal Wealth with General Bid and Ask Prices

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1 Optimal Strategie for Utility from Terminal Wealth with General Bid and Ak Price Tomaz Rogala 1 Lukaz Stettner 2 The Author 2018 Abtract In the paper we tudy utility from terminal wealth maximization with general bid and ak price tudying firt one aet cae and generalizing then reult to the multi aet cae. We how that under certain aumption continuou conditional ditribution of the aet and trict concavity of the utility function the problem can be reduced to tudy a tatic problem and then by an induction to conider multi period cae. We obtain formulae for buying, elling and no tranaction zone both in one and two aet cae. We alo how the exitence and the form of hadow price. Keyword Utility from terminal wealth General bid and ak price Shadow price Mathematic Subject Claification 93E20 91G10 1 Introduction On a given probability pace, F,F t, P we conider a dicrete time market coniting of n aet with bid S i t and ak S i t price, which are adapted to F t,for which we can ell or buy ith aet repectively at time t, where t = 0, 1...,T and i = 1,...,n. We hall aume that bid and ak price are random variable uch that 0 < S i t < S i t for t = 0, 1,...,T and i = 1,...,n. At each time t we can buy l i t or ell m i t of ith aet baing on available for u information F t. We denote by x t Reearch upported by National Science Center, Poland by Grant UMO-2016/23/B/ST1/ B Lukaz Stettner tettner@impan.pl Tomaz Rogala rogalatp@gmail.com 1 Faculty of Mathematic and Natural Science, College of Science, Cardinal Stefan Wyzyńki Univerity, Wóycickiego 1/3, Waraw, Poland 2 Intitute of Mathematic Polih Academy of Science, Sniadeckich 8, Waraw, Poland

2 our poition in a bank account and by y i t the number of ith aet in our portfolio at time t. Our principal aumption i that we do not allow hortelling nor borrowing o that both x t and y i t for t = 0, 1,...,T and i = 1,...,n hould be nonnegative. A one can ee in [10] uch retriction i typical when aet price atify o called full upport condition. To implify notation we alo aume zero interet rate on the bank account. Our purpoe i to maximize J x,y l i t, mi t = E [ Ux T + y T S T ], 1.1 where x i initial bank poition and y = y 1,...,y n are initial number of aet in our portfolio, U i a utility function which i aumed to be trictly increaing, trictly concave and continuouly differentiable and y T S T := n i=1 yt i Si T. In what follow we hall conider nonnegative coordinate x, y 1,...,y n auming that at leat one i different than 0. We denote by R + := [0, and hall ue the following notation R n+1 + := [0, n+1 \ 0,...,0, forn = 1, 2,... In the paper we are intereted in characterization of optimal trategie for 1.1 and contruction of o called hadow price i.e. the price which i between bid and ak uch that optimal value of the functional in the frictionle market with that price i the ame a in the market with bid and ak price. The interet with hadow price ha tarted with the paper [8], where hadow price wa tudied for the Black-Schole model with tranaction cot and dicounted logarithmic utility function. Exitence of hadow price for dicrete time finite market wa hown in [9]. In ome cae we are not able to find a frictionle market with price proce taking value between bid and ak price which give the ame optimal trategy a the market with tranaction cot ee [1] and [4]. Shadow price for continuou time market model ha been tudied intenively in a number of paper ee [3] and reference therein uing in mot general cae duality theory which give u an exitential reult. Dicrete time hadow price wa tudied uing duality in [4]. In the paper [10] a direct method baed on dynamic programming wa propoed. The advantage of dynamic programming method i that it allow to work on approximation method. Thi paper generalize [10], where dicrete time hadow price wa contructed uing dicrete time ytem of Bellman equation and certain geometric propertie of tranaction zone. We extend the reult of [10] howing that under quite general aumption the exitence of hadow price and contruction of optimal trategie can be retricted to the tudy of one period tatic cae. We furthermore how that contruction of optimal trategie and hadow price can be extended to the cae of variou aet multidimenional cae. Situation however i then more complicated and notation i much harder o that we retrict ourelve to the two aet cae only. The paper conit of 7 ection. In Sect. 2 we olve tatic portfolio optimization problem for one aet. In Sect. 3 we introduce induction tep conidering two time moment problem. In Sect. 4 we olve general dynamic problem with one aet. Section 5 and 6 parallel Sect. 2, 3 and 4 for the cae with two aet. An Appendix contain a number of auxiliary reult ued in the paper.

3 2 Static Two Dimenional Cae Let the function w : R 2 + R be trictly increaing with repect to both variable, trictly concave and continuouly differentiable. We hall conider it a a one period value function which depend on the value of our bank account and number of aet in our portfolio. We recall our notation that R 2 + := [0, [0, \ 0, 0. Let D :=, R 2 + : 0 < < and ˆD := ŝ R + : ŝ > 0. For every x, y,, R 2 + D define Ax, y,, := l, m R 2 + : x + m l 0, y m + l 0, 2.1 which i the et of poible invetment trategie at time 0, uch that our bank and aet account are nonnegative with bid price and ak price. For every x, y, ŝ R 2 + ˆD put Âx, y, ŝ := l, m R 2 + : x +ŝm ŝl 0, y m + l 0, 2.2 which in turn correpond to nonnegative bank and aet account with only one price ŝ without proportional tranaction cot. It i clear that for every x, y,, R 2 + D and for every ŝ [, ] we have that Ax, y,, Âx, y, ŝ. 2.3 Notice that the et Ax, y,, i convex, compact, while the et Âx, y, ŝ i cloed convex. For every x, y,, R 2 + D let wx, y,, := up wx + m l, y m + l. 2.4 l,m Ax,y,, Clearly there i a maximizer for which the upremum on the right hand ide of 2.4 i attained. For every x, y, ŝ R 2 + ˆD let ŵx, y, ŝ := up wx +ŝm ŝl, y m + l. 2.5 l,m Âx,y,ŝ We have no problem with the exitence of maximizer in ŵx, y, ŝ ince we can retrict ourelve to the maximization over the et  1 x, y, ŝ  2 x, y, ŝ, where  1 x, y, ŝ =l, 0 R 2 + : x ŝl and Â2 x, y, ŝ =0, m R 2 + : m y and each of thee et i convex and compact. One can notice that from 2.3 we get that for every x, y,, R 2 + D and for every ŝ [, ] we have that wx, y,, ŵx, y, ŝ = wx, y, ŝ, ŝ, 2.6 where we naturally extend the meaning of wx, y,, to the cae when =. In what follow we hall characterize optimal l, m for which the upremum in 2.4i attained.

4 From now on we aume that, D i fixed. Given c > 0 and ˆD conider the function from [0, c] to R defined by H c, x := w x, c x. 2.7 where the function H c, correpond to the value function with wealth c and price. Clearly, for every c > 0 we have that H c, 0 = w 0, c > H c, 0 = w 0, c and H c, c = wc, 0 = H c, c. 2.8 Moreover, for every x, y,, R 2 + D we have ŵx, y, = up u [0,x+ y] H x+y, u and ŵx, y, = up u [0,x+y] H x+y, u. 2.9 For every c > 0 and for every x [0, c] we alo have the following derivative of the function H c, and H c, at point x at x = 0orx = c we have right or left derivative repectively: H c, x = w x x, c x H c, x = w x x, c x 1 w y 1 w y x, c x x, c x, Furthermore, for every c > 0 for the left hand limit of H c, and H c, at point c we have H c, c = w xc, 0 1 w yc, 0 <w x c, 0 1 w yc, 0 = H c, c Lemma 2.1 For every c > 0 and ˆD the function H c, are trictly concave on [0, c]. Proof Thi i the conequence of the fact that for every c > 0 and ˆD the function H c, are compoition of linear and trictly concave function. Function ŵ trongly depend on the aet price. The reult below how that when the market poition x, y are poitive uch dependence i injective. Propoition 2.2 For every x, y R 2 + uch that x, y > 0 we cannot have ŵx, y, = ŵx, y, The only point where we have uch equality are: x, 0 when H x, x 0 and y, 0 when H y, 0+ 0.

5 Proof Let x, y R 2 + be uch that Since w i increaing with repect to both variable and for u [0, x, wx, x+y u <wx, x+y u, while for u x, x + y] we have wx, x+y u <wx, x+y u we therefore have ŵx, y, = ŵx, y, = wx, y Thi mean that it i optimal to do nothing for x, y both under the price a well a. Furthermore ŵx, y, ŝ = wx, y for every ŝ [, ]. Aume now that x, y > 0. Then for every ŝ [, ] the function F : [ x ŝ, y] R given by Fu := wx +ŝu, y u achieve it maximum for u = 0. Thi mean that F 0 = 0 = w x x, yŝ w y x, y which can only happen when w x x, y = 0, which contradict the fact that w wa trictly increaing. By Lemma 2.1 the function H c, i trictly concave. Therefore if H x, x 0, then H x, i increaing on [0, x].uing2.11wehavethath x, x > 0. Taking into account trict concavity of H x,, we get that the function H x, i alo trictly increaing on [0, x]. Conequently, it i clear that ŵx, 0, = ŵx, 0, = wx, 0. If H x, x <0, then the upremum of H x, on [0, x] i attained for ome x [0, x and ŵx, 0, >wx, 0. When H y, 0+ 0, then by trict concavity of H y, we know that it i decreaing on [0, y] and therefore Since H y, 0 = w0, y = ŵ0, y,. H y, 0+ = w x0, y 1 w y0, y w x 0, y 1 w y0, y = H y, 0+, then we alo have that H y, 0+ < 0 and H y,0 = w0, y = ŵ0, y,. If H y, 0+ >0, then the upremum of H y, on [0, y] i attained for ome x 0, y] and therefore ŵ0, y, >w0, y. Remark 2.3 Notice that 2.12 can not happen for x > 0 and y > 0, when wx, y i concave non necearily trictly concave, differentiable and increaing with repect to both coordinate. Strict concavity aumption will be important to tudy differentiability of ŵ.

6 For c > 0 and > 0let hc, := arg max x,y R 2 + : x+y=c wx, y h0 c, Clearly hc, = and we have h h 1 c, 1 c, = c h 0c,. Note that hc, i the optimal portfolio correponding to the wealth c and aet price. One can notice furthermore that h 0 c, = arg max x [0,c] w x, c x From the proof of Propoition 2.2 we have the following Corollary 2.4 Let c > 0 be uch that h 0 c, = c. Then h 0 c, = c. Moreover, if y > 0 i uch that h 0 y, = 0, then alo h 0 y, = 0. Proof Taking into account concavity of H c, we get h 0 c, = c only when H c, c 0. Then by 2.11 aloh c, c >0. In effect, the concavity of H c, implie that h 0 c, = c. Similarly, if y > 0 i uch that h 0 y, = 0, then H y, 0+ 0 which implie that H y, < 0 and h 0 y, = 0. Taking into account trict concavity of the function w we can how that the elector h defined in 2.14 i continuou. Namely, we have Lemma 2.5 Function h i continuou on 0, 0,. Proof Let c, 0, 0, be arbitrary and let c n, n n=1 be an arbitrary equence from 0, 0, which converge to c,. It uffice to how that We have up x [0,c n ] up w x [0,c n ] w x, c n x n up w x [0,c n ] + up x [0,c n ] x, c n x n n up w x, c x x [0,c] x, c n x + w x, n c x+ up w x [0,c] up x [0,c n ] up w x [0,c] w x, x, x, c x c x+ c x =: I n + II n. n 0. A w i continuou and c n n By continuity argument, we get that I n n c, wealohavethatii n 0. For every n N we have that h 0 c n, n c n. Thu, if for ome d R we have that h 0 c n, n n d, then it mut be d c and w h 0 c n, n, c n h 0 c n, n n n w d, c d.

7 By 2.16, thi implie that w d, c d = up x [0,c] w d, c x By trict concavity of the mapping x w d, c x, we get that h0 c, = d. Therefore, h 0 c n, n n h 0 c,. The next Corollary characterize propertie of the graph of h. Corollary 2.6 The graph of the mapping c, hc, doe not have common point except of point x, 0 R 2 + whenever H x, x 0 and 0, y R2 + whenever H y, Proof Let c 1, c 2 > 0 and 1, 2 D be uch that 1 2 and h 0 c 1, 1, c 1 h 0 c 1, 1 = h 0 c 2, 2, c 2 h 0 c 2, 2 =: x, y. 1 2 Then h 0 c 1, 1 = h 0 c 2, 2 and c 1 h 0 c 1, 1 1 = c 2 h 0 c 2, 2 2. Conequently, ŵx, y, 1 = ŵx, y, 2 = wx, y, which by Propoition 2.2 can happen only when y = 0 and H x, 1 x 0 or when x = 0 and H 2 y, We have characterized o far the function ŵ which correponded to one aet price. Now we tudy the function w which i the optimal value correponding to bid price and ak price. Lemma 2.7 For every x, y,, R 2 + D we have that wx, y,, = max up u [x,x+ y] w and wx, y,, = max up u [0,y] w Furthermore u, y + x u x + y u, u., up w u [0,x] u, y + x u, 2.17, up w x u y, u. u [y,y+ x ] 2.18 wx, y,, min ŵx, y,, ŵx, y, Proof Notice that the meaning of 2.17 and 2.18 i that tarting from x, y we can buy or ell aet for or repectively. By 2.6 we immediately get Function w and ŵ inherit concavity property of function w, which will be important for further tudie. We have Propoition 2.8 For every, D the function w,,,, ŵ,, and ŵ,, are concave on R 2 +. In particular, they are continuou on R2 +.

8 Proof The proof follow from concavity of w with repect to the firt two coordinate. Let x 1, y 1, x 2, y 2 R 2 + and ˆl 1, ˆm 1 Ax 1, y 1,,, ˆl 2, ˆm 2 Ax 2, y 2,, be maximizer in w for x 1, y 1 or x 2, y 2 repectively. For λ 0, 1 we have that λˆl λˆl 2,λˆm λ ˆm 2 Aλx λx 2,λy λy 2,, and conequently wλx λx 2,λy λy 2,, wλx λx 2 + λ ˆm λ ˆm 2 λˆl λˆl 2, λy λy 2 λ ˆm λ ˆm 2 + λˆl λˆl 2 λwx 1 + ˆm 1 ˆl 1, y 1 ˆm 1 + ˆl λwx 2 + ˆm 2 ˆl 2, y 2 ˆm 2 + ˆl 2 = λwx 1, y 1,, + 1 λwx 1, y 1,, The concavity of ŵ can be hown in the ame way. Continuity follow directly from concavity. Remark 2.9 Note that in fact we have in 2.20 trict inequality whenever x 1 + ˆm 1 ˆl 1, y 1 ˆm 1 + ˆl 1 = x 2 + ˆm 2 ˆl 2, y 2 ˆm 2 + ˆl 2, and equality if after optimal trategy we enter the ame point in R 2 +. For, D we define portfolio zone NT, := x, y R 2 + wx, y,, = wx, y, B, := x, y R 2 + wx, y,, = ŵx, y, \ NT, and S, := x, y R 2 + wx, y,, = ŵx, y, \ NT,. For, D define alo NT, := x, y R 2 + : h 0x + y, <x < h 0 x + y,. The above et have the following meaning: NT, i a no tranaction zone, i.e. the et of poition from which we do not change our portfolio, B, i a buying zone - the et poition in which we buy aet for the price until we enter the NT, zone, the et S, i a elling zone - the et of poition in which we ell aet for the price until we enter the NT, zone. The above zone can be characterized in term of the elector h 0. Namely,

9 Theorem 2.10 Let, D. Then NT, = x, y R 2 + : h 0x + y, x h 0 x + y,, B, = x, y R 2 + : x > h 0x + y,, S, = x, y R 2 + : x < h 0x + y, and for every x, y NT, we have trict inequality in Moreover, NT, i an open et and it cloure excluding point 0, 0 coincide with NT,. Proof Let x, y R 2 +.Ifwx, y,, = wx, y, then by 2.17 we are not able to increae the value of wx, y buying or elling aet. Conequently by trict concavity of w we hould have that h 0 x + y, x and h 0 x +y, x.ifx, y B,, then by 2.17 we have that x > h 0 x + y,. Ifx, y S,, then by 2.17 we have that x < h 0 x +y,.ifx h 0 x +y,, then wx, y,, = ŵx, y, and we have equality in If x h 0 x + y,, then wx, y,, = ŵx, y, and we have equality in If h 0 x + y, <x < h 0 x + y,, then wx, y,, < ŵx, y, and wx, y,, <ŵx, y, and we have trict inequality in By the continuity of the mapping c h 0 c, and c h 0 c,, we have that NT, i an open et. Clearly, it cloure coincide with NT,. Thi end the proof. Uing 2.18 we can obtain an alternative verion of the formulae for the zone in term of the elector h 1. Propoition 2.11 Let, D. Then NT, := x, y R 2 + : h 1x + y, <y < h 1 x + y,, NT, = x, y R 2 + : h 1x + y, y h 1 x + y,, B, = x, y R 2 + : y < h 1x + y,, S, = x, y R 2 + : y > h 1x + y,. We alo immediately have Corollary 2.12 For every x, y,, R 2 + D the following implication hold: and x, y B, wx, y,, = w h 0 x + y,, x + y h 0x + y x, y S, wx, y,, = w h 0 x + y,, x + y h 0x + y, For given x, y R 2 + we are looking for ŝ [, ] uch that wx, y,, = ŵx, y, ŝ. Such value if exit i called a hadow price ee [10] for more explanation..

10 It clearly depend on the value of x, y. It i rather obviou ee again to [10] that hadow price for x, y in B, i equal to, while for x, y in S, i equal to. The only problem i to find hadow price for x, y NT, i.e. in the no tranaction zone correponding to bid and ak price. Propoition 2.13 Let x, y,, R 2 + D be uch that x, y NT, and y > 0. Then there exit a unique ŝx, y, uch that x = h 0 x +ŝx, yy, ŝx, y and y = h 1 x +ŝx, yy, ŝx, y. Moreover, the function ŝ can be extended to a continuou function on 0, 0, NT,. Proof By Theorem 2.10, we have that h 0 x + y, <x < h 0 x + y,. By Lemma 2.5, the mapping h 0 x + y, i continuou. Therefore, there exit ŝx, y, uch that h 0 x +ŝx, yy, ŝx, y = x. Thi mean that ŵx, y, ŝx, y = wx, y. By Corollary 2.6, ŝx, y i unique. By Propoition 2.11 h 1 x + y, < y < h 1 x + y, and by Lemma 2.5, the mapping h 1 x +y, i continuou. Therefore, there exit x, y, uch that h 1 x + x, yy, x, y = y and ŵx, y, x, y = wx, y. By uniquene x, y =ŝx, y By uniquene again ŝx, y i continuou on NT,.Ifx n, y n n=1 i a equence from NT, which converge to the point x, y NT, \ NT,, then from Theorem 2.10 and Lemma 2.5 we have that either x = h 0 x + y, or x = h 0 x + y, y. Aume that x = h 0 x + y,. Ifforomez [, ] we have that ŝx n, y n n z, then by continuity of h we have that x n = h 0 xn +ŝx n, y n y n, ŝx n, y n n ŝx + zy, z. Therefore x =ŝx + yz, z =ŝx + y, and by Corollary 2.6 we have that z =. Remark 2.14 Propoition 2.13 ay that in our model hadow price i uniquely defined for x, y 0, 0, NT,, and furthermore i a continuou function. From Propoition 2.2 and it proof we have that ŝ i not uniquely defined only at point x, 0 whenever H x, x 0 and and y, 0 when H y, For uch point any value from the interval [, ] may erve a a hadow price. In the et B, we have ŝ =, while in the et S, we have ŝ =. In the proof of Propoition 2.2 differentiability of w wa important. Now we conider firt differentiability of ŵ and then differentiability of w. Propoition 2.15 Function ŵ i continuouly differentiable at point x, y, R 2 + ˆD. Proof For z, R + ˆD and u [0, z] define w z, u, := w Clearly, ŵx, y, = u, z u = H z, u. up w x + y, u, u [0,x+y] Aume firt that H x+y, x + y <0 and H x+y, 0+ >0. Then upremum in 2.21 i attained in the open interval 0, x + y. Therefore by Propoition 7.2 we have that the mapping

11 z, W z, := up w z, u, u [0,z] i continuouly differentiable, ince upremum i attained inide the interval [0, z] and therefore we may aume locally that upremum i over a fixed ubinterval which doe not depend explicitly on z. Conequently by 7.3 wehave W z z, = w z z, u z,, = 1 w y u z,, z u z, 2.22 and W z, = w z, u z,, = 1 2 w y u z,, z u z, 2.23 where u z, i the maximizer of w in the definition of W. Therefore function ŵx, y, = W x + y, i continuouly differentiable at point x, y, and by 2.22, 2.23 wehave ŵ x x, y, = 1 w y u x+y,, x + y u x+y,, 2.24 ŵ y x, y, = w y u x+y,, x + y u x+y,, 2.25 ŵ x, y, = 1 2 w y u x+y,, x + y u x+y,, 2.26 where u x+y, i the maximal value of u in When H x+y, x + y 0 then ŵx, y, = wx + y, 0 and when H x+y, 0+ 0 then ŵx, y, = w0, 1 x + y. When H x+y, x + y >0 or H x+y, 0+ <0 then in ome neighborhood of x, y, we have ŵx, y, = wx + y, 0 or ŵx, y, = w0, 1 x + y and differentiability follow from differentiability of w. Conequently we may have problem with differentiability only when H x+y, x + y = 0orH x+y, 0+ = 0. Then from 2.24 to2.26 by continuity of x, y, u x+y, which follow from uniquene of u x+y, we obtain continuou differentiability of ŵ at x, y,. Corollary 2.16 Function w i continuouly differentiable for every x, y NT, S, B, x, 0 : x > 0, H x, x 0 0, y : y > 0, H y, Proof When x, y NT, we have that wx, y,, = wx, y which i continuouly differentiable. For x, y S, we have that wx, y,, = ŵx, y, which i continuouly differentiable by Propoition Finally for x, y B, we have wx, y,, = ŵx, y, which i again continuouly differentiable by Propoition Conequently we may have problem with differentiability only at the boundary of NT,. Moreover in the et x, 0 : H x, x 0

12 0, y : H y, 0+ 0 we get continuou differentiability a in end of the proof of Propoition 2.15, ince it i a ubet of NT,. 3 Induction Analyi In the previou ection we tudied one period problem. Now we come to two period problem which in the equel in the next ection will be replaced by multi period problem tudied by induction. Aume on a given filtered probability pace,f,ft t=0,1, P we are given two F 1 -random variable S 1 and S 1 uch that 0 < S 1 < S 1 uch that for each x, y R 2 + the derivative of the random variable wx, y, S 1, S 1 whenever exit are integrable and that conditional law P S 1, S 1 F0 i continuou. In what follow we hall ue a regular verion of uch conditional probability which there exit by Theorem 6.3 of [7]. For x, y R 2 + put wx, y := E wx, y, S 1, S 1 F0, conidering it a regular conditional expected value. Notice that by Lemma 7.3 we can put in the place of x and y any F 0 random variable. Later on we hall conider frequently uch verion of regular conditional probability. The contruction of function wx, y i crucial for in our induction tep which we conider in the next ection. The function w i not trictly concave with repect to the firt two coordinate. A we how below the function w i already trictly concave, which allow u to ue later the reult of the Sect. 2. Propoition 3.1 Random function w i trictly concave. Proof Let x 1, y 1, x 2, y 2 R 2 + \0, 0 be uch that x 1, y 1 = x 2, y 2.From2.4 we have that there exit G := σ F 0, S 1, S 1 -meaurable random variable l1, m 1 and l 2, m 2 uch that l 1, m 1 Ax 1, y 1, S 1, S 1, l 2, m 2 Ax 2, y 2, S 1, S 1 which are optimal in the market with bid and ak price tarting from x 1, y 1 and x 2, y 2 repectively, i.e. for which we have wx 1, y 1, S 1, S 1 = wx 1 + S 1 m 1 S 1 l 1, y 1 m 1 + l 1, wx 2, y 2, S 1, S 1 = wx 2 + S 1 m 2 S 1 l 2, y 2 m 2 + l 2. Then for every λ [0, 1] we have that λ wx 1, y λ wx 2, y 2 [ = E λwx 1 + S 1 m 1 S 1 l 1, y 1 m 1 + l 1 ] + 1 λwx 2 + S 1 m 2 S 1 l 2, y 2 m 2 + l 2 F 0 [ λx1 E w + 1 λx 2 + S1 λm1 + 1 λm 2 S1 λl1 + 1 λl 2,

13 λy1 + 1 λy 2 λm1 1 λm 2 + λy1 1 λl 2 F 0 ] E [w ] λx λx 2,λy λy 2, S 1, S 1 F0 3.1 with equality whenever x 1 + S 1 m 1 S 1 l 1, y 1 m 1 + l 1 = x 2 + S 1 m 2 S 1 l 2, y 2 m 2 + l 2. Since for optimal trategie we mut have l 1 m 1 = 0 and l 2 m 2 = 0, then we conider four cae. Cae 1: m 1 = m 2 = 0 Then x 1 S 1 l 1 = x 2 S 1 l 2 and y 1 + l 1 = y 2 + l 2 o that x 1 x 2 = S 1 l 1 l 2 = S 1 y 1 y 2 and provided that y 1 = y 2 if y 1 = y 2, then l 1 = l 2 and x 1 = x 2 we have that S 1 = x 1 x 2 y 2 y 1 which, by continuity of the conditional law of S 1, can happen with conditional probability 0. Cae 2: l 1 = l 2 = 0 Then x 1 + S 1 = x 2 + S 1 m 2 and y 1 m 1 = y 2 m 2. Thi lead to x 1 x 2 = S 1 m 1 m 2 = S 1 y 2 y 1 and again for y 1 = y 2 if y 1 = y 2, then we have m 1 = m 2 and x 1 = x 2 we obtain S 1 = x 1 x 2 y 2 y 1 which, by the continuity of the conditional law S 1, can happen with conditional probability 0. Cae 3: l 1 = m 2 = 0 Then x 1 + S 1 m 1 = x 2 S 1 l 2 and y 1 m 1 = y 2 + l 2. Auming that m 1 > 0, ince then by concavity local maximum on the elling line with price S 1 coincide with global maximum in the market with one price S 1,we have that wx 1, y 1, S 1, S 1 = wx 1 + S 1 m 1, y 1 m 1 = ŵx 1, y 1, S 1 = ŵx 1 + S 1 m 1, y 1 m 1, S 1 Auming furthermore that l 2 > 0 we have that Therefore wx 2, y 2, S 1, S 1 = wx 2 S 1 l 2, y 2 + l 2 = ŵx 2, y 2, S 1 = ŵx 2 S 1 l 2, y 2 + l 2, S 1 ŵx 1 + S 1 m 1, y 1 m 1, S 1 = ŵx 1 + S 1 m 1, y 1 m 1, S 1

14 which, by Propoition 2.2, can happen only when either x 1 +S 1 m 1 = 0ory 1 m 1 = 0. If x 1 + S 1 m 1 = 0, then x 1 = 0 and m 1 = 0 and we have a contradiction. If y 1 = m 1 then y 2 + l 2 = 0 which mean that y 2 = 0 = l 2 we get a contradiction again. Aume now that m 1 = 0. Then x 2 x 1 = S 1 y 1 y 2, which can happen only with probability 0. Finally when l 2 = 0wehavex 2 x 1 = S 1 y 1 y 2 which again can happen only with probability 0. Cae 4: m 1 = l 2 = 0. Thi cae i tudied identically to the Cae 3. Summarizing, we have trict inequality in 3.1. Thi end the proof. To tart induction procedure we alo need continuou differentiability of w which i hown below. Propoition 3.2 Random function w i continuouly differentiable for x, y R 2 +, P almot urely. Proof Since w i trictly concave it i continuouly differentiable whenever it i differentiable. By Corollary 2.16 function w,,, i continuouly differentiable at point x, 0 : H x, x 0 0, y : H y, 0+ 0 and at other point x, y R 2 + except for thoe x, y R2 + for which ŵx, y, = wx, y or ŵx, y, = wx, y. Let Sx, ˆ y := ŝ > 0 : ŵx, y, ŝ = wx, y. By Propoition 2.2 for every x, y Z := R 2 + x, \ 0 : H x, x 0 0, y : H y, 0+ 0 the et Sx, ˆ y i at mot a ingleton. For every, D function w,,, i differentiable at point x, y Z, whenever / Sx, ˆ y or / Sx, ˆ y. Since the conditional law P S 1, S 2 F 0 i continuou, then function w i continuouly differentiable on Z and conidering a in the proof of Propoition 2.15 we have that it i continuouly differentiable for x, y R 2 +, P almot urely. 4 Dynamic Two Dimenional Cae In thi ection we ummarize the reult of the previou ection to tudy multi period cae uing induction. We conider the ytem of Bellman equation and w T x, y,, := Ux + y, 4.1 w T 1 x, y,, := e up l,m Ax,y,, E[w T x + m l, y m + l, S T, S T F T 1 ] 4.2 w T k x, y,, := e up l,m Ax,y,, E[w T k+1 x + m l, y m + l, S T k+1, S T k+1 F T k ], 4.3

15 for k = 1, 2,...,T, where the conditional expectation are conidered a regular conditional expectation. By Lemma 7.3 we are allowed to put in the place of x, y,, in 4.3 any nonnegative and poitive in the cae of the lat two coordinate F T k meaurable random variable. Furthermore auming uitable integrability we have that the value of regular conditional expectation in are continuou with repect to x, y and upremum over a compact et i attained o that eential upremum maybe replaced by upremum calculated for each fixed ω, ee Lemma 7.3 for detailed proof of thi nontrivial fact. Conequently, in what follow we hall write up intead of e up. We hall aume A bid and ak price S t, S t for t = 1,...,T are uch that random function w k x, y,, are well defined for k = 0, 1...,T and for x, y R 2 + random function w k x, y, S k, S k are integrable together with their derivative with repect to x and y whenever they exit. Sufficient condition for A i defined in Lemma 7.4. Furthermore we aume that B conditional law P S k+1, S k+1 Fk i continuou for k = 0, 1...,T 1. Let w k x, y := E w k+1 x, y, S k+1, S k+1 Fk, 4.4 w k x, y,, := up w k x + m l, y m + l, 4.5 l,m Ax,y,, and ŵ k x, y, ŝ := up l,m Âx,y,ŝ w k x +ŝm ŝl, y m + l. 4.6 For x, y,, R 2 + D denote by A kx, y,, the et of all F k -meaurable random variable taking value in the et Ax, y,,. Denote by x k, y k our bank poition and number of aet held at time k. By Lemma 7.3 we clearly have that w k x k, y k, S k, S k = e up w k x k + S k m S k l, y k m + l, 4.7 l,m A k x k,y k,s k,s k and w k x k, y k := E w k+1 x, y, S k+1, S k+1 F k. 4.8 Define furthermore for, D the et NT k, := x, y R 2 + : w kx, y,, = w k x, y, B k, := x, y R 2 + : w kx, y,, = ŵ k x, y, \ NT k,, S k, := x, y R 2 + : w kx, y,, = ŵ k x, y, \ NT k, which correpond repectively to no tranaction, buying or elling zone at time k.let for c > 0, 0 and k = 0, 1,...,T 1 hc,, k := arg max w k x, y 4.9 x,y: x+y=c,x 0,y 0

16 be the optimal portfolio trategy at time k given wealth value c and aet price auming that later at time k + 1,...,T 1 we have market with bid and ak h0 c,, k price S t, S t. Then hc,, k = and for > 0wehaveh h 1 c,, k 1 c,, k = c h 0 c,,k. Recall now the definition of hadow priceee [10]: thi i a tochatic proce Ŝ k x k, y k taking value between bid and ak price S k and S k uch that optimal value of utility from terminal wealth 1.1 with price proce Ŝ k x k, y k i the ame a in the cae of bid and ak price S k and S k. Theorem 4.1 Under A and B for k = 0, 1,..., T 1 and, D we have NT k, = x, y R 2 + : h 0x + y,, k x h 0 x + y,, k, B k, = x, y R 2 + : S k, = x, y R 2 + : x > h 0x + y,, k, 0x + y,, k. If at time k our bank poition i x k and number of aet held in portfolio i y k then the optimal trategy i: when x k, y k NT k S k, S k to do nothing, while when x k, y k B k S k, S k i to buy aet to reach h 1 x k + S k y k, S k paying S k for each aet and finally when x k, y k S k S k, S k i to ell aet to reach h 1 x k +S k y k, S k getting for each aet the value S k. Furthermore there exit a hadow price proce Ŝ k x k, y k, which at time k i equal to S k or S k whenever x k, y k i in S k S k, S k or B k S k, S k repectively and i equal to ŝ k x k, y k in the cae when x k, y k NT k S k, S k, where ŝ k i defined a ŝ in Propoition Proof The proof i by an induction. Notice firt that by Propoition 3.1 w k trictly concave and by Propoition 3.2 it i alo continuouly differentiable P a.. for x, y R 2 +. Conequently by Propoition 2.15 ŵ k i continuouly differentiable at x, y, ŝ R 2 + ˆD for k = 0, 1,...,T. Furthermore by Corollary 2.16 w k i continuouly differentiable for x, y NT k S k B k, where NT k i an interior of NT k.the remaining part of the proof follow from Theorem 2.10 and Propoition The exitence and the form of hadow price follow directly from Propoition 2.13 uing induction again. 5 Static Multi Two Aet Cae In thi ection we conider two aet cae: the market with two kind of aet: aet I with bid and ak price S 1 t, S1 t and aet II with bid and ak price S 2 t, S2 t repectively. Then our market poition will be denoted by the triple x, y 1, y 2 where x i a before the amount of money on the bank account while y 1 and y 2 denote number of aet I or II repectively in our portfolio. Define the et of admiible trategie: Â bb x, y 1, y 2, 1, 2 := Â b x, y 1, y 2, 1, 2 := l 1, l 2 [0, [0, : x l 1 1 l 2 2 0, 5.1 l 1, m 2 [0, [0, : x l m 2 2 0, y 2 m 2 0, 5.2

17 Â b x, y 1, y 2, 1, 2 := m 1, l 2 [0, [0, : x + m 1 1 l 2 2 0, y 1 m 1 0, 5.3 Â x, y 1, y 2, 1, 2 := m 1, m 2 [0, [0, : y 1 m 1 0, y 2 m 2 0, 5.4 which correpond to buying the firt and the econd aet trategy, buying the firt and elling the econd aet, elling the firt and buying the econd aet and elling both firt and econd aet repectively. Clearly thee et are compact. Denote by Cc, 1, 2 := x, y 1, y 2 R 3 + : c = x + y1 1 + y Thi i the et of all bank and I and II aet poition repectively, which can be achieved under the price 1 and 2 of the firt and econd aet repectively with the wealth proce equal to c. Letw : R 3 + R + be a function which i increaing with repect to it coordinate, trictly concave and differentiable. Therefore it i a continuou function. By trict continuity there i a unique in Cc, 1, 2 element maximizing w. Let Gc, 1, 2 := max wx, y 1, y 2, 5.6 x,y 1,y 2 Cc, 1, 2 and We have arg max Gc, 1, 2 := h 0 c, 1, 2, h 1 c, 1, 2, h 2 c, 1, Lemma 5.1 Function h i for i = 0, 1, 2 are continuou. Proof It follow from the fact that function w i trictly concave which implie that it upremum over a compact et C i unique, and furthermore the mapping 0, 0, 0, c, 1, 2 Cc, 1, 2 i continuou in the Haudorff metric ee the proof of theorem 2.1 in [10], or [2]. Let ŵ bb x, y 1, y 2, 1, 2 := ŵ b x, y 1, y 2, 1, 2 := ŵ b x, y 1, y 2, 1, 2 := ŵ x, y 1, y 2, 1, 2 := up wx l 1 1 l 2 2, y 1 + l 1, y 2 + l 2, 5.8 l 1,l 2 Â bb x,y 1,y 2, 1, 2 up wx l m 2 2, y 1 + l 1, y 2 m 2, 5.9 l 1,m 2 Â b x,y 1,y 2, 1, 2 up wx + m 1 1 l 2 2, y 1 m 1, y 2 + l 2, 5.10 m 1,l 2 Â b x,y 1,y 2, 1, 2 up wx + m m 2 2, y 1 m 1, y 2 m 2. m 1,m 2 Â x,y 1,y 2, 1, An interpretation of the function ŵ bb x, y 1, y 2, 1, 2 i that it i the optimal value of the function w when tarting from the poition x, y 1, y 2 we buy aet I and aet II. The meaning of the other function i imilar.

18 Let Âx, y 1, y 2, 1, 2 := l 1, m 1, l 2, m 2 [0, [0, 2 : x + m 1 l m 2 l 2 2 0, y 1 m 1 + l 1 0, y 2 m 2 + l denote the et of all admiible trategie when we tart from poition x, y 1, y 2 and 1, 2 are the price of aet I and II repectively. Define ŵx, y 1, y 2, 1, 2 := up wx + m 1 l 1 1 l 1,m 1,l 2,m 2 Âx,y 1,y 2, 1, 2 + m 2 l 2 2, y 1 m 1 + l 1, y 2 m 2 + l We have ee Propoition 2.8. Lemma 5.2 The mapping x, y 1, y 2 ŵx, y 1, y 2, 1, 2 i concave and ŵx, y 1, y 2, 1, 2 = Gx + y y 2 2, 1, Next Lemma and Propoition are three dimenional verion of Propoition 2.2. Define for given x 0, y 1 0 and y 2 0 F x,y1,y 2 1, 2 u 1, u 2 = wx + 1 u u 2, y 1 u 1, y 2 u We hall maximize F x,y1,y 2 1, 2 over u 1 and u 2 uch that u 1 y 1, u 2 y 2 and x + 1 u u 2 0. Lemma 5.3 We have ŵx, y 1, y 2, 1, 2 = wx, y 1, y when: x > 0, y 1 > 0 and y 2 > 0 and F x,y1,y 2 1, 2 u 1 0, 0 = 0, F x,y1,y 2 1, 2 u 2 0, 0 = 0, 5.17 or x > 0, y 1 > 0, y 2 = 0 and F x,y1,0 1, 2 u 1 0, 0 = 0, F x,y1,0 1, 2 u 2 0, 0 0, 5.18 or x > 0, y 1 = 0, y 2 > 0 and F x,0,y2 1, 2 u 1 0, 0 0, F x,0,y2 1, 2 u 2 0, 0 = 0, 5.19

19 or x = 0, y 1 > 0, y 2 > 0 and F 0,y1,y 2 1, 2 u 1 0+, 0 = 0, F 0,y1,y 2 1, 2 u 2 0, 0+ = 0, 5.20 or x = 0, y 1 > 0, y 2 = 0 and for any α [0, 1 ] provided that 2 > 1 or any 2 1 α [0, when 2 1 we have 1 + α 1 α 2 w x 0+, y 1, αw y 1 0, y 1, 0 + αw y 2 0, y 1, 0+ 0, 5.21 or x = 0, y 1 = 0, y 2 > 0 and for any α [0, 2 ] provided that 1 > 2 or any 1 2 α [0, when 1 2 we have α α 2 w x 0+, 0, y 2 + αw y 1 0, 0+, y αw y 2 0, 0, y 2 0, 5.22 or x > 0, y 1 = 0, y 2 = 0 and F x,0,0 1, 2 u 1 0, 0 0, F x,0,0 1, 2 u 2 0, Proof Equality 5.16 mean that it i optimal to do nothing when we are at x, y 1, y 2 and we have only price 1 and 2 for the firt and econd aet repectively. Therefore when x > 0, y 1 > 0 and y 2 > 0 partial derivative of the function F x,y1,y 2 u 1, u 2 1, 2 hould for u 1 = 0 and u 2 = 0 be equal to 0. When x > 0, y 1 > 0, y 2 = 0 function F x,y1,0 u 1, u 2 attain it maximum for u 1 = 0 and u 2 = 0, and the partial 1, 2 derivative for u 1 hould be equal to 0, while for u 2 hould be nonnegative the function hould be increaing. The cae x > 0, y 1 = 0, y 2 > 0 can be tudied in a imilar way. When x = 0, y 1 > 0, y 2 > 0 we can ell both aet and therefore we have In the cae when x = 0, y 1 > 0, y 2 = 0 we conider the function: u w αu 2 αu, y αu, 0 + αu in which α i nonnegative and uch that αu 2 αu 0. Thi function attain it maximum for u = 0 and therefore it derivative hould be nonnegative for each α within the range defined in Lemma. When x = 0, y 1 > 0, y 2 = 0 we conider the function: u w0 1 αu αu, 0 + αu, y αu which hould have nonnegative derivative for α a in the tatement of Lemma. For the cae x > 0, y 1 = 0, y 2 = 0 we have function F x,0,0 which hould have nonnegative derivative both in u 1 and u 2. 1, 2 An analog of Propoition 2.2 can be formulated a follow Propoition 5.4 We have ŵx, y 1, y 2, 1, 2 = ŵx, y 1, y 2, 1, 2 = wx, y 1, y for x > 0, y 1 > 0 and y 2 > 0 when 1, 2 = 1, 2,forx > 0, y 1 > 0 and y 2 = 0 when 1 = 1,forx > 0, y 1 = 0 and y 2 > 0 when 2 = 2,forx = 0,

20 y 1 > 0 and y 2 > 0 when 1, 2 = 1, 2,forx = 0, y 1 > 0 and y 2 = 0 when 5.21 i atified both for 1, 2 and 1, 2, and α a in Lemma 5.3, forx = 0, y 1 = 0 and y 2 > 0 when 5.22 i atified both for 1, 2 and 1, 2, and α a in Lemma 5.3, and finally for x > 0, y 1 = 0 and y 2 = 0 when 5.23 i atified both for 1, 2 and 1, 2. Proof When x > 0, y 1 > 0 and y 2 > 0 then we have 5.17 both for 1, 2 and 1, 2. Since F x,y1,y 2 1, 2 u 1 0, 0 = 1 w x x, y 1, y 2 w y 1 x, y 1, y 2 = and the ame hold for 1 we therefore have 1 = 1. From the partial derivative with repect to u 2 we get 2 = 2. In the cae x > 0, y 1 > 0 and y 2 = 0orx > 0, y 1 = 0 and y 2 > 0 from the partial derivative with repect to u 1 or u 2 repectively we get either 1 = 1 or 2 = 2. In the cae x = 0, y 1 > 0 and y 2 > 0from5.20 we have 1 w x 0+, y 1, y 2 w y 1 0+, y 1, y 2 = 0 2 w x 0+, y 1, y 2 w y 2 0+, y 1, y 2 = both for 1, 2 and 1, 2 from which we obtain 1, 2 = 1, 2.Theremaining part of Propoition that i the cae when two variable are equal to 0 follow directly from Lemma 5.3. Let Ax, y 1, y 2, 1, 1, 2, 2 := l 1, m 1, l 2, m 2 [0, [0, 2 : x + m 1 1 l m 2 2 l 2 2 0, y 1 m 1 + l 1 0, y 2 m 2 + l be the et of admiible portfolio trategie when we have bid and ak price 1, 1 and 2, 2 for aet I and aet II repectively. Define the correponding to uch trategie one period value function wx, y 1, y 2, 1, 1, 2, 2 := up wx + m 1 1 l m 2 2 l 2 2, y 1 l 1,m 1,l 2,m 2 Ax,y 1,y 2, 1, 1, 2, 2 m 1 + l 1, y 2 m 2 + l The following lemma decribe our deciion rule we ue in the cae of w Lemma 5.5 We have [ wx, y 1, y 2, 1, 1, 2, 2 = max ŵ bb x, y 1, y 2, 1, 2, ŵ b x, y 1, y 2, 1, 2, ] ŵ b x, y 1, y 2, 1, 2, ŵ x, y 1, y 2, 1,

21 Proof In fact, tarting from x, y 1, y 2 we have four poible choice of our trategie: buy firt and econd aet, buy firt and ell econd, ell firt and buy econd or ell both aet, taking into account that in each cae we can do nothing, o that no tranaction are included in thi cheme. Furthermore Lemma 5.6 We have [ wx, y 1, y 2, 1, 1, 2, 2 min ŵx, y 1, y 2, 1, 2, ŵx, y 1, y 2, 1, 2, ] ŵx, y 1, y 2, 1, 2, ŵx, y 1, y 2, 1, Proof Clearly x + m 1 l m 2 l 2 2 x + m 1 1 l m 2 2 l 2 2 and therefore Ax, y 1, y 2, 1, 1, 2, 2 Âx, y 1, y 2, 1, Similarlywehave Ax, y 1, y 2, 1, 1, 2, 2 Âx, y 1, y 2, 1, 2, Ax, y 1, y 2, 1, 1, 2, 2 Âx, y 1, y 2, 1, 2, Ax, y 1, y 2, 1, 1, 2, 2 Âx, y 1, y 2, 1, Now by 5.31 and monotonicity of w with repect to each coordinate wx, y 1, y 2, 1, 1, 2, 2 := up l 1,m 1,l 2,m 2 Ax,y 1,y 2, 1, 1, 2, 2 wx + m 1 1 l m 2 2 l 2 2, y 1 m 1 + l 1, y 2 m 2 + l 2 up l 1,m 1,l 2,m 2 Âx,y 1,y 2, 1, 2 wx + m 1 l m 2 l 2 2, y 1 m 1 + l 1, y 2 m 2 + l 2 = ŵx, y 1, y 2, 1, and imilarly uing 5.32 we obtain wx, y 1, y 2, 1, 1, 2, 2 ŵx, y 1, y 2, 1, 2, wx, y 1, y 2, 1, 1, 2, 2 ŵx, y 1, y 2, 1, 2, wx, y 1, y 2, 1, 1, 2, 2 ŵx, y 1, y 2, 1, 2, which complete the proof. We now define no tranaction zone, buying-no tranaction, no tranaction - buying, elling - no tranaction, no tranaction - elling, buying-buying, buying-elling,

22 elling-buying and elling-elling zone repectively, to implify notation we kip the dependence of thee zone on the value of 1, 1, 2, 2 and the range R 3 + of x, y 1, y 2 NT := x, y 1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = wx, y 1, y BNT := x, y1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = up l 1 0: x l wx l1 1, y 1 + l 1, y 2 NT NTB := x, y1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = up l 2 0: x l wx l2 2, y 1, y 2 + l 2 \ NT SNT := x, y1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = up m 1 0: m 1 y 1 wx + m 1 1, y 1 m 1, y 2 \ NT NTS := x, y1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = up wx + m 2 2, y 1, y 2 m 2 \ NT BB := m 2 0: m 2 y 2 x, y 1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = ŵ bb x, y 1, y 2, 1, 2 \ BNT NTB NT BS := x, y 1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = ŵ b x, y 1, y 2, 1, 2 \ BNT NTS NT SB := x, y 1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = ŵ b x, y 1, y 2, 1, 2 \ SNT NTB NT SS := x, y 1, y 2 : wx, y 1, y 2, 1, 1, 2, 2 = ŵ x, y 1, y 2, 1, 2 \ SNT NTS NT 5.35

23 Theorem 5.7 The optimal trategie in the et BB BS SB SS are to reach the point h 0 c, 1, 2, h 1 c, 1, 2, h 2 c, 1, 2, 5.36 where c i the wealth correponding to price 1, 2 which depend on the zone BB, BS, SB, SS, that i we have BB = x, y 1, y 2 : y 1 < h 1 x + y y 2 2, 1, 2, y 2 < h 2 x + y y 2 2, 1, 2, 5.37 BS = x, y 1, y 2 : y 1 < h 1 x + y y 2 2, 1, 2, y 2 > h 2 x + y y 2 2, 1, 2, 5.38 SB = x, y 1, y 2 : y 1 > h 1 x + y y 2 2, 1, 2, y 2 < h 2 x + y y 2 2, 1, 2, 5.39 SS = x, y 1, y 2 : y 1 > h 1 x + y y 2 2, 1, 2, y 2 > h 2 x + y y 2 2, 1, When x, y 1, y 2 NT it i optimal do not change our portfolio. In the et BNT, NTB, SNT, NTS the control i reduced to one aet cae tudied in Theorem The et NT, BNT, NTB, SNT, NTS, BB, BS, SB and SS are dijoint and cover all nonzero portfolio cae i.e. all nonnegative portfolio except of trivial zero portfolio. Proof Notice firt that when x, y 1, y 2 BB then by the formula 5.34 it i optimal to buy both aet. It i in the cae when y 1 < h 1 x + y y 2 2, 1, 2 and y 2 < h 2 x + y y 2 2, 1, 2, which i in fact the formula for BB in Other formulae follow from imilar conideration. The form of optimal portfolio follow directly from optimal trategie for function w. Directly from the form of 5.37 we ee that the et BB, BS, SB and SS are dijoint. The other et are dijoint almot by the definition In analogy to Propoition 2.13 conider now hadow price for two aet cae. Propoition 5.8 If x, y 1, y 2 NT and x 1, y 1, y 2 > 0 there exit unique price ŝ 1 x, y 1, y 2 and ŝ 2 x, y 1, y 2 uch that 1 ŝ 1 x, y 1, y 2 1, 2 ŝ 2 x, y 1, y 2 2 and ŵx, y 1, y 2, ŝ 1 x, y 1, y 2, ŝ 2, y 1, y 2 = wx, y 1, y The price ŝ 1 x, y 1, y 2 and ŝ 2 x, y 1, y 2 for x, y 1, y 2 NT are called hadow price. Proof If x, y 1, y 2 NT then wx, y 1, y 2, 1, 1, 2, 2 = wx, y 1, y 2.Forc > 0, > 0let h 1 c,, y 1 := h 2 c,, y 2 := arg max wx, y 1, y x,y 2 : x+y 2 =c,x 0,y 2 0 arg max wx, y 1, y x,y 1 : x+y 1 =c,x 0,y 1 0

24 Extending the proof of Lemma 2.5 we ee that h 1 and h 2 are continuou function. Conequently by Theorem 2.10 we have that h 2 0 x + 1 y 1, 1, y 2 x h 2 0 x + 1 y 1, 1, y 2 and uing Propoition 2.11 and 2.13 we obtain the exitence of ŝ 1 x, y 1, y 2 uch that x = h 2 0 x +ŝ1 x, y 1, y 2 y 1, ŝ 1 x, y 1, y 2, y 2 and y 1 = h 2 1 x +ŝ1 x, y 1, y 2 y 1, ŝ 1 x, y 1, y 2, y 2 and wx, y 1, y 2, ŝ 1 x, y 1, y 2, ŝ 1 x, y 1, y 2, 2, 2 = wx, y 1, y 2. But thi mean that x, y 2 i in a no tranaction zone in the market with bid and ak price 2, 2 with fixed y 1.Uing Theorem 2.10 again we have h 1 0 x + 2 y 2, 2, y 1 x h 1 0 x + 2 y 2, 2, y 1 and by Propoition 2.11 and 2.13 we obtain the exitence of ŝ 2 x, y 1, y 2 uch that x = h 1 0 x +ŝ2 x, y 1, y 2 y 1, ŝ 2 x, y 1, y 2, y 1 and y 2 = h 1 1 x +ŝ2 x, y 1, y 2 y 1, ŝ 2 x, y 1, y 2, y 1 and finally we have Multidimenional Induction Step and Dynamic Portfolio We hall formulate here major tep and ketch main difference in the proof, which allow u to tudy the cae of two aet in a imilar way a in the one aet cae. Propoition 6.1 Function ŵ i continuouly differentiable at point x, y 1, y 2, 1, 2 R 3 + ˆD 2 with repect to firt three coordinate. Proof We follow the proof of Propoition Forz > 0let Ɣz := u 1, u 2 : u 1 0, u 2 0, u 1 + u 2 z. Define w z, u 1, u 2, 1, 2 := wu 1, u 2, z u 1 u Then ŵx, y 1, y 2, 1, 2 = up u 1,u 2 Ɣx+ 1 y y 2 w x + 1 y y 2, u 1, u 2, 1, and conider the function W z, 1, 2 := up w z, u 1, u 2, 1, 2. u 1,u 2 Ɣz When function w attain it maximum over Ɣz for u 1,z, 1, 2 > 0, u 2,z, 1, 2 > 0 uch that u 1,z, 1, 2 +u 2,z, 1,2 < z then differentiability of W we again locally may aume that upremum i attained in a fixed compact ubet of Ɣz follow from Propoition 7.2. Moreover ince ŵx, y 1, y 2, 1, 2 = W x + 1 y y 2, 1, 2,by7.3 partial derivative of ŵ are in fact partial derivative of w with u 1,x+ 1 y y 2, 1, 2 > 0, u 2,x+ 1 y y 2, 1, 2 > 0. When the maximum of w over Cz, 1, 2 i attained at point x 0, ȳ 1 0, ȳ 2 = 0, we have ŵx, y 1, y 2, 1, 2 = w x, ȳ 1, 0, 1, 2 = W x + y y 2 2, where W z, 1 := up u [0,z] wu, z u, 0. Continuou differentiability of W with repect to z follow from the proof of 1 Propoition By6.2 clearly alo ŵ i continuouly differentiable with repect to x, y 1, y 2.

25 The cae when w over Cz, 1, 2 i attained at point x = 0, ȳ 1 0, ȳ 2 0 and at point x 0, ȳ 1 = 0, ȳ 2 0 can be hown in a imilar way uing the ame argument. Furthermore we have Corollary 6.2 Function w i continuouly differentiable with repect to firt three coordinate at point x, y 1, y 2, 1, 1, 2, 2 R 3 + D2 for x, y 1, y 2 BB BS SB SS a well a in the interior of the et NT, BNT, NTB, SNT, NTS. Proof The et BB BS SB SS i open and therefore for e.g. x, y 1, y 2 BB while for x, y 1, y 2 BS wx, y 1, y 2, 1, 1, 2, 2 = ŵx, y 1, y 2, 1, 2 wx, y 1, y 2, 1, 1, 2, 2 = ŵx, y 1, y 2, 1, 2 o that continuou differentiability of w follow from continuou differentiability of ŵ. In the cae x, y 1, y 2 SB and x, y 1, y 2 SS we have the ame conideration. In the cae when x, y 1, y 2 i in the interior of NT we have that wx, y 1, y 2, 1, 1, 2, 2 = wx, y 1, y 2 and continuou differentiability of w follow from differentiability of w. In the cae when x, y 1, y 2 i in the interior of BNT we have that wx, y 1, y 2, 1, 1, 2, 2 = ŵx, y 1, y 2, 1, 2 and again continuou differentiability of w follow from continuou differentiability of ŵ. The cae when x, y 1, y 2 i in the interior of SNT or NTS can be hown in the ame way. Aume on a given filtered probability pace,f,f t t=0,1, P we have four F 1 -random variable S 1 1, S1 1, S2 1, S2 1 uch that 0 < S1 1 < S1 1 and 0 < S2 1 < S 2 1 uch that for each x, y1, y 2 R 3 + the derivative of the random variable wx, y, S 1 1, S1 1, S2 1, S2 1 whenever exit are integrable. Furthermore aume that conditional law P S 1 1, S1 1, S2 1, S2 1 F0 i continuou. For x, y 1, y 2 R 3 + put wx, y 1, y 2 := E wx, y 1, y 2, S 1, S 1, S 2, S 2 F0, conidering it a a regular conditional expected value uing Lemma 7.3. Wehave Propoition 6.3 Random function w i trictly concave. Proof We ue the argument of Propoition 3.1. Namely we have to how that for two different point x 1, y1 1, y2 1 and x 2, y2 1, y2 2 we have to exclude P a.e. the cae when there exit l1 1, m1 1, l2 1, m2 1 Ax 1, y1 1, y2 1, S1 1, S1 1, S2 1, S2 1 and l1 2, m1 2, l2 2, m2 2 Ax 2, y2 1, y2 2, S1 1, S1 1, S2 1, S2 1 uch that

26 x 1 + S 1 1 m1 1 S1 1 l1 1 + S2 1 m2 1 S2 1 l2 1, y1 1 m1 1 + l1 1, y2 1 m2 1 + l2 1 = x 2 + S 1 1 m1 2 S1 1 l1 2 + S2 1 m2 2 S2 1 l2 2, y1 2 m1 2 + l1 2, y2 2 m2 2 + l2 2. auming additionally that l1 1 m1 1 = 0, l1 2 m1 2 = 0, l2 1 m2 1 = 0 and l2 2 m2 2 = 0. We hall conider three general cae: Cae 1. In both cae i.e. when we tart with initial poition x 1, y1 1, y2 1 and x 2, y2 1, y2 2 we make the ame kind of deciion: that i we buy or ell the firt aet and we buy or ell the econd aet. In thi cae S 1 1, S2 1,orS1 1, S2 1,orS1 1, S2 1 or S1 1, S2 1 lie on at mot two dimenional hyperpace which can happen with probability 0 by continuity of the law. Cae 2. Our invetment trategie for x 1, y1 1, y2 1 and x 2, y2 1, y2 2 differ in the cae of one aet. To be more precie conider the cae m 1 1 = 0, l2 1 = 0 and m1 2 = 0, m 2 2 = 0. Then we have x 1 l1 1S1 1 + m2 1 S2 1 = x 2 l2 1S1 1 l2 2 S2 1, y1 1 + l1 1 = y1 2 + l1 2, and y1 2 m2 1 = y2 2 + l2 2.Ifm2 1 > 0 and l2 2 > 0 then ŵx 1 l1 1 S1 1 + m2 1 S2 1, y1 1 + l1 1, y2 1 m2 1, S1 1, S2 1 = ŵx 1 l1 1 S1 1 + m2 1 S2 1, y1 1 + l1 1, y2 1 m2 1, S1 1, S from which by Propoition 5.4 we have that a x 1 l1 1S1 1 + m2 1 S2 1 = 0, or b y1 1 + l1 1 = 0orcy2 1 m2 1 = 0. If y1 1 + l1 1 > 0 and y2 1 m2 1 > 0 then a can not happen by Propoition 5.4. Therefore it remain to conider the cae b and c. In the cae b we have y1 1 = 0 = l1 1, and then alo y1 1 = 0 = l1 2. Conequently we have x 1 m 2 1 S2 1 = x 2 l2 2S2 1 and y2 1 m2 1 = y2 2 + l2 2 and we can continue a in the proof of Propoition 3.1. In the cae c we alo have that y2 2 = l2 2 = 0 and then x 1 x 2 + S 1 1 y1 1 y1 2 + S2 1 y2 1 = 0 which can happen with probability 0 ince it would mean that S 1 1, S2 1 lie on the hyperplane. It remain to conider the cae m 2 1 = 0 and l2 2 = 0. When m2 1 = 0 then y2 1 = y2 2 + l2 2 and l2 2 = y2 1 y2 2 o that we have x 1 x 2 + S 1 1 y1 1 y1 2 + y2 1 y2 2 S2 1 = 0, which can happen with probability 0. When l2 2 = 0 then m2 1 = y2 1 y2 2 and x 1 x 2 + S 1 1 y1 1 y1 2 + S2 1 y2 1 = 0, which again can happen with probability 0. Cae 3. Our invetment trategie for x 1, y1 1, y2 1 and x 2, y2 1, y2 2 differ for each aet. Thi i the cae when e.g. we have m 1 1 = 0, l2 1 = 0 and l1 2 = 0, m2 2 = 0. We then have x 1 l1 1S1 1 +m2 1 S2 1 = x 2 m 1 2 S1 1 +l2 2 S2 1, y1 1 +l1 1 = y1 2 m1 2, and y2 1 m2 1 = y2 2 +l2 2. When m 2 1 > 0 and l2 2 > 0 then we have three cae: a x 1 l1 1S1 1 + m2 1 S2 1 = 0, b y1 1 + l1 1 = 0orcy2 1 m2 1 = 0. When y1 1 + l1 1 > 0 and y2 1 m2 1 > 0 the cae a by Propoition 5.4 can not happen. In the cae b we have y1 1 = l1 1 = 0 and y1 2 = m1 2 and we can continue a in the proof of Propoition 3.1. In the cae c we have y1 2 = m2 1 and y2 2 = l2 2 = 0 and we come to the equation x 1S 1 1 x 1 + y1 2S2 1 + y2 2 S2 1 = 0, which can happen with probability 0. The cae m 2 1 = 0orl2 2 = 0 lead to the cae tudied in the proof of Propoition 3.1. To finih induction tep a in the two dimenional cae we need the following

27 Propoition 6.4 Random function w i continuouly differentiable for x, y 1, y 2 R 3 + almot urely. Proof Notice that w maybe not differentiable at point x, y 1, y 2 whenever we have ŵx, y 1, y 2, 1, 2 = wx, y 1, y which correpond to the boundary of NT. Then in the cae: x > 0, y 1 > 0, y 2 > 0, x = 0, y 1 > 0, y 2 > 0, x > 0, y 1 = 0, y 2 > 0, x > 0, y 1 > 0, y 2 = 0, x > 0, y 1 = 0, y 2 = 0 uing Propoition 5.4 we ee that conditional law of value of 1, 2 for which we have 6.4 i equal to 0, by the continuity of the law P S 1 1, S1 1, S2 1, S2 1 F0. In the cae when ŵ0, y 1, 0, 1, 2 = wx, y 1, 0 or ŵ0, 0, y 2, 1, 2 = wx, 0, y 2 do not determine uniquely 1 or 2 repectively but in that cae continuou differentiability of w follow a in Propoition 3.2. Other cae of poible violation of differentiability concern ituation, when our poition in one aet i at the boundary of no tranaction zone while the other aet we are in buying or elling zone. Such cae practically reduce to one aet cae, which wa tudied in Propoition 3.2. To continue contruction of dynamical portfolio and to prove a two aet analog of Theorem 4.1 we hall need two aet verion of the aumption A and B concerning integrability of uitable function w k and continuity of the conditional law of P S 1 k+1, S1 k+1, S2 k+1, S2 k+1 Fk for k = 0,...,T 1. Open Acce Thi article i ditributed under the term of the Creative Common Attribution 4.0 International Licene which permit unretricted ue, ditribution, and reproduction in any medium, provided you give appropriate credit to the original author and the ource, provide a link to the Creative Common licene, and indicate if change were made. Appendix Let f : R n+k + R for n, k = 1, 2,... be a continuou function, which i continuouly differentiable with repect to the firt n variable with the meaning that it derivative are continuou function of their all coordinate. Let U R k + be a compact et. Lemma 7.1 For x 1,...,x n R n + and i = 1, 2,...,n we have lim up h 0 u U f x 1,...,x i 1, x i + h, x i+1,...,x n, u f x 1,...,x n, u f x 1,...,x n, u h x i = 0. h