Optimal Strategies for Utility from Terminal Wealth with General Bid and Ask Prices
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Abstract
In the paper we study utility from terminal wealth maximization with general bid and ask prices studying first one asset case and generalizing then results to the multi asset case. We show that under certain assumptions (continuous conditional distribution of the assets) and strict concavity of the utility function the problem can be reduced to study a static problem and then by an induction to consider multi period case. We obtain formulae for buying, selling and no transaction zones both in one and two asset cases. We also show the existence and the form of shadow prices.
Keywords
Utility from terminal wealth General bid and ask prices Shadow priceMathematics Subject Classification
93E20 91G101 Introduction
2 Static Two Dimensional Case
Let the function \(w : {\mathbb {R}}_{+}^2 \longrightarrow {\mathbb {R}}\) be strictly increasing with respect to both variables, strictly concave and continuously differentiable. We shall consider it as a one period value function which depends on the value of our bank account and number of assets in our portfolio. We recall our notation that \( {\mathbb {R}}_{+}^2:=[0,\infty )\times [0,\infty )\setminus \left\{ (0,0)\right\} \). Let \({\mathbb {D}} := \{ (\underline{s}, \overline{s}) \in {\mathbb {R}}_{+}^{2} : \quad 0< \underline{s} < \overline{s} \}\) and \({\hat{{\mathbb {D}}}} := \{ {\hat{s}} \in {\mathbb {R}}_{+} : \quad {\hat{s}} > 0 \}\).
From now on we assume that \((\underline{s}, \overline{s}) \in {\mathbb {D}}\) is fixed.
Lemma 2.1
For every \(c > 0\) and \(s\in {\hat{{\mathbb {D}}}}\) the functions \(H_{c,s}\) are strictly concave on [0, c].
Proof
This is the consequence of the fact that for every \(c > 0\) and \(s\in {\hat{{\mathbb {D}}}}\) the functions \(H_{c,s}\) are compositions of linear and strictly concave functions. \(\square \)
Functions \({\hat{w}}\) strongly depend on the asset price s. The result below shows that when the market positions (x, y) are positive such dependence is injective.
Proposition 2.2
Proof
Remark 2.3
Notice that (2.12) can not happen for \(x>0\) and \(y>0\), when w(x, y) is concave (non necessarily strictly concave), differentiable and increasing with respect to both coordinates. Strict concavity assumption will be important to study differentiability of \({\hat{w}}\).
Corollary 2.4
Let \(c > 0\) be such that \(h_0 (c,\underline{s}) = c\). Then \(h_0 (c,\overline{s}) = c\). Moreover, if \(y > 0\) is such that \(h_0(\overline{s} y,\overline{s}) = 0\), then also \(h_0 (\underline{s} y, \underline{s}) = 0\).
Proof
Taking into account concavity of \(H_{c,\underline{s}}\) we get \(h_0 (c,\underline{s}) = c\) only when \(H_{c,\underline{s}}'(c) \geqslant 0\). Then by (2.11) also \(H_{c,\overline{s}}'(c) > 0\). In effect, the concavity of \(H_{c,\overline{s}}\) implies that \(h_0 (c,\overline{s}) = c\).
Similarly, if \(y > 0\) is such that \(h_0(\overline{s} y,\overline{s}) = 0\), then \(H_{\overline{s} y,\overline{s}}(0+) \leqslant 0\) which implies that \(H_{\underline{s} y,\underline{s}} < 0\) and \(h_0 (\underline{s} y,\underline{s}) = 0\). \(\square \)
Taking into account strict concavity of the function w we can show that the selector h defined in (2.14) is continuous. Namely, we have
Lemma 2.5
Function h is continuous on \((0, \infty ) \times (0, \infty )\).
Proof
The next Corollary characterizes properties of the graph of h.
Corollary 2.6
The graph of the mapping \((c, s) \longmapsto h(c, s)\) does not have common points except of points \((x, 0) \in {\mathbb {R}}_{+}^{2}\) whenever \(H_{x,s}'(x) \geqslant 0\) and \((0, y) \in {\mathbb {R}}_{+}^{2}\) whenever \(H_{s y,s}'(0+) \leqslant 0\).
Proof
Consequently, \({\hat{w}}(x, y, s_{1}) = {\hat{w}}(x, y, s_{2}) = w(x, y)\), which by Proposition 2.2 can happen only when \(y = 0\) and \(H_{x, s_{1}}'(x) \geqslant 0\) or when \(x = 0\) and \(H_{s_{2} y, s_{2}}(0+) \leqslant 0\). \(\square \)
We have characterized so far the function \({\hat{w}}\) which corresponded to one asset price s. Now we study the function \({\overline{w}}\) which is the optimal value corresponding to bid price \(\underline{s}\) and ask price \(\overline{s}\).
Lemma 2.7
Proof
Notice that the meaning of (2.17) and (2.18) is that starting from (x, y) we can buy or sell assets for \(\overline{s}\) or \(\underline{s}\) respectively. By (2.6) we immediately get (2.19). \(\square \)
Functions \({\bar{w}}\) and \({\hat{w}}\) inherit concavity property of function w, which will be important for further studies. We have
Proposition 2.8
For every \((\underline{s}, \overline{s}) \in {\mathbb {D}}\) the functions \({\overline{w}}(\cdot , \cdot , \underline{s}, \overline{s}), {\hat{w}}(\cdot , \cdot , \underline{s})\) and \({\hat{w}}(\cdot , \cdot , \overline{s})\) are concave on \({\mathbb {R}}_{+}^{2}\). In particular, they are continuous on \({\mathbb {R}}_{+}^{2}\).
Proof
Remark 2.9
Theorem 2.10
Proof
Let \((x, y) \in {\mathbb {R}}_{+}^{2}\). If \({\overline{w}}(x, y, \underline{s}, \overline{s}) = w(x, y)\), then by (2.17) we are not able to increase the value of w(x, y) buying or selling assets. Consequently by strict concavity of w we should have that \(h_0 (x + \underline{s} y, \underline{s}) \leqslant x\) and \(h_0 (x + \overline{s} y, \overline{s}) \geqslant x\). If \((x, y) \in \mathbf {B}(\underline{s}, \overline{s})\), then by (2.17) we have that \(x > h_0 (x + \overline{s} y, \overline{s})\). If \((x, y) \in \mathbf {S}(\underline{s}, \overline{s})\), then by (2.17) we have that \(x < h_0 (x + \underline{s} y, \underline{s})\). If \(x \geqslant h_0 (x + \overline{s} y, \overline{s})\), then \({\overline{w}}(x, y, \underline{s}, \overline{s}) = {\hat{w}}(x, y, \overline{s})\) and we have equality in (2.19). If \(x \leqslant h_0 (x + \underline{s} y, \underline{s})\), then \({\overline{w}}(x, y, \underline{s}, \overline{s}) = {\hat{w}}(x, y, \underline{s})\) and we have equality in (2.19). If \(h_0 (x + \underline{s} y, \underline{s})< x < h_0 (x + \overline{s} y, \overline{s})\), then \({\overline{w}}(x, y, \underline{s}, \overline{s}) < {\hat{w}}(x, y, \underline{s})\) and \({\overline{w}}(x, y, \underline{s}, \overline{s}) < {\hat{w}}(x, y, \overline{s})\) and we have strict inequality in (2.19). By the continuity of the mappings \(c \longmapsto h_0 (c, \underline{s})\) and \(c \longmapsto h_0 (c, \overline{s})\), we have that \(\mathbf {NT}^{\circ }(\underline{s}, \overline{s})\) is an open set. Clearly, its closure coincides with \(\mathbf {NT}(\underline{s}, \overline{s})\). This ends the proof. \(\square \)
Using (2.18) we can obtain an alternative version of the formulae for the zones in terms of the selector \(h_1\).
Proposition 2.11
We also immediately have
Corollary 2.12
For given \((x,y) \in {\mathbb {R}}_{+}^{2}\) we are looking for \({\hat{s}}\in [\underline{s},\underline{s}]\) such that \({\overline{w}}(x, y, \underline{s}, \overline{s})={\hat{w}}(x,y,{\hat{s}})\). Such value if exists is called a shadow price (see [10] for more explanation). It clearly depends on the value of (x, y). It is rather obvious (see again to [10]) that shadow price for (x, y) in \(\mathbf {B} (\underline{s}, \overline{s})\) is equal to \(\overline{s}\), while for (x, y) in \(\mathbf {S}(\underline{s}, \overline{s})\) is equal to \(\underline{s}\). The only problem is to find shadow price for \((x,y)\in \mathbf {NT} (\underline{s}, \overline{s})\) i.e. in the no transaction zone corresponding to bid \(\underline{s}\) and ask \(\overline{s}\) prices.
Proposition 2.13
Let \((x, y, \underline{s}, \overline{s}) \in {\mathbb {R}}_{+}^{2} \times {\mathbb {D}}\) be such that \((x, y) \in \mathbf {NT}^{\circ }(\underline{s}, \overline{s})\) and \(y > 0\). Then there exists a unique \({\hat{s}}(x, y) \in (\underline{s}, \overline{s})\) such that \(x = h_0 \big ( x + {\hat{s}}(x, y) y, {\hat{s}}(x, y) \big )\) and \(y = h_1 \big ( x + {\hat{s}}(x, y) y, {\hat{s}}(x, y) \big )\). Moreover, the function \({\hat{s}}\) can be extended to a continuous function on \((0, \infty ) \times (0, \infty ) \cap \mathbf {NT}(\underline{s}, \overline{s})\).
Proof
Remark 2.14
Proposition 2.13 says that in our model shadow price is uniquely defined for \((x,y)\in (0, \infty ) \times (0, \infty ) \cap \mathbf {NT}(\underline{s}, \overline{s})\), and furthermore is a continuous function. From Proposition 2.2 and its proof we have that \({\hat{s}}\) is not uniquely defined only at points (x, 0) whenever \(H_{x,\underline{s}}'(x)\geqslant 0\) and and (y, 0) when \(H_{y\overline{s},\overline{s}}'(0+)\leqslant 0\). For such points any value from the interval \([\underline{s}, \overline{s}]\) may serve as a shadow price. In the set \(\mathbf {B}(\underline{s}, \overline{s})\) we have \({\hat{s}}=\overline{s}\), while in the set \(\mathbf {S}(\underline{s}, \overline{s})\) we have \({\hat{s}}=\underline{s}\).
In the proof of Proposition 2.2 differentiability of w was important. Now we consider first differentiability of \({\hat{w}}\) and then differentiability of \({\overline{w}}\).
Proposition 2.15
Function \({\hat{w}}\) is continuously differentiable at point \((x, y, {s}) \in {\mathbb {R}}_{+}^{2} \times {\hat{{\mathbb {D}}}}\).
Proof
For \((z, s) \in {\mathbb {R}}_{+} \times {\hat{{\mathbb {D}}}}\) and \(u \in [0, z]\) define \(w^{*}(z, u, s) := w \Big ( u, \frac{z  u}{s} \Big )=H_{z,s}(u)\).
When \(H_{x+sy,{s}}'(x+sy) \geqslant 0\) then \({\hat{w}}(x, y, s)=w(x+sy,0)\) and when \(H_{x+ys,{s}}'(0+) \leqslant 0\) then \({\hat{w}}(x, y, s)=w(0,{1\over s}x+y)\). When \(H_{x+sy,{s}}'(x+sy) > 0\) or \(H_{x+ys,{s}}'(0+) < 0\) then in some neighborhood of (x, y, s) we have \({\hat{w}}(x, y, s)=w(x+sy,0)\) or \({\hat{w}}(x, y, s)=w(0,{1\over s}x+y)\) and differentiability follows from differentiability of w. Consequently we may have problem with differentiability only when \(H_{x+sy,{s}}'(x+sy) = 0\) or \(H_{x+ys,{s}}'(0+) = 0\). Then from (2.24) to (2.26) by continuity of \((x,y,s)\mapsto u_{x+sy,s}\) (which follows from uniqueness of \(u_{x+sy,s}\)) we obtain continuous differentiability of \({\hat{w}}\) at (x, y, s). \(\square \)
Corollary 2.16
Proof
When \((x,y) \in \mathbf {NT}^{\circ }(\underline{s},\overline{s})\) we have that \({\overline{w}}(x,y,\underline{s},\overline{s})=w(x,y)\) which is continuously differentiable. For \((x,y) \in \mathbf {S}(\underline{s},\overline{s})\) we have that \({\overline{w}}(x,y,\underline{s},\overline{s})={\hat{w}}(x,y,\underline{s})\) which is continuously differentiable by Proposition 2.15. Finally for \((x,y) \in \mathbf {B}(\underline{s},\overline{s})\) we have \({\overline{w}}(x,y,\underline{s},\overline{s})={\hat{w}}(x,y,\overline{s})\) which is again continuously differentiable by Proposition 2.15. Consequently we may have problem with differentiability only at the boundary of \(\mathbf {NT}(\underline{s},\overline{s})\). Moreover in the sets \(\left\{ (x,0): H_{x,\underline{s}}'(x)\geqslant 0\right\} \cup \left\{ (0,y): H_{y\overline{s},\overline{s}}'(0+)\leqslant 0\right\} \) we get continuous differentiability as in end of the proof of Proposition 2.15, since it is a subset of \(\mathbf {NT}(\underline{s},\overline{s})\). \(\square \)
3 Induction Analysis
In the previous section we studied one period problem. Now we come to two period problem which in the sequel (in the next section) will be replaced by multi period problem studied by induction. Assume on a given filtered probability space \(\big ( \Omega , {\mathcal {F}}, ({\mathcal {F}}_{t})_{t = 0, 1}, \mathbb {P} \big )\) we are given two \({\mathcal {F}}_{1}\)random variables \({\underline{S}}_{1}\) and \({\overline{S}}_{1}\) such that \(0< {\underline{S}}_{1} < {\overline{S}}_{1}\) such that for each \((x,y)\in {\mathbb {R}}_{+}^{2}\) the derivatives of the random variable \({\overline{w}}(x,y,{\underline{S}}_{1}, {\overline{S}}_{1})\) whenever exist are integrable and that conditional law \(\mathbb {P} \big ( ({\underline{S}}_{1}, {\overline{S}}_{1}) \in \cdot \big  {\mathcal {F}}_{0} \big )\) is continuous. In what follows we shall use a regular version of such conditional probability (which there exists by Theorem 6.3 of [7]).
Proposition 3.1
Random function \({\tilde{w}}\) is strictly concave.
Proof
Case 1: \(\quad m_{1} = m_{2} = 0\)
Case 2: \(\quad l_{1} = l_{2} = 0\)
Case 3: \(\quad l_{1} = m_{2} = 0\)
Case 4: \(\quad m_{1} = l_{2} = 0\). This case is studied identically to the Case 3.
Summarizing, we have strict inequality in (3.1). This ends the proof. \(\square \)
To start induction procedure we also need continuous differentiability of \({\tilde{w}}\) which is shown below.
Proposition 3.2
Random function \({\tilde{w}}\) is continuously differentiable for \((x,y)\in {\mathbb {R}}_{+}^{2}\), \(\mathbb {P}\) almost surely.
Proof
4 Dynamic Two Dimensional Case
 (A)

bid and ask prices \(({\underline{S}}_{t}, {\overline{S}}_{t})\) for \(t=1,\ldots ,T\) are such that random functions \({\overline{w}}_k(x,y,\underline{s}, \overline{s})\) are well defined for \(k=0,1\ldots ,T\) and for \((x,y)\in {\mathbb {R}}_{+}^{2}\) random functions \({\overline{w}}_k(x,y,{\underline{S}}_{k}, {\overline{S}}_{k})\) are integrable together with their derivatives with respect to x and y (whenever they exist).
 (B)

conditional law \(\mathbb {P} \big ( ({\underline{S}}_{k+1}, {\overline{S}}_{k+1}) \in \cdot \big  {\mathcal {F}}_{k} \big )\) is continuous for \(k=0,1\ldots ,T1\).
Theorem 4.1
Proof
The proof is by an induction. Notice first that by Proposition 3.1 \({\tilde{w}}_k\) strictly concave and by Proposition 3.2 it is also continuously differentiable \(\mathbb {P}\) a.s. for \((x,y)\in {\mathbb {R}}_{+}^{2}\). Consequently by Proposition 2.15 \({\hat{w}}_k\) is continuously differentiable at \((x, y, {\hat{s}}) \in {\mathbb {R}}_{+}^{2} \times {\hat{{\mathbb {D}}}}\) for \(k=0,1,\ldots ,T\). Furthermore by Corollary 2.16 \({\overline{w}}_k\) is continuously differentiable for \((x,y)\in \mathbf {NT}_k^\circ \cup \mathbf {S}_k \cup \mathbf {B}_k\), where \(\mathbf {NT}_k^\circ \) is an interior of \(\mathbf {NT}_k\). The remaining part of the proof follows from Theorem 2.10 and Proposition 2.11. The existence and the form of shadow price follows directly from Proposition 2.13 using induction again. \(\square \)
5 Static Multi (Two) Asset Case
Lemma 5.1
Functions \(h_i\) for \(i=0,1,2\) are continuous.
Proof
It follows from the fact that function w is strictly concave which implies that its supremum over a compact set \({\mathbb {C}}\) is unique, and furthermore the mapping \((0,\infty )\times (0,\infty ) \times (0,\infty )\ni (c,s^1,s^2)\mapsto {\mathbb {C}}(c,s^1,s^2)\) is continuous in the Hausdorff metric (see the proof of theorem 2.1 in [10], or [2]). \(\square \)
Lemma 5.2
Lemma 5.3
Proof
Equality (5.16) means that it is optimal to do nothing when we are at \((x,y^1,y^2)\) and we have only prices \(s^1\) and \(s^2\) for the first and second asset respectively. Therefore when \(x>0\), \(y^1>0\) and \(y^2>0\) partial derivatives of the function \(F_{s^1,s^2}^{x,y^1,y^2}(u^1,u^2)\) should for \(u^1=0\) and \(u^2=0\) be equal to 0. When \(x>0, y^1>0, y^2=0\) function \(F_{s^1,s^2}^{x,y^1,0}(u^1,u^2)\) attains its maximum for \(u^1=0\) and \(u^2=0\), and the partial derivative for \(u^1\) should be equal to 0, while for \(u^2\) should be nonnegative (the function should be increasing). The case \(x>0, y^1=0, y^2>0\) can be studied in a similar way. When \(x=0, y^1>0, y^2>0\) we can sell both assets and therefore we have (5.20). In the case when \(x=0, y^1>0, y^2=0\) we consider the function: \(u\mapsto w(0+s^1(1+\alpha )us^2\alpha u, y^1(1+\alpha )u,0+\alpha u)\) in which \(\alpha \) is nonnegative and such that \(s^1(1+\alpha )us^2\alpha u\geqslant 0\). This function attains its maximum for \(u=0\) and therefore its derivative should be nonnegative for each \(\alpha \) within the range defined in Lemma. When \(x=0, y^1>0, y^2=0\) we consider the function: \(u\mapsto w(0s^1 \alpha u+s^2(1+\alpha ) u, 0+\alpha u, y^2(1+\alpha ) u)\) which should have nonnegative derivative for \(\alpha \) as in the statement of Lemma. For the case \(x>0, y^1=0, y^2=0\) we have function \( F_{s^1,s^2}^{x,0,0}\) which should have nonnegative derivatives both in \(u^1\) and \(u^2\). \(\square \)
An analog of Proposition 2.2 can be formulated as follows
Proposition 5.4
Proof
Lemma 5.5
Proof
In fact, starting from \((x,y^1,y^2)\) we have four possible choices of our strategies: buy first and second asset, buy first and sell second, sell first and buy second or sell both assets, taking into account that in each case we can do nothing, so that no transactions are included in this scheme. \(\square \)
Furthermore
Lemma 5.6
Proof
Theorem 5.7
Proof
Notice first that when \((x,y^1,y^2)\in \mathbf {BB}\) then by the formula (5.34) it is optimal to buy both assets. It is in the case when \( y^1 < h_1(x+y^1\overline{s}^1+y^2\overline{s}^2,\overline{s}^1,\overline{s}^2)\) and \(y^2 < h_2(x+y^1\overline{s}^1+y^2\overline{s}^2,\overline{s}^1,\overline{s}^2),\) which is in fact the formula for \(\mathbf {BB}\) in (5.37). Other formulae follows from similar consideration. The form of optimal portfolios follow directly from optimal strategies for function w. Directly from the form of (5.37) we see that the sets \(\mathbf {BB}\), \(\mathbf {BS}\), \(\mathbf {SB}\) and \(\mathbf {SS}\) are disjoint. The other sets are disjoint almost by the definition (5.34). \(\square \)
In analogy to Proposition 2.13 consider now shadow price for two assets case.
Proposition 5.8
The prices \({\hat{s}}^1(x,y^1,y^2)\) and \({\hat{s}}^2(x,y^1,y^2)\) for \((x,y^1,y^2)\in \mathbf {NT}\) are called shadow price.
Proof
6 Multidimensional Induction Step and Dynamic Portfolio
We shall formulate here major steps and sketch main differences in the proofs, which allow us to study the case of two assets in a similar way as in the one asset case.
Proposition 6.1
Function \({\hat{w}}\) is continuously differentiable at points \((x, y^1, y^2, s^1, s^2) \in {\mathbb {R}}_{+}^{3} \times {\hat{{\mathbb {D}}}}^2\) with respect to first three coordinates .
Proof
The cases when w over \({\mathbb {C}}(z,s^1,s^2)\) is attained at point \({\bar{x}}= 0, {\bar{y}}^1 \geqslant 0, {\bar{y}}^2 \geqslant 0 \) and at point \({\bar{x}}\geqslant 0, {\bar{y}}^1 = 0, {\bar{y}}^2 \geqslant 0 \) can be shown in a similar way using the same arguments. \(\square \)
Furthermore we have
Corollary 6.2
Function \({\overline{w}}\) is continuously differentiable with respect to first three coordinates at points \((x, y^1, y^2, \underline{s}^1,\overline{s}^1, \underline{s}^2, \overline{s}^2) \in {\mathbb {R}}_{+}^{3} \times {{\mathbb {D}}}^2\) for \((x,y^1,y^2)\in \mathbf {BB} \cup \mathbf {BS}\cup \mathbf {SB}\cup \mathbf {SS}\) as well as in the interiors of the sets \(\mathbf {NT}\), \(\mathbf {BNT}\), \(\mathbf {NTB}\), \(\mathbf {SNT}\), \(\mathbf {NTS}\).
Proof
Assume on a given filtered probability space \(\big ( \Omega , {\mathcal {F}}, ({\mathcal {F}}_{t})_{t = 0, 1}, \mathbb {P} \big )\) we have four \({\mathcal {F}}_{1}\)random variables \({\underline{S}}_{1}^1\), \({\overline{S}}_{1}^1\), \({\underline{S}}_{1}^2\), \({\overline{S}}_{1}^2\) such that \(0< {\underline{S}}_{1}^1 < {\overline{S}}_{1}^1\) and \(0< {\underline{S}}_{1}^2 < {\overline{S}}_{1}^2\) such that for each \((x,y^1,y^2)\in {\mathbb {R}}_{+}^{3}\) the derivatives of the random variable \({\overline{w}}(x,y,{\underline{S}}_{1}^1, {\overline{S}}_{1}^1, {\underline{S}}_{1}^2, {\overline{S}}_{1}^2)\) whenever exist are integrable. Furthermore assume that conditional law \(\mathbb {P} \big ( ({\underline{S}}_{1}^1, {\overline{S}}_{1}^1,{\underline{S}}_{1}^2, {\overline{S}}_{1}^2) \in \cdot \big  {\mathcal {F}}_{0} \big )\) is continuous.
Proposition 6.3
Random function \({\tilde{w}}\) is strictly concave.
Proof
Case 1. In both cases (i.e. when we start with initial position \((x_1,y_1^1,y_1^2)\) and \((x_2,y_2^1,y_2^2)\) we make the same kind of decisions: that is we buy or sell the first asset and we buy or sell the second asset. In this case \({\overline{S}}^1_1, {\overline{S}}^2_1\), or \({\overline{S}}^1_1, {\underline{S}}^2_1\), or \( {\underline{S}}^1_1,{\overline{S}}^2_1\) or \( {\underline{S}}^1_1,{\underline{S}}^2_1\) lie on at most two dimensional hyperspace which can happen with probability 0 (by continuity of the laws).
Case 3. Our investment strategies for \((x_1,y_1^1,y_1^2)\) and \((x_2,y_2^1,y_2^2)\) differ for each asset. This is the case when e.g. we have \(m_1^1=0\), \(l_1^2=0\) and \(l_2^1=0\), \(m_2^2=0\). We then have \(x_1l_1^1{\overline{S}}_1^1+m_1^2{\underline{S}}_1^2=x_2m_2^1{\underline{S}}_1^1+l_2^2{\overline{S}}_1^2\), \(y_1^1+l_1^1=y_2^1m_2^1\), and \(y_1^2m_1^2=y_2^2+l_2^2\). When \(m_1^2>0\) and \(l_2^2>0\) then we have three cases: (a) \(x_1l_1^1{\overline{S}}_1^1+m_1^2{\underline{S}}_1^2=0\), (b) \(y_1^1+l_1^1=0\) or (c) \(y_1^2m_1^2=0\). When \(y_1^1+l_1^1>0\) and \(y_1^2m_1^2>0\) the case (a) by Proposition 5.4 can not happen. In the case (b) we have \(y_1^1=l_1^1=0\) and \(y_2^1=m_2^1\) and we can continue as in the proof of Proposition 3.1. In the case (c) we have \(y_1^2=m_1^2\) and \(y_2^2=l_2^2=0\) and we come to the equation \(x_1{\overline{S}}_1^1x_1+y_1^2{\underline{S}}_1^2+y_2^2 {\underline{S}}_1^2=0\), which can happen with probability 0. The cases \(m_1^2=0\) or \(l_2^2=0\) lead to the cases studied in the proof of Proposition 3.1. \(\square \)
To finish induction step as in the two dimensional case we need the following
Proposition 6.4
Random function \({\tilde{w}}\) is continuously differentiable for \((x,y^1, y^2)\in {\mathbb {R}}_{+}^{3}\) almost surely.
Proof
To continue construction of dynamical portfolio and to prove a two asset analog of Theorem 4.1 we shall need two asset versions of the assumptions (A) and (B) concerning integrability of suitable functions \({\overline{w}}_k\) and continuity of the conditional laws of \(\mathbb {P} \big ( ({\underline{S}}_{k+1}^1, {\overline{S}}_{k+1}^1,{\underline{S}}_{k+1}^2, {\overline{S}}_{k+1}^2) \in \cdot \big  {\mathcal {F}}_{k} \big )\) for \(k=0,\ldots ,T1\).
Notes
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