Quadratic minimization with portfolio and terminal wealth constraints

Abstract

We address a problem of stochastic optimal control drawn from the area of mathematical finance. The goal is to minimize the expected value of a general quadratic loss function of the wealth at close of trade when there is a specified convex constraint on the portfolio over the trading interval, together with a specified almost-sure lower-bound on the wealth at close of trade. We use a variational approach of Rockafellar which leads naturally to an appropriate vector space of dual variables, a dual functional on the space of dual variables such that the dual problem of maximizing the dual functional is guaranteed to have a solution (i.e. a Lagrange multiplier) when a simple and natural Slater condition holds for the terminal wealth constraint, and obtain necessary and sufficient conditions for optimality of a candidate wealth process. The dual variables are pairs, each comprising an Itô process paired with a member of the adjoint of the space of essentially bounded random variables measurable with respect to the event $\sigma$-algebra at close of trade. The necessary and sufficient conditions are used to construct an optimal portfolio in terms of the Lagrange multiplier. The dual problem simplifies to maximization of a concave function over the real line when the portfolio is unconstrained but the terminal wealth constraint is maintained.

Appendix: Proofs

We collect in this appendix the proofs of several propositions in the main text. We shall need the following result (see Proposition I-1 on p.387 of Bismut (1973)):

Proposition 22

For $X\equiv (X_0,\dot{X},\Lambda _X)\in \mathbb {B}$ and $Y\equiv (Y_0,\dot{Y},\Lambda _Y)\in \mathbb {B}$ (recall Remark 4), put

$$\begin{aligned}&\mathbb {M}(X,Y)(t)\,:=\, X(t)Y(t)\,-\,X_0Y_0 \nonumber \\&\quad - \int _0^t \{X(\tau )\dot{Y}(\tau )+\dot{X}(\tau )Y(\tau )+ \Lambda _X^{\prime }(\tau )\Lambda _Y(\tau )\}\,\text {d}\tau ,\quad t\in [0,T]. \end{aligned}$$

Then $\{(\mathbb {M}(X,Y)(t),\mathcal {F}_t),\,t\in [0,T]\}$ is a continuous martingale with $\mathbb {M}(X,Y)(0)=0$.

Proof of Proposition 2

Fix some $X \in \mathbb {B}- \mathbb {A}_{1}$. From (34) we have either $X \not \in \mathbb {B}_{2}$ or $X \not \in \mathbb {A}$. Suppose $X \not \in \mathbb {B}_{2}$: then, from (34), we have $P [ X(T) - B< u_{2}] > 0$ for each $u_{2}\in L_{\infty }$, so that (26) gives $F(X, u) = +\infty$ for all $u \in \mathbb {U}$. On the other hand, when $X \not \in \mathbb {A}$, then (26) again gives $F(X,u) = +\infty$ for all $u\in \mathbb {U}$. Since $\langle (u_1,u_{2}), (Y,Z) \rangle \in \mathbb {R}$ for all $(u_1, u_{2}) \in \mathbb {U}$ and $(Y,Z)\in \mathbb {Y}$, we conclude from (33) that (i) $K(X,(Y,Z)) = +\infty$, for all $X \in \mathbb {B}- \mathbb {A}_{1}$ and $(Y,Z) \in \mathbb {Y}$. Now fix $X \in \mathbb {A}_{1}$: from (33), (32), (26), for each $(Y,Z) \in \mathbb {Y}$ we have

$$\begin{aligned} K(X,(Y,Z)) = \inf _{u_1\in L_{2}} \left\{ \text {E}[ u_1Y(T)] + \text {E}[ J(X(T)-u_1) ] \right\} + \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} \left\{ Z(u_{2})\right\} . \end{aligned}$$

(107)

From Remark 12 one has (ii) $\inf _{\{u_{2}\in L_{\infty }, \; u_{2}\le X(T) - B\}} \left\{ Z(u_{2})\right\} = -\infty$, for all $Z\in L_{\infty }^{*}$ such that $Z\not \le 0$. Moreover, for each $(X,Y) \in \mathbb {B}\times \mathbb {B}_{1}$ we also have

$$\begin{aligned}&\inf _{u_1\in L_{2}} \left\{ \text {E}[ u_1Y(T)] + \text {E}J(X(T)-u_1) \right\} = \text {E}[ X(T) Y(T) ]\nonumber \\&\qquad - \sup _{v_1\in L_{2}} \left\{ \text {E}[v_1Y(T)] - \text {E}J(v_1) \right\} \nonumber \\&\quad = \text {E}[ X(T) Y(T) ] - \text {E}[ J^{*}(Y(T)) ] \in \mathbb {R}, \end{aligned}$$

(108)

and (36) follows from (108), (ii), (107) and (i). Note that the first equality at (108) follows upon defining $v_1:=X(T) - u_1$, and using $X(T) \in L_{2}$. The second equality at (108) follows from Thrm. 2 on p.532–533 of Rockafellar (1968), since $L_{2}$ is decomposable (see p.532 of Rockafellar (1968)), while (35) and (8), together with Condition 2(iii), ensure that $J(\cdot )$ and $J^{*}(\cdot )$ are normal convex integrands (see Lemma 2 on p.528 of Rockafellar (1968)), and that $J(u)$ and $J^{*}(u)$ are $P$-integrable for each $u\in L_{2}$. $\square$

Proof of Proposition 3

Fix some $(X,(Y,Z)) \in \mathbb {B}\times \mathbb {Y}$. Since $\text {E}[X(T)Y(T)]$ and $\text {E}J^{*}(Y(T))$ are $\mathbb {R}$-valued (recall Remark 16) from (39) and (36) we get

$$\begin{aligned}&g(Y,Z) = K(X,(Y,Z)) \in \mathbb {R}\nonumber \\&\quad \iff \left\{ \begin{array}{l} (1)\; X\in \mathbb {A}_{1}, \; (2) \; Z\le 0, \quad (3) \; \varkappa (Y,Z) \in \mathbb {R},\\ (4)\; \text {E}[X(T)Y(T) ] + \varkappa (Y,Z) + \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} Z(u_{2})= 0. \end{array} \right. \end{aligned}$$

(109)

Moreover, from (15) and (36), we have $f(X) = K(X,(Y,Z))\in \mathbb {R}$ $\iff$ (1) $X(T)-B\ge 0$, (2) $X\in \mathbb {A}_{1}$, (3) $Z\le 0$, (4) $\text {E}[ J(X(T)) + J^{*}(Y(T)) - X(T)Y(T) ]$ $= \inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}} Z(u_{2})$ $\iff$ $(1^\prime )$ $X(T)-B\ge 0$, $(2^\prime )$ $X\in \mathbb {A}_{1}$, $(3^\prime )$ $Z\le 0$, $(4^\prime )$ $\text {E}[ J(X(T)) + J^{*}(Y(T)) - X(T)Y(T) ] = 0$, $(5^\prime )$ $\inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}} Z(u_{2}) = 0$ (note that the second equivalence follows from (35), which ensures $J(X(T)) + J^{*}(Y(T)) - X(T)Y(T) \ge 0$ $\text {a.s.}$, together with the fact that $X(T)-B\ge 0$ ensures $\inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}} Z(u_{2}) \le 0$). Moreover, from the non-negativity of the integrand at $(4^\prime )$, we see that $(4^\prime )$ holds if and only if $J(X(T)) + J^{*}(Y(T)) - X(T)Y(T) = 0$ $\text {a.s.}$ if and only if $X(T) = (\partial J^{*})(Y(T))$ (see Propos. I.5.1 and Cor. I.5.2 on p.21-22 of Ekeland and Témam (1976)). We then get

$$\begin{aligned}&f(X) = K(X,(Y,Z)) \in \mathbb {R}\nonumber \\&\quad \iff \left\{ \begin{array}{ll} (1) X(T) -B\ge 0, \;\; (2)\; X\in \mathbb {A}_{1}, &{} (3) \; Z\le 0,\\ (4) X(T) = (\partial J^{*})(Y(T)), &{} (5) \; \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} Z(u_{2})= 0. \end{array} \right. \qquad \qquad \end{aligned}$$

(110)

Moreover, from (15), we know that $f(X) \in ( -\infty , \infty ]$, for all $X \in \mathbb {B}$. Also, from (38) and $\mathbb {A}_{1}\not = \emptyset$ (recall (34)) we have $\varkappa (Y,Z) \in (-\infty ,\infty ]$, for all $(Y,Z)\in \mathbb {Y}$; since $J^{*}(Y(T))$ is $P$-integrable for each $Y\in \mathbb {B}_{1}$ (see Remark 16), from (39) we obtain $g(Y,Z) \in [-\infty ,\infty )$, for all $(Y,Z)\in \mathbb {Y}$. Thus, for each $X\in \mathbb {B}$ and each $(Y,Z)\in \mathbb {Y}$, we obtain from (40) the equivalence that $f(X) = g(Y,Z)$ $\iff$ $f(X) = K(X,(Y,Z)) \in \mathbb {R}$ and $g(Y,Z) = K(X,(Y,Z)) \in \mathbb {R}$. This equivalence, together with (110) and (109), gives (41). $\square$

Proof of Proposition 4

We have “separation of infima” for the regular and singular parts of $Z$, namely

$$\begin{aligned} \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} Z(u_{2})= \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} \text {E}[ Z_{r}u_{2}]+ \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} Z^{\circ }(u_{2}),\;\;\;\;X\in \mathbb {B}_{2}, \;\; Z\in L_{\infty }^{*}, \end{aligned}$$

(111)

(recall (34)), as follows immediately from Thm. 1 of Rockafellar (1971). Consider the first term on the right of (111) when $Z\le 0$, in which case (recall Remark 13) we have $Z_{r}\le 0$ $\text {a.s.}$ Fix some $X \in \mathbb {B}_{2}$; then, from (34) and the $P$-essential boundedness of $B$ (see Condition 2(iii)), both $Z_{r}(X(T)-B)$ and $Z_{r}X(T)$ are majorized by the integrable random variable $\alpha Z_{r}$ (for some $\alpha \in \mathbb {R}$; recall that $Z_{r}\in L_{1}$), thus $\text {E}[ Z_{r}(X(T)-B)]$ and $\text {E}[ Z_{r}X(T) ]$ exist in $[-\infty ,\infty )$ (see the remark just prior to Cor. 4.1.1 in Chow and Teicher (1988)), and (i) $\text {E}[ Z_{r}(X(T)-B)] = \text {E}[ Z_{r}X(T)] - \text {E}[ Z_{r}B]$ (see Thrm. 4.1.1(v) of Chow and Teicher (1988)). Moreover, since $Z_{r}\le 0$ $\text {a.s.}$, and $X(T) -B$ is $\text {a.s.}$ lower-bounded by some $\alpha \in \mathbb {R}$ (from (34)), it is easily seen from the monotone convergence theorem that one has (ii) $\inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}}\text {E}[Z_{r}u_{2}] = \text {E}[Z_{r}(X(T)-B)] = \text {E}[Z_{r}X(T) ] - \text {E}[ Z_{r}B]$ (the second equality following from (i)). From (ii) and (111) we get

$$\begin{aligned} \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} Z(u_{2})= \text {E}[Z_{r}X(T)] -\text {E}[Z_{r}B] + \inf _{ \begin{array}{c} {u_{2}\in L_{\infty }} \\ {u_{2}\le X(T) - B} \end{array}} Z^{\circ }(u_{2}), \;\; X\in \mathbb {B}_{2}, \;\;\;Z\in \mathbb {G}, \end{aligned}$$

(112)

(recall Notation 6). Upon combining (112), (39), (38), we obtain (42).

As for (43), this is a matter of showing equivalence of (43)(4) and (41)(4), since we know from Remark 13 that (43)(3) and (41)(3) are equivalent. Fix a pair $(X,(Y,Z))\in \mathbb {B}\times \mathbb {Y}$ such that (iii) $X(T) - B\ge 0$ and (iv) $Z\le 0$. From (iii), (iv) and Remark 13, we get that (v) $\inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}}Z^{\circ }(u_{2}) \le 0$ and $\text {E}[ Z_{r}(X(T) - B)] \le 0$. From (v) and (112) we have $\inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}}Z(u_{2}) = 0$ $\iff$ $\text {E}[ Z_{r}(X(T) - B)] + \inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}}Z^{\circ }(u_{2}) = 0$ $\iff$ $\inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}}Z^{\circ }(u_{2}) = \text {E}[ Z_{r}(X(T) - B)] =0$ $\iff$ $\inf _{\{u_{2}\in L_{\infty }, u_{2}\le X(T) - B\}}Z^{\circ }(u_{2}) =0$ and $Z_{r}(X(T) - B) =0$ $\text {a.s.}$ (since, from (iii), (iv), and Remark 13, we have $Z_{r}(X(T) - B) \le 0$). This gives the equivalence of (43)(4) and (41)(4) as required. $\square$

Proof of Proposition 8

We evaluate the supremum at (46). In view of Remark 5(b), for each $\pi \in \Pi _{2}$ we obtain (i) $X^{\pi }\in \mathbb {B}$ with (ii) $X^{\pi }(0) = x_0$, $\dot{X}^{\pi } = r X^{\pi }+ \pi ^{\prime } \sigma \theta$, and $\Lambda _{X^{\pi }}^{\prime } = \pi ^{\prime } \sigma$. From (ii) and (18), we get (iii) $\mathbb {M}(X^{\pi },Y)(T) = X^{\pi }(T)Y(T) - x_0Y_{0} - \int _{0}^{T} \; \pi ^{\prime }(\tau ) \sigma (\tau ) [ \theta (\tau ) Y(\tau ) + \Lambda _{Y}(\tau ) ]\,\text {d}\tau$, for all $\pi \in \Pi _{2}$ and $Y\in \mathbb {B}_{1}$, and, from (i) and Proposition 22, we get (iv) $\text {E}[ \mathbb {M}(X^{\pi },Y)(T) ] = 0$, for all $\pi \in \Pi _{2}$ and $Y\in \mathbb {B}_{1}$. It then follows from (iv) and (iii) that

$$\begin{aligned} \text {E}[ X^{\pi }(T) Y(T) ] {=} x_0Y_{0} {+}\text {E}\int _{0}^{T}\pi ^{\prime }(\tau )\sigma (\tau ) [ \theta (\tau ) Y(\tau )+ \Lambda _{Y}(\tau )] \,\text {d}\tau , \pi \in \Pi _{2}, Y\in \mathbb {B}_{1}.\nonumber \\ \end{aligned}$$

(113)

From (7), (14), Notation 7(a), and (113), we get

$$\begin{aligned} \sup _{X\in \mathbb {A}} \text {E}[-X(T)Y(T)]&= \sup _{\begin{array}{c} {\pi \in \Pi _{2}}\\ {\pi \in K \;\;\text {a.e.}} \end{array}} \text {E}[ -X^{\pi }(T) Y(T) ]\nonumber \\&= \sup _{\pi \in \Pi _{2}} \{ \text {E}[-X^{\pi }(T)Y(T)] - \text {E}\int _{0}^{T}\delta _{\mathbb {R}^{N}}(\pi (\tau )|K)\,\text {d}\tau \} \nonumber \\&= -x_0Y_0 + \sup _{\pi \in \Pi _{2}} \text {E}\int _{0}^{T}\; \{ \pi ^{\prime }(\tau )\sigma (\tau ) [-\theta (\tau ) Y(\tau ) - \Lambda _{Y}(\tau )] \nonumber \\&\qquad \qquad \qquad \qquad \qquad - \delta _{\mathbb {R}^{N}}(\pi (\tau )|K)\} \,\text {d}\tau , \end{aligned}$$

(114)

for each $Y\in \mathbb {B}_{1}$. We next evaluate the supremum at the right of (114). The indicator function $\delta _{\mathbb {R}^{N}}(\cdot |K)$ (see Notation 7(a)) is a normal convex integrand by Lemma 1 on p.528 of Rockafellar (1968). From Notation 7(b), and the fact that $0 \in K$, the functions $\delta _{\mathbb {R}^{N}}(\vartheta (\cdot )|K)$ and $\delta ^{*}_{\mathbb {R}^{N}}(\vartheta (\cdot )|K)$ are identically zero (hence $\lambda \otimes P$-integrable) on $[0,T]\times \Omega$, when $\vartheta$ is the zero element of $\Pi _{2}$. Since $\Pi _{2}$ is decomposable (see p.532 of Rockafellar (1968)), and $\delta ^{*}_{\mathbb {R}^{N}}(\cdot |K)$ is the convex conjugate of $\delta _{\mathbb {R}^{N}}(\cdot |K)$, we have

$$\begin{aligned} \sup _{\pi \in \Pi _{2}} \text {E}\int _{0}^{T}\; \{ \pi ^{\prime }(\tau )\vartheta (\tau ) - \delta _{\mathbb {R}^{N}}(\pi (\tau )|K)\} \,\text {d}\tau = \text {E}\int _{0}^{T}\; \delta ^{*}_{\mathbb {R}^{N}}(\vartheta (\tau )|K) \,\text {d}\tau , \;\;\;\; \vartheta \in \Pi _{2}, \end{aligned}$$

(115)

(as follows upon identifying $f(\cdot ):=\delta _{\mathbb {R}^{N}}(\cdot |K)$, and $L = L^{*} :=\Pi _{2}$, in Thrm. 2 on p.532–533 of Rockafellar (1968)). Now fix $Y\in \mathbb {B}_{1}$, and put $\vartheta (t,\omega ) :=- \sigma (t,\omega ) [ \theta (t,\omega ) Y(t,\omega ) + \Lambda _{Y}(t,\omega )]$, $(t,\omega )\in [0,T]\times \Omega$; since $\Lambda _{Y}\in \Pi _{2}$ (see (18), (12)), it follows from the boundedness of $\theta$ (see Remark 4), the bound given by Remark 1, and Remark 5(a), that $\vartheta \in \Pi _{2}$. Using this choice of $\vartheta$ in (115), and then combining with (114), we obtain (46). $\square$

Proof of Proposition 9

Put $\alpha _1 :=\inf _{X\in \mathbb {A}_{1}} \text {E}[X(T) Y(T)]$ and $\alpha _2 :=\inf _{X\in \mathbb {A}} \text {E}$ $[X(T) Y(T)]$ for some $Y \in \mathbb {B}_{1}$. Since $\mathbb {A}_{1}\subset \mathbb {A}$, (recall (34)), we have $\alpha _1 \ge \alpha _2$. To show the opposite inequality fix some $X \in \mathbb {A}$. From (14) and (7) we have (i) $X = X^{\pi }$ for some $\pi \in \mathcal {A}$. Define the $\mathcal {F}_t$-stopping times $T_n:=\inf \{ t \in [0,T] \; | \; S_0^{-1}(t) X(t) \le -n\} \wedge T$. Since $0 \in K$ (see Condition 2(i)) we have (ii) $\pi _{n}\in \mathcal {A}$ for $\pi _{n}$ defined by $\pi _{n}(t) :=\pi (t) I\{ t \le T_n\}$. Now put (iii) $X_n:=X^{\pi _{n}}$, for each $n = 1,2,\ldots$ (recall (5)). Now (iv) $S_0^{-1}(t) X_n(t) = S_0^{-1}(t\wedge T_n) X(t\wedge T_n)$, $t\in [0,T]$, as follows from (ii), (iii) and (5). From the definition of $T_n$ we find (v) $S_0^{-1}(T_n) X(T_n) \ge - n$. Upon combining (iv) and (v), we see that $S_0^{-1}(T) X_n(T) \ge -n$, and thus (vi) $X_n(T) - B\ge -n S_0(T) - B$. Now $S_0(T)$ and $B$ are $P$-essentially bounded (from (2) with Conditions 1 and 2(iii)), so it follows from (vi) and (34) that (vii) $X_n\in \mathbb {B}_{2}$. Moreover, from (iii) and (ii) we have $X_n\in \mathbb {A}$, and from this with (vii) and (34) we obtain (viii) $X_n\in \mathbb {A}_{1}$ for all $n = 1,2,\ldots$. Now of course $\lim _{n \rightarrow \infty }T_n= T$, and from this together with (iv) we get (ix) $\lim _{n \rightarrow \infty }X_n(T) = X(T)$ a.s. and therefore (x) $\lim _{n \rightarrow \infty }\text {E}[ X_n(T) Y(T) ] = \text {E}[ X(T) Y(T) ]$. To see how (x) follows put

$$\begin{aligned}&\zeta :=S_0(T) \left\{ x_0+ \int _{0}^{T} |S_0^{-1}(\tau )\pi ^{\prime }(\tau ) \sigma (\tau )\theta (\tau )|\,\text {d}\tau \right. \nonumber \\&\left. \quad + \max _{t \le T} \left| \int _{0}^{t} S_0^{-1}(\tau ) \pi ^{\prime }(\tau )\sigma (\tau ) \,\text {d}W (\tau ) \right| \right\} . \end{aligned}$$

From (iii), (ii) and (5), we have (xi) $| X_n(T)| \le \zeta$ for all $n = 1,2,\ldots$, and, from Condition 1, the bounds given by Remark 1, and Doob’s maximal $L^{2}$-inequality, it is straightforward to check that (xii) $\zeta \in L_{2}$. Moreover, (xiii) $Y(T) \in L_{2}$ (since $Y\in \mathbb {B}_{1}$), so it follows from (xii) and (xiii) that (xiv) $\zeta |Y(T)| \in L_{1}$. In view of (xi) and (xiv), we have (xv) $| X_n(T) Y(T) | \le \zeta |Y(T) | \in L_{1}$ for all $n = 1,2,\ldots$, and (x) follows from (ix), (xv), and dominated convergence. Finally, from (viii), (x) and the fact that $X\in \mathbb {A}$ is arbitrary, we obtain $\alpha _1 \le \alpha _2$, as required. $\square$

Proof of Proposition 11

(a) follows upon expanding $X(t) = H^{-1}(t) [ X(0) + \int _{0}^{t}\; \psi ^{\prime }(\tau ) \,\text {d}W(\tau )]$ by Itô’s formula and using Remark 25(b). An argument which is identical to that for Lemma 5.1 of Labbé and Heunis (2007) shows that $\text {E}[ \sup _{t\in [0,T]} |X(t)|^{2}] < \infty$, and this fact, together with an argument identical to that for Lemma 5.2 of Labbé and Heunis (2007), establishes $\pi \in \Pi _{2}$; now $X \in \mathbb {B}$ follows from this, together with Condition 1 and the expansion of $\,\text {d}X(t)$ given by (a). $\square$

Proof of Proposition 19

Fix $(Y,Z) \in \mathbb {Y}$ with $Z\le 0$, $Z^{\circ }= 0$, and $Z_{r}\in L_{1}- L_{2}$. Then, from Remark 13, we have (i) $Z_{r}\le 0$ $\text {a.s.}$. Now put (ii) $\zeta _{n}:=(-Z_{r}) \wedge n \ge 0$ $\text {a.s.}$ for all $n = 1,2,\ldots$ Then (iii) $\zeta _{n}\in L_{\infty }\subset L_{2}$, and, since $Z_{r}\not \in L_{2}$, the monotone convergence theorem gives (iv) $\lim _{n \rightarrow \infty }\text {E}[\zeta _{n}^{2}] = \infty$. Now put (v) $\left|| \zeta _{n}\right||_{2} :=( \text {E}[\zeta _{n}^{2}])^{1/2}$ and $\eta _{n}:=\zeta _{n}/ \left|| \zeta _{n}\right||_{2} \ge 0$, for all $n=1,2,\ldots$ Then, from (iii), we have that (vi) $\eta _{n}\in L_{\infty }\subset L_{2}$ with $\left|| \eta _{n}\right||_{2} :=( \text {E}[\eta _{n}^{2}])^{1/2} = 1$, for all $n = 1,2,\ldots$ Next, identifying $\xi$ with $\eta _{n}$ at (v), use Proposition 11 to define the $\mathbb {R}$-valued process $X_{n}$, and the $\mathbb {R}^{N}$-valued processes $\psi _n \in {\mathcal {F}}^{*}$, $\pi _n \in {\mathcal {F}}^{*}$. Then (vii) $X_{n}(T) = \eta _{n}\ge 0$, and (viii) $X_{n}(0) = \text {E}[\eta _{n}H(T)]$. From Proposition 11(a), we have

$$\begin{aligned} \,\text {d}X_{n}(t) = \{ r(t) X_{n}(t) + \pi ^{\prime }_n(t) \sigma (t) \theta (t) \} \,\text {d}t + \pi ^{\prime }_n(t) \sigma (t) \,\text {d}W(t). \end{aligned}$$

(116)

Next define the $\mathbb {R}$-valued process $\hat{X}_n$ by

$$\begin{aligned} \,\text {d}\hat{X}_n(t) = \{ r(t) \hat{X}_n(t) + \pi ^{\prime }_n(t) \sigma (t) \theta (t) \} \,\text {d}t + \pi ^{\prime }_n(t) \sigma (t) \,\text {d}W(t), \;\;\; \hat{X}_n(0) = x_0. \end{aligned}$$

(117)

From (117), (116) and (viii), we get (ix) $\hat{X}_n(T) - X_n(T) = S_0(T) [x_0- \text {E}[\eta _{n}H(T)]]$. In light of (ix) and (vii) we see that (x) $\hat{X}_n(T) \ge S_0(T) [x_0- \text {E}[\eta _{n}H(T)]]$. Since $S_0(T)$ and $B$ are $P$-essentially bounded (as follows from Condition 1 and Condition 2(iii)), from (x) we get (xi) $\hat{X}_n \in \mathbb {B}_{2}$ (recall (34)). Moreover, since $K = \mathbb {R}^{N}$, we have $\mathcal {A}= \Pi _{2}$ (recall (7)); but $\pi _n \in \Pi _{2}$ (from Proposition 11(b)), so it follows from (117) that (xii) $\hat{X}_n \in \mathbb {A}$ (recall (14)). In view of (xii), (xi) and (34), we obtain $\hat{X}_n \in \mathbb {A}_{1}$, and therefore

$$\begin{aligned} \inf _{X \in \mathbb {A}_{1}} \text {E}[ X(T) ( Y(T) + Z_{r})] \le \text {E}[ \hat{X}_n(T) ( Y(T) + Z_{r})]. \end{aligned}$$

(118)

From (vii) and (ix) we find that the right side of (118) is given by

$$\begin{aligned} \text {E}[ \hat{X}_n(T) (Y(T) + Z_{r}) ]&= \text {E}[ \eta _{n}Z_{r}] + \text {E}[\eta _{n}Y(T)]\nonumber \\&+ x_0\text {E}[S_0(T)(Y(T)+Z_{r})]\\&- \text {E}[ \eta _{n}H(T) ] \text {E}[ S_0(T) (Y(T)+Z_{r}) ]. \nonumber \end{aligned}$$

(119)

From (vi) and the Cauchy-Schwarz inequality we get (xiii) $| \text {E}[ \eta _{n}H(T)] | \le (\text {E}[H^{2}(T)])^{1/2} < \infty$ and (xiv) $| \text {E}[ \eta _{n}Y(T)] | \le ( \text {E}[Y^{2}(T)])^{1/2} < \infty$. Moreover, (xv) $\text {E}[ \eta _{n}Z_{r}] = \text {E}[ \zeta _{n}Z_{r}] / \left|| \zeta _{n}\right||_{2} \le - \text {E}[ \zeta _{n}^{2} ] / \left||\zeta _{n}\right||_{2} = - \left|| \zeta _{n}\right||_{2}$ (the first equality at (xv) follows from (v), while the inequality follows from (ii)), and then, from (xv) and (iv), we obtain (xvi) $\lim _{n \rightarrow \infty }\text {E}[ \eta _{n}Z_{r}] = -\infty$. From (xvi), (xiv) and (xiii), one sees that the right side of (119) must converge to $-\infty$ as $n \rightarrow \infty$; but the quantity on the right of (119) is an upper-bound for the infimum on the left of (118) for each $n = 1,2,\ldots$, and thus

$$\begin{aligned} \inf _{X \in \mathbb {A}_{1}} \text {E}[ X(T) ( Y(T) + Z_{r})] = - \infty . \end{aligned}$$

(120)

Since $Z^{\circ }= 0$, from (120) and (42) we get $g(Y,Z) = - \infty$, as required. $\square$

Proof of Proposition 20

The function $\alpha \rightarrow \tilde{h}(\alpha H(T,\omega ),\omega )$ is concave and continuous for $\alpha \in \mathbb {R}$ (see (89)), thus, from dominated convergence, it is easily seen that (i) $\alpha \rightarrow \varPsi (\alpha )$ is continuous and clearly concave on $\alpha \in \mathbb {R}$. From (89) and (86) we have $\tilde{h}(y,\omega ) \le h((c + a B)(\omega ), \omega )$ for all $y \in \mathbb {R}$, and from this with Condition 2(iii) and (35), we get (ii) $\text {E}\tilde{h}(\alpha H(T) ) \le \text {E}h( c + a B) < \infty$ for all $\alpha \in \mathbb {R}$. From Condition 4, with $K :=\mathbb {R}^{N}$, there is some $\tilde{\pi }\in \Pi _{2}$ and $\epsilon \in (0,\infty )$ such that (iii) $X^{\tilde{\pi }}(T) H(T) \ge BH(T) + \epsilon H(T)$. But $\{ X^{\tilde{\pi }}(t) H(t) \}$ is a $\mathcal {F}_t$-martingale (from (53), (4) and Itô’s formula), thus $\text {E}[ X^{\tilde{\pi }}(T) H(T) ] = x_0$, and then from (iii) we get (iv) $x_0> \text {E}[ BH(T) ]$. Now it follows from (iv), (ii), and (90) that (v) $\lim _{\alpha \rightarrow - \infty } \varPsi (\alpha ) = - \infty$. Since $H(T) > 0$ $\text {a.s.}$ we have (vi) $P [ \alpha _0 H(T) \ge c + a B] > 0$ for some $\alpha _{0} > 0$. Moreover, from (89), (86), (35), and Condition 2(iii), it is easily seen that there are some non-negative constants $k_1, k_2 \in \mathbb {R}$ such that (vii) $\text {E}\tilde{h}(\alpha H(T)) \le k_1 + | \alpha | k_2 - (\alpha ^{2} / 2) \text {E}[ H^{2}(T) / a ; \; \alpha _0 H(T) \ge c + aB]$ for all $\alpha \ge \alpha _{0}$. From (vi) we also have (viii) $\text {E}[ H^{2}(T) / a ; \; \alpha _0 H(T) \ge c + aB] > 0$. Now it follows from (viii), (vii) and (90), that (ix) $\lim _{\alpha \rightarrow \infty } \varPsi (\alpha ) = - \infty$, and existence of a maximizer $\bar{\alpha }\in \mathbb {R}$ of $\varPsi (\cdot )$ follows from (ix), (v), and (i). $\square$