Quadratic minimization with portfolio and intertemporal wealth constraints

Abstract

We address a problem of stochastic optimal control motivated by portfolio optimization in mathematical finance, the goal of which 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, together with a specified almost-sure lower-bound on intertemporal wealth over the full trading interval. A precursor to the present work, by Heunis (Ann Financ 11:243–282, 2015), addressed the simpler problem of minimizing a general quadratic loss function with a convex portfolio constraint and a stipulated almost-sure lower-bound on the wealth only at close of trade. In the parlance of optimal control the problem that we shall address here exhibits the combination of a control constraint (i.e. the portfolio constraint) together with an almost-sure intertemporal state constraint (on the wealth over the full trading interval). Optimal control problems with this combination of constraints are well known to be quite challenging even in the deterministic case, and of course become still more so when one deals with these same constraints in a stochastic setting. We nevertheless find that an ingenious variational approach of Rockafellar (Conjugate duality and optimization, CBMS-NSF series no. 16, SIAM, 1974), which played a key role in the precursor work noted above, is fully equal to the challenges posed by this problem, and leads naturally to an appropriate vector space of dual variables, together with 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 (or Lagrange multiplier) when the problem constraints satisfy a simple and natural Slater condition. We then establish necessary and sufficient conditions for the optimality of a candidate wealth process in terms of the Lagrange multiplier, and use these conditions to construct an optimal portfolio.

Appendix: Proofs

In this appendix we collect the proofs of several technical propositions in Sects. 2–5.

1.1 Proof of Proposition 1

Fix any $\pi \in \varPi _{2}$. Upon expansion of $X^{\pi }H$ by Ito’s formula, using (10) and (6), one sees that $X^{\pi }H$ is a continuous $\{ \mathcal {F}_t \}$-local martingale. Moreover, from (14) and (7), together with the Cauchy–Schwarz inequality, one finds that $E\left[ {\sup _{t \in {[0, T]}} \left| { H(t)X^{\pi }(t)}\right| }\right] < \infty$, so that (i) $X^{\pi }H$ is actually a continuous $\{ \mathcal {F}_t \}$-martingale (see Theorem 51 on p. 38 of Protter (2004)). Now suppose that $X^{\pi }(T)\geqslant b \,\text {a.s.}$ From the positivity of H it is immediate that (ii) $E\left[ { H(T)X^{\pi }(T)} \,\big |\, \mathcal {F}_t\right] \geqslant E\left[ { H(T)b} \,\big |\, \mathcal {F}_t\right] , t \in {[0, T]}, \,\text {a.s.}$ From (i), (ii) and (28) we obtain

$$\begin{aligned} X^{\pi }(t)= H^{-1}(t)E\left[ { H(T)X^{\pi }(T)} \,\big |\, \mathcal {F}_t\right] \overset{}{\geqslant } H^{-1}(t)E\left[ { H(T)b} \,\big |\, \mathcal {F}_t\right] \overset{}{=} B(t),\nonumber \\ \end{aligned}$$

(133)

for all $t\in {[0, T]}, \,\text {a.s.}$ $\square$

1.2 Proof of Proposition 2

(a) Fix some $\pi \in \mathcal {A}\backslash \mathcal {A}_1$. From (73) we have (i) $P\left( {\{X^{\pi }+\alpha \geqslant B\}}\right) <1$ for all $\alpha \in \mathbb {R}$. Then one has (ii) $P\left( {\{X^{\pi }+V \geqslant B\}}\right) \leqslant P\left( {\{X^{\pi }+\left\| {V}\right\| _{\infty }\geqslant B\}}\right) < 1$ for all $V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})$ (the first inequality follows from (38) and the second from (i)). Then (iii) $F(\pi , (u, V))=+\infty$ for all $(u, V)\in \mathbb {U}$ (as follows from (ii) and (52)). In view of (iii), (70) and $\left\langle (u, V), (Y, Z)\right\rangle \in \mathbb {R}$ for all $(u, V)\in \mathbb {U}$ and $(Y,Z)\in \mathbb {Y}$, we get (iv) $K(\pi , (Y,Z)) = +\infty$ for all $(Y,Z)\in \mathbb {Y}$ when $\pi \in \mathcal {A}\backslash \mathcal {A}_1$. On the other hand, when $\pi \in \varPi _{2}\backslash \mathcal {A}$, then it is immediate from (52) that again $F(\pi , (u, V))=+\infty$ for all $(u, V)\in \mathbb {U}$, and, just as before, we have (v) $K(\pi , (Y,Z)) = +\infty$ for all $(Y,Z)\in \mathbb {Y}$ when $\pi \in \varPi _{2}\backslash \mathcal {A}$. Now (iv) and (v), with $\mathcal {A}_1 \subset \mathcal {A}$, give the last of the alternatives at (75).

(b) Now fix $\pi \in \mathcal {A}_1$. Then of course $\pi \in \mathcal {A}$, and it follows from (70), (67), and (52), that

$$\begin{aligned} K(\pi ,(Y, Z))= & {} \inf _{u\in \mathcal {L}_2{(\mathcal {F}_T)}}\{E\left[ {uY(T)+J(X^{\pi }(T)-u)}\right] \}\nonumber \\&\quad +\inf _{ \begin{array}{c} {V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})} \\ {X^{\pi }+ V \geqslant B} \end{array}} \{Z(V)\}, \quad (Y,Z) \in \mathbb {Y}. \end{aligned}$$

(134)

Now the V-infimum on the right of (134) takes the value $- \infty$ when $Z \not \geqslant 0$. To see that this is the case fix $Z \in \mathcal {L}_\infty ^*(\mathcal {F}_{T}; \mathcal {C})$ with $Z \not \geqslant 0$. Then $Z(\tilde{V})<0$ for some $\tilde{V}\in (\mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C}))^+$, from which it is immediate that (vi) $\inf _{V\in (\mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C}))^+}\{Z(V)\} = -\infty$. Moreover, since $\pi \in \mathcal {A}_1$, it follows from (73) that (vii) there is some $\hat{\alpha }\in \mathbb {R}$ such that $X^{\pi }+\hat{\alpha }\geqslant B$, and in view of (vii) we then have the set inclusion $\left\{ {V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})}\; | \;{V \geqslant \hat{\alpha }}\right\} \subset \left\{ {V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})}\; | \;{X^{\pi }+ V \geqslant B}\right\}$, from which we get

$$\begin{aligned} \inf _{ \begin{array}{c} {V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})} \\ {X^{\pi }+ V \geqslant B} \end{array}} \{Z(V)\} \quad&{\leqslant }\quad \inf _{ \begin{array}{c} {V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})} \\ {V\geqslant \hat{\alpha }} \end{array}}\{Z(V)\} = Z(\hat{\alpha })\nonumber \\&+ \inf _{{V}\in (\mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C}))^+}\{Z({V})\} = - \infty , \end{aligned}$$

(135)

the final equality at (135) following from (vi) and the fact that $Z(\hat{\alpha }) \in \mathbb {R}$ (regarding $\hat{\alpha }$ as a constant member of $\mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})$). Now the second of the alternatives at (75) (for $\pi \in \mathcal {A}_1, Z \not \geqslant 0$) is clear from (135) and (134). Finally take $\pi \in \mathcal {A}_1$ and $Z \geqslant 0$. We have $X^{\pi }(T), Y(T)\in \mathcal {L}_2{(\mathcal {F}_T)}$ (see Remark 18(1)), and therefore (exactly as at (108) of Heunis (2015)) one finds

$$\begin{aligned}&\inf _{u\in \mathcal {L}_2{(\mathcal {F}_T)}}\left\{ E\left[ {uY(T)+J(X^{\pi }(T)-u)}\right] \right\} \nonumber \\&\quad = E\left[ {X^{\pi }(T)Y(T)}\right] -E\left[ {J^*(Y(T))}\right] ,\;\; Y \in \widetilde{\mathbb {B}}_{1}. \end{aligned}$$

(136)

The first of the alternatives at (75) follows from (134) and (136).$\square$

1.3 Proof of Proposition 6

From Remark 21 we have

$$\begin{aligned} g(\bar{Y}-\varepsilon \bar{Y}, \bar{Z}- \varepsilon \bar{Z}) \leqslant g(\bar{Y}, \bar{Z}) \,\,\text{ and }\,\, \bar{Z}- \varepsilon \bar{Z}\geqslant 0, \quad \varepsilon \in [0,1]. \quad \end{aligned}$$

(137)

In view of (137) and (77), we get

$$\begin{aligned}&\varkappa (\bar{Y}-\varepsilon \bar{Y}, \bar{Z}- \varepsilon \bar{Z}) + E\left[ {J^*(\bar{Y}(T)-\varepsilon \bar{Y}(T))}\right] \nonumber \\&\quad \geqslant \varkappa (\bar{Y}, \bar{Z})+E\left[ {J^*(\bar{Y}(T))}\right] , \;\; \varepsilon \in [0,1], \end{aligned}$$

(138)

and, from (78), one finds

$$\begin{aligned} \varkappa (\bar{Y}-\varepsilon \bar{Y}, \bar{Z}- \varepsilon \bar{Z}) = (1-\varepsilon )\varkappa (\bar{Y}, \bar{Z}), \quad \varepsilon \in [0,1]. \quad \end{aligned}$$

(139)

Upon combining (139) and (138), together with $\varkappa (\bar{Y}, \bar{Z})\in \mathbb {R}$ (see Remark 21), we have

$$\begin{aligned} \varkappa (\bar{Y}, \bar{Z}) + E\left[ {\frac{J^*(\bar{Y}(T))-J^*(\bar{Y}(T)- \varepsilon \bar{Y}(T))}{\varepsilon }}\right] \leqslant 0, \quad \varepsilon \in (0,1). \end{aligned}$$

(140)

In view of Condition 4, (69), and (74), one can use the dominated convergence theorem and $\epsilon \rightarrow 0$ at (140) to obtain

$$\begin{aligned} \varkappa (\bar{Y}, \bar{Z}) + E\left[ {\partial J^*(\bar{Y}(T)) \bar{Y}(T)}\right] \leqslant 0.\quad \end{aligned}$$

(141)

From Remark 21 one again has

$$\begin{aligned} g(\bar{Y}+\varepsilon Y, \bar{Z}+ \varepsilon Z) \leqslant g(\bar{Y}, \bar{Z}) \,\,\text{ for } \text{ all } \,\,(Y, Z)\in \mathbb {Y}\,\,\text{ and }\,\,\varepsilon \in (0,\infty ). \end{aligned}$$

(142)

If $Z\geqslant 0$ then $\bar{Z}+ \varepsilon Z\geqslant 0$ for all $\varepsilon \in (0,\infty )$ (since $\bar{Z}\geqslant 0$ from Remark 21) and therefore, from (142) and (77), we obtain

$$\begin{aligned} \varkappa (\bar{Y}+\varepsilon Y, \bar{Z}+ \varepsilon Z) + E\left[ {J^*(\bar{Y}(T)+\varepsilon Y(T))}\right] \overset{}{\geqslant } \varkappa (\bar{Y}, \bar{Z}) + E\left[ {J^*(\bar{Y}(T))}\right] , \nonumber \\ \end{aligned}$$

(143)

for all $(Y, Z)\in \mathbb {Y}$ with $Z\geqslant 0$ and $\varepsilon \in (0, \infty )$. Moreover, from (78), one obtains

$$\begin{aligned}&\varkappa (\bar{Y}+\varepsilon Y, \bar{Z}+ \varepsilon Z) = \sup _{\pi \in \mathcal {A}_1}\bigg \{-E\left[ {X^{\pi }(T)(\bar{Y}(T)+\varepsilon Y(T))}\right] \nonumber \\&\qquad \left. -\inf _{V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})}\left\{ {(\bar{Z}+ \varepsilon Z)(V)}\; \Big | \;{X^{\pi }+ V \geqslant B}\right\} \right\} \quad \nonumber \\&\quad \overset{}{\leqslant } \sup _{\pi \in \mathcal {A}_1}\left\{ -E\left[ {X^{\pi }(T)\bar{Y}(T)}\right] -E\left[ {\varepsilon X^{\pi }(T)Y(T)}\right] \right. \nonumber \\&\qquad \left. - \inf _{V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})}\left\{ {\bar{Z}(V)}\; \Big | \;{X^{\pi }+ V \geqslant B}\right\} -\inf _{V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})}\left\{ {\varepsilon Z(V)}\; \Big | \;{X^{\pi }+ V \geqslant B}\right\} \right\} \nonumber \\&\quad \overset{}{\leqslant } \sup _{\pi \in \mathcal {A}_1}\left\{ -E\left[ {X^{\pi }(T)\bar{Y}(T)}\right] - \inf _{V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})}\left\{ {\bar{Z}(V)}\; \Big | \;{X^{\pi }+ V \geqslant B}\right\} \right\} \nonumber \\&\qquad +\sup _{\pi \in \mathcal {A}_1}\left\{ -\varepsilon E\left[ {X^{\pi }(T)Y(T)}\right] -\varepsilon \inf _{V\in \mathcal {L}_\infty (\mathcal {F}_{T}; \mathcal {C})}\left\{ {Z(V)}\; \Big | \;{X^{\pi }+ V \geqslant B}\right\} \right\} \nonumber \\&\quad \overset{}{=}\varkappa (\bar{Y}, \bar{Z}) + \varepsilon \varkappa (Y, Z), \quad \text{ for } \text{ all }\,\, (Y, Z)\in \mathbb {Y}\,\,\text{ with }\,\, Z\geqslant 0\,\, \text{ and }\,\,\varepsilon \in (0,\infty ). \end{aligned}$$

(144)

Combining (144) and (143), together with $\varkappa (\bar{Y}, \bar{Z})\in \mathbb {R}$ (see Remark 21), we get

$$\begin{aligned} \varkappa (Y, Z) + E\left[ {\frac{J^*(\bar{Y}(T)+\varepsilon Y(T))-J^*(\bar{Y}(T))}{\varepsilon }}\right] \geqslant 0, \quad \varepsilon \in (0,\infty ),\; Z\geqslant 0. \nonumber \\ \end{aligned}$$

(145)

Now take $\varepsilon \rightarrow 0$ at (145) and use the dominated convergence theorem to obtain (82). Upon combining (82) with (141) and $\bar{Z}\geqslant 0$, we also get (83). $\square$

1.4 Proof of Proposition 8

This is immediate from Heunis (2015, see Proposition 8 on p. 260 and eqn. (14) on p. 250). $\square$

1.5 Proof of Proposition 9

Fix some $(y, \nu , \varrho ) \in \mathbb {R}\times \varPi _{2}\times \mathcal {L}_2(\{\mathcal {F}_t\}; \mathcal {BV}_0^r)$, and define

$$\begin{aligned} \eta _{1}(t) := \int _0^tS_0(\tau )\varrho (\text {d}\tau ), \quad \eta _{2}(t):=S_0(t)\nu (t) - \theta (t)\eta _{1}(t)-y\theta (t), \quad t \in {[0, T]}.\nonumber \\\end{aligned}$$

(146)

Since $\varrho \in \mathcal {L}_2(\{\mathcal {F}_t\}; \mathcal {BV}_0^r)$, and r and $\theta$ are uniformly bounded (see Condition 1 and Remark 2), it is immediate that $\eta _1 \in \varPi _{2}$ and therefore $\eta _2 \in \varPi _{2}$. It then follows from Labbé et al. (2007, Lemma 5.4, p. 90) that

$$\begin{aligned} \eta _{2}(t)= \xi (t)+ \theta (t)\int _0^t\xi '(s)\text {d}W(s)\;\; \,\text {a.e.}\,\, \text{ on }\,\, \varOmega \times {[0, T]}\;\; \text{ for } \text{ some } \text{ unique }\,\, \xi \in \varPi _{2}.\nonumber \\\end{aligned}$$

(147)

Combining (146) and (147), we have

$$\begin{aligned} S_0(t)\nu (t)= & {} y\theta (t)+ \xi (t)+ \theta (t)\int _0^t\xi '(\tau )\text {d}W(\tau ) + \theta (t)\int _0^tS_0(\tau )\varrho (\text {d}\tau ),\nonumber \\&\,\text {a.e.}\,\, \text{ on }\,\, \varOmega \times {[0, T]}. \end{aligned}$$

(148)

Now put

$$\begin{aligned} \gamma (t):= [S_0(t)]^{-1}\xi (t), \quad t\in {[0, T]}. \end{aligned}$$

(149)

Since $\xi \in \varPi _{2}$ it is immediate that $\gamma \in \varPi _{2}$, and it follows from (148) and (149) that

$$\begin{aligned} \nu (t)= \gamma (t)+ \frac{\theta (t)}{S_0(t)} \left\{ y + \int _0^tS_0(\tau ) \gamma '(\tau ) \text {d}W(\tau ) + \int _0^tS_0(\tau )\varrho (\text {d}\tau )\right\} \quad \,\text {a.e.}\quad \end{aligned}$$

(150)

Now (88) follows from (150) and (64).$\square$

1.6 Proof of Proposition 10

Fix some $(y, \gamma , \varrho ) \in \mathbb {R}\times \varPi _{2}\times \mathcal {L}_2(\{\mathcal {F}_t\}; \mathcal {BV}_0^r)$, and put

$$\begin{aligned} Y(t) := \widetilde{\varXi }(y, \gamma , \varrho )(t), \quad t \in {[0, T]}, \quad \text{ so } \text{ that } \quad Y \in \widetilde{\mathbb {B}}_{1}\;\; \text{(see } \text{(65)) }. \end{aligned}$$

(151)

Then Y satisfies the SDE (63). Now put

$$\begin{aligned} M(t) := \int _0^t\left[ X(s)\gamma '(s)+ Y(s-) \pi '(s)\sigma (s)\right] \text {d}W(s), \quad t \in {[0, T]}. \end{aligned}$$

(152)

Since the process X is continuous, expansion of XY by Itô’s formula, with (89) and (63), gives

$$\begin{aligned} M(t)= & {} X(t)Y(t)- X(0) y - \int _0^tX(s)\varrho (\text {d}s)\nonumber \\&- \int _0^t\pi '(s)\sigma (s)\left[ Y(s)\theta (s)+ \gamma (s)\right] \text {d}s, \quad t \in {[0, T]}. \end{aligned}$$

(153)

From (152) one sees that M is a continuous $\{ \mathcal {F}_t \}$-local martingale with $M(0) = 0$. We are now going to see that M is a martingale. For this, it is enough to establish

$$\begin{aligned} E\left[ {\sup _{t\in {[0, T]}}\left| {{M}(t)}\right| }\right] < \infty , \end{aligned}$$

(154)

(see Theorem 51 on p. 38 of Protter 2004). In view of (153)

$$\begin{aligned} \sup _{t\in {[0, T]}}\left| {{M}(t)}\right|\leqslant & {} \sup _{t\in {[0, T]}}\left| {X(t)Y(t)}\right| + \left| {X(0) y}\right| + \sup _{t\in {[0, T]}}\left| {\int _0^tX(s)\varrho (\text {d}s)}\right| \nonumber \\&+\, \int _0^T\left| {\pi '(s)\sigma (s)\left[ Y(s)\theta (s)+ \gamma (s)\right] }\right| \text {d}s. \end{aligned}$$

(155)

From (89) we have (exactly as at (14)),

$$\begin{aligned} E\left[ {\sup _{t\in {[0, T]}}\left| {X(t)}\right| ^2}\right] < \infty , \end{aligned}$$

(156)

and then it follows from the Cauchy–Schwarz inequality, together with (156), (151) and (69), that

$$\begin{aligned} E\left[ {\sup _{t\in {[0, T]}}\left| {X(t)Y(t)}\right| }\right] \leqslant E\left[ {\sup _{t\in {[0, T]}}\left| {X(t)}\right| ^2}\right] ^{1/2} E\left[ {\sup _{t\in {[0, T]}}\left| {Y(t)}\right| ^2}\right] ^{1/2} < \infty .\nonumber \\ \end{aligned}$$

(157)

Again from the Cauchy–Schwarz inequality, (156) and $\varrho \in \mathcal {L}_2(\{\mathcal {F}_t\}; \mathcal {BV}_0^r)$ (see Notation 10(5)),

$$\begin{aligned} E\left[ {\sup _{t\in {[0, T]}}\left| {\int _0^tX(s)\varrho (\text {d}s)}\right| }\right]\leqslant & {} E\left[ {(\sup _{s\in {[0, T]}}\left| {X(s)}\right| ) \left\| {\varrho }\right\| _T}\right] \nonumber \\\leqslant & {} E\left[ {\sup _{s\in {[0, T]}}\left| {X(s)}\right| ^2}\right] ^{1/2} E\left[ {\left\| {\varrho }\right\| _T^2}\right] ^{1/2} < \infty ,\nonumber \\ \end{aligned}$$

(158)

and it is immediate from $\pi , \gamma \in \varPi _{2}$, (151), (69) and the uniform boundedness of $\theta$ and $\sigma$ (see Remark 2 and Condition 1) that

$$\begin{aligned} E\left[ { \int _0^T\left| {\pi '(s)\sigma (s)\left[ Y(s)\theta (s)+ \gamma (s)\right] }\right| \text {d}s}\right] < \infty . \end{aligned}$$

(159)

Now (154) follows from (155) and (157)–(159), so that M is indeed a martingale, and therefore $E\left[ { M(T) }\right] = 0$. Now (90) follows upon taking expectations on each side of (153) with $t := T$.$\square$