1 Introduction

The purpose of this paper is to use stochastic control theory to obtain new results on the connections between the primal, utility maximization portfolio problem and its convex dual, both in the non-robust and the robust (worst case scenario/multiple-priors) setting. This approach allows us to get more detailed information about the connection between the primal and the dual problem. In particular, we show that the optimal wealth process of the primal problem coincides with the optimal adjoint process for the dual problem. This generalizes results that have been obtained earlier by using convex duality theory.

First, let us briefly recall the main results from the duality method in utility maximization, as presented in e.g. [8]: Let \(U:[0,\infty ] \rightarrow \mathbb {R}\) be a given utility function, assumed to be strictly increasing, strictly concave, continuously differentiable \((C^1)\) and satisfying the Inada conditions:

$$\begin{aligned} U{^\prime }(0)&= \lim _{x \rightarrow 0^+} U{^\prime }(x) = \infty \\ U{^\prime }(\infty )&= \lim _{x \rightarrow \infty } U{^\prime }(x) = 0. \end{aligned}$$

Let \(S(t) = S(t,\omega ) \; ; \; 0 \le t \le T\), \(\omega \in \Omega \), represent the discounted unit price of a risky asset at time t in a financial market. We assume that S(t) is a semimartingale on a filtered probability space \((\Omega , \mathcal {F}, \mathbb {F}:=\{\mathcal {F}_t\}_{0 \le t \le T}, P)\). Let \(\varphi (t)\) be an \(\mathbb {F}\)-predictable portfolio process, giving the number of units held of the risky asset at time t. If \(\varphi (t)\) is self-financing, the corresponding wealth process \(X(t) := X^x_\varphi (t)\) is given by

$$\begin{aligned} X(t) = x + \int _0^t \varphi (s) dS(s) \; ; \; 0 \le t \le T, \end{aligned}$$
(1.1)

where \(T \ge 0\) is a fixed terminal time and \(x > 0\) is the initial value of the wealth. We say that \(\varphi \) is admissible and write \(\varphi \in \mathcal {A}\) if the integral in (1.1) converges and

$$\begin{aligned} X_\varphi ^x(t) > 0 \text { for all } t \in [0,T], \text { a.s.}. \end{aligned}$$
(1.2)

The classical optimal portfolio problem is to find \(\varphi ^* \in \mathcal {A}\) (called an optimal portfolio) such that

$$\begin{aligned} u(x) := \sup _{\varphi \in \mathcal {A}} E [U(X^x_\varphi (T))] = E[U(X^x_{\varphi ^*}(T))]. \end{aligned}$$
(1.3)

The duality approach to this problem is as follows: Let

$$\begin{aligned} V(y) := \sup _{x > 0} \{U(x) - xy\} \; ; \; y > 0 \end{aligned}$$
(1.4)

be the convex conjugate function of U. Then it is well-known that V is strictly convex, decreasing, \(C^1\) and satisfies

$$\begin{aligned} V{^\prime }(0) = - \infty , \; V{^\prime }(\infty ) = 0,\; V(0) = U(\infty ) \; \text { and } V(\infty ) = U(0). \end{aligned}$$
(1.5)

Moreover,

$$\begin{aligned} U(x) = \inf _{y > 0} \{ V(y) + xy\} \; ; \; x > 0, \end{aligned}$$
(1.6)

and

$$\begin{aligned} U{^\prime }(x) = y \Leftrightarrow x = -V{^\prime }(y). \end{aligned}$$
(1.7)

Let \(\mathcal M\) be the set of probability measures Q which are equivalent local martingale measures (ELMM), in the sense that Q is equivalent to P and S(t) is a local martingale with respect to Q. We assume that \(\mathcal M\ne \emptyset \), which means absence of arbitrage opportunities on the financial market. The dual problem to (1.3) is for given \(y >0\) to find \(Q^* \in \mathcal M\) (called an optimal scenario measure) such that

$$\begin{aligned} v(y):= \inf _{Q \in \mathcal M} E \left[ V \left( y \frac{ dQ}{dP}\right) \right] = E \left[ V\left( y \frac{dQ^*}{dP}\right) \right] . \end{aligned}$$
(1.8)

One of the main results in [8] is that, under some conditions, \(\varphi ^*\) and \(Q^*\) both exist and they are related by

$$\begin{aligned} U{^\prime }(X_{\varphi ^*}^x(T)) =y \frac{dQ^*}{dP} \quad \text { with } y = u{^\prime }(x) \end{aligned}$$
(1.9)

i.e.

$$\begin{aligned} X_{\varphi ^*}^x(T) = - V{^\prime } \left( y \frac{dQ^*}{dP}\right) \quad \text { with } x = -v{^\prime }(y). \end{aligned}$$
(1.10)

In this paper we extend this result to a robust (model uncertainty) optimal portfolio problem and in a dynamic setting, by using stochastic control theory. We work in the slightly more special market setting with a risky asset price S(t) described by an Itô-Lévy process. This enables us to use the machinery of the maximum principle and backward stochastic differential equations (BSDE) driven by Brownian motion B(t) and a compensated Poisson random measure \(\tilde{N}(dt,d\zeta ) \; ; \; t \ge 0 \; ; \; \zeta \in \mathbb {R}_0 := \mathbb {R}\backslash \{0\}\). (We refer to e.g. [13] for more information about the maximum principle). Our approach has the advantage that it gives a dynamic relation between the optimal scenario in the dual formulation and the optimal portfolio in the primal formulation:

We start by the case without uncertainty (Sect. 2) for pedagogical purposes. In particular, in Sect. 2.2 we extend (1.10) to a dynamic identity between processes, and prove that

$$\begin{aligned} \hat{X}(t) = \hat{p}_2(t) ; t \in [0,T] \end{aligned}$$
(1.11)

where \(\hat{X}(t)\) is the optimal wealth process and \(\hat{p}_2(t)\) is the adjoint process for the dual problem. When \(t=T\) this gives the classical duality result above, namely

$$\begin{aligned} \hat{X}(T) = - V{^\prime } \left( y \frac{dQ^*}{dP}\right) (= \hat{p}_2(T) ). \end{aligned}$$
(1.12)

A similar result is obtained for the optimal density process for the dual problem.

As a step on the way, we prove in Sect. 2.1 a result of independent interest, namely that the existence of an optimal scenario is equivalent to the replicability of a related T-claim.

Then in Sect. 3 we extend the discussion to robust (model uncertainty) optimal portfolio problems. More precisely, we formulate robust versions of the primal problem (1.3) and of the dual problem (1.8), we establish a corresponding dynamic identity between processes and we show explicitly how to get from the solution of one to the solution of the other.

This paper addresses duality of robust utility maximization problems entirely by means of stochastic control methods, but there are several papers of related interest based on convex duality methods, see e.g. the survey paper [4] and the references therein. We also refer the reader to [16] where the author uses convex duality to study utility maximization under model uncertainty (multiple prior) and obtains a BSDE characterization of the optimal wealth process in markets driven by Brownian motion. In [6], a robust dual characterization of the robust primal utility maximization problem is obtained by convex duality methods. The dual formulation obtained is similar to ours, but there is no BSDE connection.

None of the above papers deal with a dynamic duality.

2 Dynamic Duality in Utility Maximization

2.1 Optimal Portfolio, Optimal Scenario and Replicability

We now specialize the setting described in Sect. 1 as follows: Suppose the financial market has a risk free asset with unit price \(S_0(t)=1\) for all t and a risky asset with price S(t) given by

$$\begin{aligned} \left\{ \begin{array}{ll} dS(t) = \displaystyle S(t^-)\left( b(t)dt+ \sigma (t) dB(t) + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \tilde{N}(dt,d\zeta ) \right) \; ; \; 0 \le t \le T\\ S(0) >0 \end{array}\right. \end{aligned}$$
(2.1)

where \(b(t), \sigma (t)\) and \(\gamma (t,\zeta )\) are predictable processes satisfying \(\gamma > -1\) and

$$\begin{aligned} E \left[ \int _0^T \left\{ | b(t)| + \sigma ^2(t) + \int _\mathbb {R}\gamma ^2(t,\zeta ) \nu (d\zeta ) \right\} dt \right] < \infty . \end{aligned}$$
(2.2)

Here B(t) and \(\tilde{N}(dt,d\zeta ):= N(dt,d\zeta ) -\nu (d\zeta )dt\) is a Brownian motion and an independent compensated Poisson random measure, respectively, on a filtered probability space \((\Omega , \mathcal {F}, \mathbb {F}:=\{\mathcal {F}_t\}_{0 \le t \le T}, P)\) satisfying the usual conditions, P is a reference probability measure and \(\nu \) is the Lévy measure of N.

Let \(\varphi (t)\) be a self financing portfolio and let \(X(t) := X^x_ \varphi (t) \) be the corresponding wealth process given by

$$\begin{aligned} \left\{ \begin{array}{ll} dX(t) = \varphi (t) S(t^-)\left[ b(t) dt + \sigma (t) dB(t) + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \tilde{N}(dt,d\zeta )\right] \; ; \; 0 \le t \le T \\ X(0) = x > 0. \end{array}\right. \end{aligned}$$
(2.3)

Definition 2.1

(Admissible Portfolios) Let \(\varphi \) be an \(\mathbb {F}\)-predictable, S-integrable process. We say that \(\varphi \) is admissible if

$$\begin{aligned}&X (t) > 0 \text { for all } t \in [0,T], \text { a.s.} \nonumber \\&E \left[ \int _0^T \varphi (t)^2 S(t)^2 \left\{ b(t) ^2 +\sigma ^2(t) + \displaystyle \int _\mathbb {R}\gamma ^2(t,\zeta ) \nu (d \zeta ) \right\} dt \right] < \infty , \nonumber \\&E\left[ \int _0^T | X(t) | ^{2}dt\right] < \infty \end{aligned}$$
(2.4)
$$\begin{aligned}&E\left[ U{^\prime }(X(T)) ^{2}\right] < \infty . \end{aligned}$$
(2.5)

We denote by \(\mathcal {A}\) the set of admissible portfolios. Conditions (2.4), (2.5) are needed for the application of the maximum principles. See Appendix A.

As in (1.3), for given \(x>0\), we want to find \(\varphi ^* \in \mathcal {A}\) such that

$$\begin{aligned} u(x) := \sup _{\varphi \in \mathcal {A}} E [U(X_\varphi ^x(T))] = E[U(X_{\varphi ^*}^x(T))]. \end{aligned}$$
(2.6)

We consider the family \(\mathcal M\) of equivalent local martingale measures (ELMM) that can be represented by means of the family of positive measures \(Q = Q_\theta \) of the form

$$\begin{aligned} dQ_\theta (\omega ) = G_\theta (t) dP(\omega ) \text { on } \mathcal {F}_t, \end{aligned}$$
(2.7)

where

$$\begin{aligned} \left\{ \begin{array}{ll} \Big \{dG_\theta (t) = G_\theta (t^-) \left[ \theta _0(t) dB(t) + \displaystyle \int _\mathbb {R}\theta _1(t,\zeta ) \tilde{N}(dt,d\zeta )\right] \; ; \; 0 \le t \le T \\ G_\theta (0) = y > 0, \end{array}\right. \end{aligned}$$
(2.8)

and \(\theta = (\theta _0, \theta _1)\) is a predictable process satisfying the conditions

$$\begin{aligned} E\left[ \int _0^T \left\{ \theta ^2_0(t) + \displaystyle \int _\mathbb {R}\theta ^2_1 (t,\zeta ) \nu (d\zeta ) \right\} dt \right] < \infty , \; \theta _1(t,\zeta ) > -1 \;\; \text { a.s.} \end{aligned}$$
(2.9)

and

$$\begin{aligned} b(t) + \sigma (t) \theta _0(t) + \int _\mathbb {R}\gamma (t,\zeta ) \theta _1 (t,\zeta ) \nu (d\zeta ) = 0 \; ; \; t \in [0,T]. \end{aligned}$$
(2.10)

If \(y=1\) this condition implies that \(Q_\theta \) is an ELMM for this market. See e.g. [13, Chap. 1].

Remark 2.2

The set \(\mathcal M\) with \(y=1\) is contained in the set ELMM. Note that there are ELMM’s which are not of the above form. However, \(\mathcal M\) is the family we choose to work with, and all our results are proved for this family of measures.

We let \(\Theta \) denote the set of all \(\mathbb {F}\)-predictable processes \(\theta = (\theta _0, \theta _1)\) satisfying (2.9)–(2.10).

The dual problem corresponding to (1.8) is for given \(y >0\) to find \(\hat{\theta }\in \Theta \) and v(y) such that

$$\begin{aligned} - v(y) := \sup _{\theta \in \Theta } E [-V(G^y_\theta (T))] = E[-V(G^y_{\hat{\theta }}(T))]. \end{aligned}$$
(2.11)

Such a process \(\hat{\theta }\) is called an optimal scenario.

We will use two stochastic maximum principles for stochastic control to study the problem (2.11) and relate it to (2.6). We refer to Appendix A for a presentation of these principles and to [17] for more information about backward stochastic differential equations (BSDEs) with jumps.

We recall the existence and uniqueness result for BSDEs with jumps, due to [20] (see [20]). If \(T>0\), \(F \in L^2(\mathcal{F}_T)\), and g is a Lipschitz driver, then there exists a unique solution \(\in S^2 \times H^2 \times H_{\nu }\) of the BSDE with jumps

$$\begin{aligned} \left\{ \begin{array}{ll} dp(t) &{}= -g(t,p(t),q(t),r(t,\cdot ))dt + q(t)dB(t) + \int _{\mathbb {R}} r(t,\zeta ) \tilde{N}(dt,d\zeta )\; ; \; 0 \le t \le T \\ p(T) &{}= F, \end{array}\right. \end{aligned}$$

where

  • \(S^2\) is the set of real-valued càdlàg adapted processes \(\phi \) with \(E(\sup _{0\le t \le T} |\phi _t |^2) < \infty .\)

  • \(H^2\) is the set of real-valued predictable processes \(\phi \) such that \( E \left[ \left( \int _0 ^T \phi _t ^2 dt\right) \right] < \infty ,\)

  • \(H_{\nu }^2\) is the set of predictable processes \(\ell \) such that \(E\left[ \left( \int _0 ^T (\int _{\mathbb {R}} |\ell (t,\zeta ) |^2 \nu (d\zeta ) ) \,dt\right) \right] < \infty .\)

From now on, when we say that a process triple \((p(t),q(t),r(t,\zeta ))\) satisfies a BSDE of the form (), it is tacitly understood that \((p,q,r) \in S^2 \times H^2 \times H_{\nu }\).

We first prove two auxiliary results, the first of which may be regarded as a special case of Proposition 4.4 in [5].

Proposition 2.3

(Primal problem and associated constrained FBSDE) Let \(\hat{\varphi }\) in \(\mathcal {A}\). Then \(\hat{\varphi } \) is optimal for the primal problem (2.6) if and only if the (unique) solution \((X_{\hat{\varphi }}^x, (\hat{p}_1,\hat{q}_1,\hat{r}_1))\) of the FBSDE consisting of the SDE (2.3) and the BSDE

(2.12)

also satisfies the equation

$$\begin{aligned} b(t)\hat{p}_1(t) + \sigma (t) \hat{q}_1(t) + \int _\mathbb {R}\gamma (t,\zeta ) \hat{r}_1 (t,\zeta ) \nu (d\zeta ) = 0 \; ; \; t \in [0,T], \end{aligned}$$
(2.13)

where \((\hat{p}_1,\hat{q}_1,\hat{r}_1)\) \( \in \) \( S^2 \times H^2 \times H_{\nu }^2.\)

Proof

(i) The Hamiltonian corresponding to the primal problem is given by

$$\begin{aligned} H_1(t,x, \varphi , p,q,r) = \varphi S(t^-) ( b(t) p + \sigma (t) q + \int _\mathbb {R}\gamma (t, \zeta ) r(\zeta ) \nu (d\zeta ) ). \end{aligned}$$
(2.14)

Assume \(\hat{\varphi } \in \mathcal {A}\) is optimal for the primal problem (2.6). Then by the necessary maximum principle (Theorem 4.2), we have

$$\begin{aligned} \frac{\partial H_1}{ \partial \varphi }(t,x,\varphi , \hat{p}_1(t), \hat{q}_1(t), \hat{r}_1(t, \cdot )) \mid _{\varphi = \hat{\varphi }(t)} = 0, \end{aligned}$$

where \((\hat{p}_1, \hat{q}_1, \hat{r}_1)\) satisfies (2.12), since \( \frac{\partial H_1}{ \partial x}(t,x,\varphi , \hat{p}_1(t), \hat{q}_1(t), \hat{r}_1(t, \cdot )) = 0.\) This implies (2.13).

(ii) Conversely, suppose the solution \((\hat{p}_1, \hat{q}_1, \hat{r}_1)\) of the BSDE (2.12) satisfies (2.13). Then \(\hat{\varphi }\), with the associated \((\hat{p}_1, \hat{q}_1, \hat{r}_1)\) satisfies the conditions for the sufficient maximum principle (Theorem 4.1) with the additional feature of a constraint. See (2.17) below. We conclude that \(\hat{\varphi }\) is optimal. \(\square \)

Remark 2.4

The BSDE (2.12) is linear, and hence it is well known that it has a unique solution (p, q, r) for every choice of \(X_{\varphi }^x(T)\). See e.g. [17],[18]. We are seeking \( \hat{\varphi }\) such that the corresponding solution \((\hat{p},\hat{q},\hat{r})\) of (2.12) also satisfies (2.13).

Remark 2.5

By (2.12) we have \(\hat{p}_1(t) = E [U{^\prime }(X_{\hat{\varphi }}^x(T)) \mid \mathcal {F}_t]>0\) for all t in [0, T], and if we divide equation (2.13) throughout by \(\hat{p}_1(t)\) we get

$$\begin{aligned} b(t) + \sigma (t) \hat{\theta }_0(t) + \int _{\mathbb {R}} \gamma (t,\zeta ) \hat{\theta }_1(t,\zeta ) \nu (d\zeta ) = 0 \; ; \; t \in [0,T], \end{aligned}$$
(2.15)

where

$$\begin{aligned} \hat{\theta }_0(t): = \frac{\hat{q}_1(t)}{\hat{p}_1(t)} \; ; \;\; \hat{\theta }_1(t,\zeta ) : = \frac{\hat{r}_1(t,\zeta )}{\hat{p}_1(t)} \; , \; t \in [0,T]. \end{aligned}$$
(2.16)

By the Girsanov theorem this is saying that if we define the measure \( Q_{(\hat{\theta }_0,\hat{\theta }_1)} \) as in (2.7), (2.8) with \(y=1\), then \( Q_{(\hat{\theta }_0,\hat{\theta }_1)} \) is an ELMM for the market described by (2.1).

We now turn to the dual problem (2.11). We may regard it as a stochastic control problem in the control process \(\theta \) with the constraint (2.10). To solve this problem we use the well-known Lagrange multiplier technique. Thus we define the Hamiltonian \(H_2^L\) by

$$\begin{aligned} H_2^L(\theta _0, \theta _1,L)&:= g \theta _0 q + g \int _\mathbb {R}\theta _1(\zeta ) r(\zeta ) \nu (d\zeta ) \nonumber \\&\quad + L(t) \left( b(t) + \sigma (t) \theta _0 + \int _\mathbb {R}\gamma (t,\zeta ) \theta _1(\zeta ) \nu (d\zeta )\right) , \end{aligned}$$
(2.17)

where L(t) is the Lagrange multiplier process. Maximizing \(H_2^L\) over all \(\theta _0\) and \(\theta _1\) gives the following first order conditions for the optimal \( \hat{\theta }\) and the associated adjoint processes \(\hat{q}_2\), \(\hat{r}_2\) :

$$\begin{aligned} G_{\hat{\theta }}(t) \hat{q}_2(t) + L(t) \sigma (t) = 0; \; \; G_{\hat{\theta }}(t) \hat{r}_2(t,\cdot ) + L(t) \gamma (t,\cdot ) = 0. \end{aligned}$$

Since \( G_{\hat{\theta }}(t) \ne 0\), we can write these as follows:

$$\begin{aligned} \hat{q}_2(t) = - \frac{L(t)}{G_{\hat{\theta }}(t)} \sigma (t); \;\; \hat{r}_2(t,\zeta ) = - \frac{L(t)}{G_{\hat{\theta }(t)}} \gamma (t,\zeta ). \end{aligned}$$
(2.18)

The adjoint equation becomes:

(2.19)

In view of (2.10) this can be written

(2.20)

Note that

$$\begin{aligned} \text {If } \sigma (t) \ne 0 \text { then } - \frac{L(t)}{G_{\hat{\theta }}(t)} = \frac{ \hat{q}_2(t)}{\sigma (t)} \end{aligned}$$
(2.21)
$$\begin{aligned} \text {If } \gamma (t,\zeta ) \ne 0 \text { then } - \frac{L(t)}{G_{\hat{\theta }}(t)} = \frac{ \hat{r}_2(t,\zeta )}{\gamma (t,\zeta )}. \end{aligned}$$
(2.22)

In particular note that if \(\gamma (t,\zeta ) \ne 0,\) then

$$\begin{aligned} \hat{\lambda }(t):=- \frac{ \hat{r}_2(t,\zeta )}{\gamma (t,\zeta )} \end{aligned}$$
(2.23)

does not depend on \(\zeta \).

If \(\sigma (t) = \gamma (t,\zeta ) = 0\), then by (2.18) we have \( \hat{q}_2(t) = \hat{r}_2 (t,\zeta ) = 0\) and hence we have \(d\hat{p}_2(t) = 0\). From (2.18) we get the following constraint:

$$\begin{aligned} - \hat{q}_2(t) \gamma (t,\zeta ) + \sigma (t) \hat{r}_2(t, \zeta ), \quad 0 \le t \le T . \end{aligned}$$
(2.24)

We have proved:

Proposition 2.6

(Dual problem and associated constrained FBSDE) Let \(\hat{\theta }\in \Theta \). Then \(\hat{\theta }\) is an optimal scenario for the dual problem (2.11) if and only if the solution \((G_{\hat{\theta }}\), \((\hat{p}_2, \hat{q}_2, \hat{r}_2)) \) of the FBSDE consisting of the FSDE (2.8) and the BSDE

$$\begin{aligned} \left\{ \begin{array}{ll} d \hat{p}_2(t) &{} = \displaystyle K(\hat{q}_2, \hat{r}_2)(t) \left[ b(t) dt + \sigma (t) dB(t) + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \tilde{N}(dt,d\zeta )\right] \\ \hat{p}_2 (T)&{} = - V{^\prime }(G^y_{\hat{\theta }}(T)) \end{array}\right. \end{aligned}$$
(2.25)

with

$$\begin{aligned} K(\hat{q}_2, \hat{r}_2)(t) := \frac{\hat{q}_2(t)}{\sigma (t)} \chi _{\sigma (t) \ne 0} + \hat{\lambda }(t) \chi _{\sigma (t) = 0, \gamma (t,\zeta ) \ne 0} \end{aligned}$$
(2.26)

also satisfies (2.24), where \((\hat{p}_2, \hat{q}_2, \hat{r}_2) \in S^2 \times H^2 \times H_{\nu }^2\).

The sufficient part follows from the fact that the functions \(g \rightarrow - V(g)\) and

$$\begin{aligned} g \rightarrow \sup _{\theta _0,\theta _1} {H}_2^L(t,g,\theta _0,\theta _1,\hat{p}_2(t), \hat{q}_2(t), \hat{r}_2(t,\cdot )) \end{aligned}$$

are concave.

We deduce as a by-product the following results of independent interest which relates the existence of a solution of the dual problem to the replication of a related T-claim.

Proposition 2.7

For given \(y>0\) and \(\hat{\theta }\in \Theta \) the following are equivalent:

  1. (i)
    $$\begin{aligned} \sup _{\theta \in \Theta } E [-V(G_\theta ^y(T))] = E[-V(G_{\hat{\theta }}^y (T))] < \infty . \end{aligned}$$
  2. (ii)

    The claim \(F:= - V{^\prime } (G_{\hat{\theta }} ^y(T))\) is replicable, with initial value \(x = \hat{p}_2(0)\), where \((\hat{p}_2, \hat{q}_2, \hat{r}_2)\) solves

    (2.27)

Moreover, if (i) or (ii) holds, then

$$\begin{aligned} \hat{\varphi }(t) : = \frac{K(\hat{q}_2,\hat{r}_2)(t)}{S(t^-)} \end{aligned}$$
(2.28)

is a replicating portfolio for \(F:= -V{^\prime }(G_{\hat{\theta }}^y(T))\), where \((\hat{p}_2,\hat{q}_2,\hat{r}_2)\) is the solution of the BSDE (2.27).

Proof

(i) \(\Rightarrow \) (ii): We have already proved that (i) implies (2.27). This equation states that the contingent claim \(F:= - V{^\prime }(G_{\hat{\theta }}^y(T))\) is replicable, with replicating portfolio \(\hat{\varphi }(t)\) given by (2.28) and initial value \(x = \hat{p}_2(0)\). Note that \(\hat{p}_2(t) > 0\) for all t, since V is strictly decreasing so \(- V{^\prime }(G_{\hat{\theta }}^y(T)) > 0\).

(ii) \(\Rightarrow \) (i): Suppose

\(F:= - V{^\prime } (G_{\hat{\theta }}^y(T))\) is replicable with initial value \(x = \hat{p}_2(0)\), and let \(\varphi \in \mathcal {A}\) be a replicating portfolio. Then \(X(t) = X^x_{\varphi }(t)\) satisfies the equation

(2.29)

Define

$$\begin{aligned} \hat{p}(t) := X(t), \hat{q}(t) := \varphi (t) \sigma (t) S(t^-) \text { and } \hat{r}(t,\zeta ) := \varphi (t) \gamma (t,\zeta ) S(t^-). \end{aligned}$$
(2.30)

They satisfy the relation (2.24). Moreover, by (2.30) we get

$$\begin{aligned} \varphi (t)S(t^-) = K(q,r)(t). \end{aligned}$$
(2.31)

Therefore, from (2.29) we get that \((\hat{p},\hat{q}, \hat{r})\) satisfies the BSDE

(2.32)

We conclude that \( \hat{p}(t) = X(t) = \hat{p}_2(t)\). Hence (i) holds, by Proposition 2.6.

The last statement follows from (2.31). \(\square \)

2.2 Relations Between Optimal Scenario and Optimal Portfolio

We proceed to show that the method above actually gives a connection between an optimal scenario \(\hat{\theta }\in \Theta \) for the dual problem (2.11) and an optimal portfolio \(\hat{\varphi }\in \mathcal {A}\) for the primal problem (2.6).

Theorem 2.8

(a) Suppose \(\hat{\varphi }\in \mathcal {A}\) is optimal for the primal problem (2.6).

Let \((\hat{p}_1(t), \hat{q}_1(t), \hat{r}_1(t,\zeta ))\) be the associated adjoint processes, solution of the constrained BSDE (2.12)–(2.13). Define

$$\begin{aligned} \hat{\theta }_0(t) = \frac{\hat{q}_1(t)}{\hat{p}_1(t^-)}, \;\;\; \hat{\theta }_1(t,\zeta ) = \frac{\hat{r}_1(t,\zeta )}{\hat{p}_1(t^-)}. \end{aligned}$$
(2.33)

Suppose

$$\begin{aligned} E\left[ \int _0^T \{\hat{\theta }_0^2(t) + \int _\mathbb {R}\hat{\theta }_1^2(t,\zeta )\nu (d\zeta ) \}dt\right] < \infty ; \; \; \hat{\theta }_1 >-1. \end{aligned}$$
(2.34)

Then \(\hat{\theta }= (\hat{\theta }_0, \hat{\theta }_1) \in \Theta \) is optimal for the dual problem (2.11) with initial value \(y = \hat{p}_1(0)\). Moreover, with \(y = \hat{p}_1(0)\),

$$\begin{aligned} G_{\hat{\theta }}^y(t) =\hat{p}_1(t) ; \quad t \in [0,T] . \end{aligned}$$
(2.35)

In particular

$$\begin{aligned} G_{\hat{\theta }}^y(T) = U{^\prime }(X_{\hat{\varphi }}^x(T)). \end{aligned}$$
(2.36)

(b) Conversely, suppose \(\hat{\theta }= (\hat{\theta }_0, \hat{\theta }_1) \in \Theta \) is optimal for the dual problem (2.11). Let \((\hat{p}_2(t), \hat{q}_2(t), \hat{r}_2(t,\zeta ))\) be the associated adjoint processes, solution of the BSDE (2.25) with the constraint (2.24). Suppose the portfolio

$$\begin{aligned} \hat{\varphi }(t) : = \frac{K(\hat{q}_2,\hat{r}_2)(t)}{S(t^-)} \end{aligned}$$
(2.37)

is admissible. Then \(\hat{\varphi }\) is an optimal portfolio for the primal problem (2.6) with initial value \(x = \hat{p}_2(0)\). Moreover, with \(x = \hat{p}_2(0)\),

$$\begin{aligned} X_{\hat{\varphi }}^x(t) = \hat{p}_2(t) ; \quad t \in [0,T]. \end{aligned}$$
(2.38)

In particular

$$\begin{aligned} X_{\hat{\varphi }}^x(T) = - V{^\prime } (G_{\hat{\theta }}^y(T)). \end{aligned}$$
(2.39)

Proof

(a) Suppose \(\hat{\varphi }\) is optimal for problem (2.6) with initial value x. Then, by Proposition 2.3, the adjoint processes \(\hat{p}_1(t), \hat{q}_1(t), \hat{r}_1(t,\zeta )\) for Problem (2.6) satisfy (2.12)–(2.13). Consider the process \(\hat{\theta }(t)\) defined in (2.33) and suppose (2.34) holds. Then \(\hat{\theta }\in \Theta \) and (2.12) can be written

(2.40)

Therefore \(\hat{p}_1(t) \equiv G_{\hat{\theta }}^y(t) \) (see (2.8)) if we put \(y :=\hat{p}_1(0)>0\), and we have, by (1.7)

$$\begin{aligned} U{^\prime }(X_{\hat{\varphi }}^x(T)) = G_{\tilde{\theta }}^y(T), \text { i.e. } X_{\hat{\varphi }}^x(T) = - V{^\prime }(G_{\tilde{\theta }}^y(T)). \end{aligned}$$
(2.41)

Now define

$$\begin{aligned} \hat{p}_2(t) := X_{\hat{\varphi }}^x(t), \hat{q}_2(t) := \hat{\varphi }(t) \sigma (t) S(t^-) \text { and } \hat{r}_2(t,\zeta ) := \hat{\varphi }(t) \gamma (t,\zeta ) S(t^-). \end{aligned}$$
(2.42)

Then \( (\hat{p}_2, \hat{q}_2, \hat{r}_2)\) satisfy the conditions of Proposition 2.6 which imply that \(\hat{\theta }\) is optimal for problem (2.11).

(b) Suppose \(\hat{\theta }\in \Theta \) is optimal for problem (2.11) with initial value y. Let \(\hat{p}_2(t), \hat{q}_2(t), \hat{r}_2(t,\cdot )\) be the associated adjoint processes, solution of the BSDE (2.25) with the constraint (2.24). Then they satisfy the equation

(2.43)

Define

$$\begin{aligned} \tilde{\varphi }(t) := \hat{\varphi }(t) : = \frac{K(\hat{q}_2,\hat{r}_2)(t)}{S(t^-)}, \end{aligned}$$
(2.44)

and assume \(\tilde{\varphi }(t) \) is admissible. Then \( \hat{p}_2(t) \equiv X_{\tilde{\varphi }}^x(t) \) for \(x = \hat{p}_2(0)\). In particular

$$\begin{aligned} X_{\tilde{\varphi }}^x(T) = - V{^\prime }(G_{\hat{\theta }}^y(T)), \text { i.e. } G_{\hat{\theta }}^y(T) = U{^\prime } (X_{\tilde{\varphi }}^x(T)). \end{aligned}$$
(2.45)

Therefore \(G_{\hat{\theta }}^y(t)= G_{\hat{\theta }}(t)\) satisfies the equation

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle dG_{\hat{\theta }}(t) = G_{\hat{\theta }}(t^-) \left[ \hat{\theta }_0(t) dB(t) + \displaystyle \int _\mathbb {R}\hat{\theta }_1(t,\zeta ) \tilde{N}(dt,d\zeta )\right] \; ; \; 0 \le t \le T \\ G_{\hat{\theta }}(T) = U{^\prime }(X_{\tilde{\varphi }}^x(T)). \end{array}\right. \end{aligned}$$
(2.46)

Define now

$$\begin{aligned} p_1(t) := G_{\hat{\theta }}(t), q_1(t) := G_{\hat{\theta }}(t) \hat{\theta }_0(t), r_1(t,\zeta ) := G_{\hat{\theta }}(t) \hat{\theta }_1(t,\zeta ). \end{aligned}$$
(2.47)

Then by (2.46) \((p_1,q_1,r_1)\) solves the BSDE

(2.48)

Moreover, since \(\hat{\theta }\in \Theta \), it satisfies (2.10), that is

$$\begin{aligned} b(t) + \sigma (t) \hat{\theta }_0(t) + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \hat{\theta }_1(t,\zeta ) \nu (d\zeta ) = 0 \; ; \; 0 \le t \le T \end{aligned}$$
(2.49)

i.e., \((p_1,q_1,r_1)\) satisfies the equation

$$\begin{aligned} b(t) + \sigma (t) \frac{q_1(t)}{p_1(t)} + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \frac{r_1(t,\zeta )}{p_1(t)} \nu (d\zeta ) = 0 \; ; \; 0 \le t \le T. \end{aligned}$$
(2.50)

It follows from Proposition 2.3 that \(\hat{\varphi }:= \tilde{\varphi }\) is an optimal portfolio for problem (2.6) with initial value \(x = \hat{p}_2(0)\). \(\square \)

Remark 2.9

Conditions of the above theorem have to be verified in each specific case. They hold for examples in in Examples 2.10 and 3.6. Note that the integrability condition in (2.34) hold whenever the utility function U satisfies the condition

$$\begin{aligned} U{^\prime } \text { is bounded and bounded away from } 0. \end{aligned}$$
(2.51)

Indeed this implies that \(p_1(t)\) which is equal to \(E [U{^\prime }(X_{\hat{\varphi }}^x(T)) \mid \mathcal {F}_t]\) is bounded away from 0 and that (\(q_1\), \(r_1\)) belongs to \( H^2 \times H_{\nu }\). Therefore \(\frac{1}{p_1}\) is bounded and (\(\frac{q_1}{p_1}\) , \(\frac{r_1}{p_1}\)) belong to \(H^2 \times H_{\nu }\). Condition (2.51) does not hold a priori for the most commonly studied utility functions, e.g. the logarithmic or the power functions, but any given utility function can be perturbed slightly such that it holds, simply by modifying it arbitrary near 0 or arbitrary near infinity, if necessary.

Example 2.10

As an illustration of Theorem 2.8 let us apply it to the situation when \(\sigma =0\), \(\gamma (t,\zeta ) = \gamma (t,1) > 0\) and N(t) is the Poisson process with intensity \(\lambda > 0\). Then \(\nu (d \zeta ) = \lambda \delta _1(d\zeta )\), where \(\delta _1\) is Dirac measure at 1, and hence

$$\begin{aligned} \displaystyle \int _\mathbb {R}\gamma (t, \zeta ) \tilde{N}(dt, d\zeta ) = \gamma (t,1) (dN(t)- \lambda dt) := \gamma (t,1) d \tilde{N}(t), \end{aligned}$$
(2.52)

and (2.1) and (2.3) become, respectively,

$$\begin{aligned} dS(t) = S(t^-)[b(t) dt + \gamma (t,1) d \tilde{N}(t)] \; ; \; S(0) > 0 \end{aligned}$$
(2.53)

and

$$\begin{aligned} dX(t) = \varphi (t) S(t^-) [b(t) dt + \gamma (t,1) d \tilde{N}(t)] \; ; \; X(0) = x > 0. \end{aligned}$$
(2.54)

Assume that b(t) and \(\gamma (t,1)\) are bounded predictable processes and that there exists a constant \(C <1\) such that

$$\begin{aligned} \frac{|b(t)|}{\lambda |\gamma (t,1)|} \le C \; ; \; 0 \le t \le T. \end{aligned}$$
(2.55)

Then \(\Theta \) has just one element \(\theta _1(t,1)\), given by

$$\begin{aligned} \theta _1(t,1) = - \frac{b(t)}{\lambda \gamma (t,1)} \end{aligned}$$
(2.56)

and hence by (2.10) and the Itô formula,

$$\begin{aligned} G^y_{\theta _1}(t)&= y \exp \Big ( \int _0^t \ln \big (1 - \frac{b(s)}{\lambda \gamma (s,1)}\big ) d \tilde{N}(s) \nonumber \\&\quad + \lambda \int _0^t \left\{ \ln \Big ( 1 - \frac{b(s)}{\lambda \gamma (s,1)}\Big ) + \frac{b(s)}{\lambda \gamma (s,1)} \right\} ds \Big ) \; ; \; 0 \le t \le T. \end{aligned}$$
(2.57)

By Theorem 2.8b) we get that

$$\begin{aligned} \hat{\varphi }(t) := \frac{ \hat{r}_2(t,1)}{\gamma (t,1) S(t^-)} \end{aligned}$$
(2.58)

is an optimal portfolio for the primal problem (2.8), where \((\hat{p}_2(t), \hat{r}_2(t,1))\) solves the BSDE (2.25), which in our case gets the form

$$\begin{aligned} \left\{ \begin{array}{ll} d\hat{p}_2(t) &{} = \frac{\hat{r}_2(t,1)}{\gamma (t,1)} b(t) dt + \hat{r}_2(t,1) d \tilde{N}(t) \; ; \; 0 \le t \le T \\ \hat{p}_2 (T) &{} = - V{^\prime }(G^y_{\theta _1}(T)). \end{array}\right. \end{aligned}$$
(2.59)

To solve this BSDE we try a solution of the form

$$\begin{aligned} \hat{r}_2(t,1) = \hat{p}_2(t) \psi (t), \end{aligned}$$
(2.60)

for some predictable process \(\psi \), and get the solution

$$\begin{aligned} \hat{p}_2(t)&= \hat{p}_2(0) \exp \Big ( \int _0^t \ln (1 + \psi (s)) d \tilde{N}(s) \nonumber \\&\quad + \int _0^t \{ \lambda (\ln (1 + \psi (s)) - \psi (s)) + \frac{b(s)}{\gamma (s,1)} \psi (s) \} ds\Big ). \end{aligned}$$
(2.61)

In particular, if \(U(x) = \ln x\), then \(V(y) = - \ln y-1\) and \(V{^\prime }(y) = - \frac{1}{y}\). Hence (2.59) implies that

$$\begin{aligned} \hat{p}_2(0)&\exp \left( \int _0^T \ln (1 + \psi (s)) d \tilde{N}(s) \right. \nonumber \\&\quad + \int _0^T \left\{ \lambda ( \ln (1 + \psi (s)) - \psi (s)) + \frac{b(s)}{\gamma (s,1)} \psi (s) \right\} dt \nonumber \\&= \frac{1}{y} \exp \left( - \int _0^T \ln \left( 1 - \frac{b(s)}{\lambda \gamma (s,1)}\right) d \tilde{N}(s) \right. \nonumber \\&\left. \quad - \lambda \int _0^T \left\{ \ln \left( 1 - \frac{b(s)}{\lambda \gamma (s,1)}\right) + \frac{b(s)}{\lambda \gamma (s,1)}\right\} ds \right) . \end{aligned}$$
(2.62)

Choose

$$\begin{aligned} \hat{p}_2(0) = \frac{1}{y} \end{aligned}$$
(2.63)

and choose \(\psi (s)\) such that the \(d\tilde{N}\)-integrals of (2.62) coincide, i.e.

$$\begin{aligned} \ln (1 + \psi (s)) = - \ln \left( 1 - \frac{b(s)}{\lambda \gamma (s,1)}\right) \end{aligned}$$

i.e.

$$\begin{aligned} \psi (s) = \frac{b(s)}{\lambda \gamma (s,1) - b(s)}. \end{aligned}$$
(2.64)

Then we see that also the ds-integrals coincide, i.e.

$$\begin{aligned} \lambda (\ln (1 + \psi (s)) - \psi (s)) + \frac{b(s)}{\gamma (s,1)} \psi (s) = - \lambda \left( \ln \left( 1 - \frac{b(s)}{\lambda \gamma (s,1)}\right) + \frac{b(s)}{\lambda \gamma (s,1)} \right) . \end{aligned}$$

Hence, with this choice of \(\psi \), we see that \(d(\hat{p}_2(t)) = d ( \frac{1}{G_{\theta _1}^y})(t).\) The process

$$\begin{aligned} \hat{p}_2 = \frac{1}{G_{\theta _1}^y}\; ; \; \hat{r}_2 = \hat{p}_2 \psi \end{aligned}$$

with \(\psi \) given by (2.64) solves BSDE (2.59). Moreover (2.24) holds trivially. We conclude by (2.58) that the optimal portfolio \(\hat{\varphi }(t)\) for problem (2.8) with \(U(x) = \ln x\) is

$$\begin{aligned} \hat{\varphi }(t) = \frac{\hat{p}_2(t) b(t)}{\gamma (t,1) S(t^-)(\lambda \gamma (t,1) - b(t))}\end{aligned}$$
(2.65)

which means that the optimal fraction \(\hat{\pi }(t)\) to be placed in the risky asset is, using (2.38)

$$\begin{aligned} \hat{\pi }(t) = \frac{\hat{\varphi }(t) S(t^-)}{X_{\hat{\varphi }}(t)} = \frac{b(t)}{\gamma (t,1)(\lambda \gamma (t,1) - b(t))}. \end{aligned}$$
(2.66)

Remark 2.11

To check that \(\hat{\varphi }\) is admissible, we have to verify that (2.4) and (2.5) hold for \(\varphi =\hat{\varphi }\). To this end, we see that condition (2.55) suffices.

3 Robust Duality

In this section we extend our study to a robust optimal portfolio problem and its dual.

3.1 Model Uncertainty Setup

To get a representation of model uncertainty, we consider a family of probability measures \(R = R^\kappa \) \(\sim P\), with Radon-Nikodym derivative on \(\mathcal{F}_t\) given by

$$\begin{aligned} \frac{d(R^\kappa \mid \mathcal{F}_t)}{d(P \mid \mathcal{F}_t)} = Z^\kappa _t \end{aligned}$$

where, for \(0 \le t \le T\), \(Z^\kappa _t\) is a martingale of the form

$$\begin{aligned} dZ^\kappa _t = Z^\kappa _{t^-} [\kappa _0(t) dB_t + \int _\mathbb {R}\kappa _1(t,\zeta ) \tilde{N}(dt,d\zeta )] \;; \quad Z^\kappa _0 = 1. \end{aligned}$$

Let \(\mathbb K\) denote a given set of admissible scenario controls \(\kappa = (\kappa _0, \kappa _1)\), \(\mathcal {F}_{t}\)-predictable, s.t. \(\kappa _1(t,z) \ge -1 + \epsilon \), and \(E \left[ \int _0^T \{ |\kappa _0^2(t) | + \int _\mathbb {R}\kappa _1^2(t,z) \nu (dz)\} dt\right] < \infty \).

By the Girsanov theorem, using the measure \(R^\kappa \) instead of the original measure P in the computations involving the price process S(t), is equivalent to using the original measure P in the computations involving the perturbed price process \(S_\mu (t)\) instead of P(t), where \(S_\mu (t)\) is given by

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle dS_\mu (t) =S_\mu (t^-)[ (b (t) + \mu (t))dt + \sigma (t) dB(t ) + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \tilde{N}(dt, d \zeta )] \\ S_\mu (0) >0, \end{array}\right. \end{aligned}$$
(3.1)

with

$$\begin{aligned} \mu (t )= - \sigma (t )\kappa _0(t) - \int _\mathbb {R}\gamma (t,\zeta ) \kappa _1(t,\zeta ) \nu (d\zeta )dt. \end{aligned}$$
(3.2)

Accordingly, we now define \(\mathbb M\) to be the set of all predictable processes \(\mu \) such that

$$\begin{aligned} E \left[ \int _0^T |\mu (t)| dt\right] < \infty , \end{aligned}$$

and we consider the perturbed process

(3.3)

for \(\mu \in \mathbb M\). Let \(X = X_{\varphi ,\mu }^x\) be the corresponding wealth process given by

$$\begin{aligned} \left\{ \begin{array}{ll} dX(t) \displaystyle = \varphi (t)S_\mu (t^-) [(b(t) + \mu (t))dt + \sigma (t) dB(t) + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \tilde{N}(dt,d\zeta )] \; ; \; 0 \le t \le T \\ X(0) = x > 0, \end{array}\right. \end{aligned}$$
(3.4)

where \(\varphi \) is an admissible portfolio, that is it belongs to the set \(\mathcal {A}\) of \(\mathbb {F}\)-predictable processes such that

(3.5)

for all \( \mu \in \mathbb M.\)

3.2 The Robust Primal and Dual Problems

Let \(\rho : \mathbb {R}\rightarrow \mathbb {R}\) be a convex penalty function, assumed to be \(\mathcal C^1\), and U a utility function as in Sect. 1. We assume that \(\rho (\mu )\) has a minimum at \(\mu = 0\) and that \(\rho (0) = 0\). Then \(\rho (\mu )\) can be interpreted as a penalization for choosing \(\mu \ne 0\).

Definition 3.1

The robust primal problem is, for given \(x>0\), to find \((\hat{\varphi }, \hat{\mu }) \in \mathcal {A}\times \mathbb M \) such that

$$\begin{aligned} \inf _{\mu \in \mathbb M} \sup _{\varphi \in \mathcal {A}} I(\varphi , \mu ) = I(\hat{\varphi }, \hat{\mu }) = \sup _{\varphi \in \mathcal {A}} \inf _{\mu \in \mathbb M} I(\varphi , \mu ), \end{aligned}$$
(3.6)

where

$$\begin{aligned} I(\varphi , \mu ) = E \left[ U(X_{\varphi , \mu }^x(T)) + \int _0^T \rho (\mu (t))dt \right] . \end{aligned}$$
(3.7)

The problem (3.6) is a stochastic differential game. To handle this, we use an extension of the maximum principle to games, as presented in, e.g., [15]. We obtain the following characterization of a solution (saddle point) of (3.6):

Proposition 3.2

(Robust primal problem and associated constrained FBSDE)A pair \((\hat{\varphi }, \hat{\mu }) \in \mathcal {A}\times \mathbb M\) is a solution of the robust primal problem (3.6) if and only if the solution \((X_{\hat{\varphi },\hat{\mu }}^x\), \((p_1,q_1, r_1))\) of the FBSDE consisting of the SDE (3.4) and the BSDE

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle dp_1(t) = q_1(t) dB(t) + \displaystyle \int _\mathbb {R}r_1(t,\zeta ) \tilde{N}(dt, d\zeta ) \; ; \; 0 \le t \le T \\ p_1(T) = U{^\prime }(X_{\hat{\varphi },\hat{\mu }}^x(T)) \end{array}\right. \end{aligned}$$
(3.8)

also satisfies

$$\begin{aligned} (b(t) + \hat{\mu }(t)) p_1(t) + \sigma (t) q_1(t) + \int _\mathbb {R}\gamma (t,\zeta ) r_1(t,\zeta ) \nu (d\zeta ) = 0 \; ; \; t \in [0,T] \end{aligned}$$
(3.9)
$$\begin{aligned} \rho {^\prime }(\hat{\mu }(t)) + \hat{\varphi }(t) S_{\hat{\mu }}(t^-) p_1(t) = 0 \; ; \; t \in [0,T], \end{aligned}$$
(3.10)

where \((p_1, q_1, r_1) \in S^2 \times H^2 \times H_{\nu }^2.\)

Proof

Define the Hamiltonian by

$$\begin{aligned} H_1(t,x,\varphi ,\mu ,p,q,r)= & {} \rho (\mu ) + \varphi S_\mu (t^-)\nonumber \\&\times \left[ (b(t) + \mu )p + \sigma (t)q + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) r(\zeta )\nu (d\zeta )\right] .\nonumber \\ \end{aligned}$$
(3.11)

The associated BSDE for the adjoint processes \((p_1,q_1,r_1)\) is (3.8).

The first order conditions for a maximum point \(\hat{\varphi }\) and a minimum point \(\hat{\mu }\), respectively, for the Hamiltonian are given by (3.9) and (3.10). Since \(H_1\) is concave with respect to \(\varphi \) and convex with respect to \(\mu \), these first order conditions are also sufficient for \(\hat{\varphi }\) and \(\hat{\mu }\) to be a maximum point and a minimum point, respectively. \(\square \)

We now study a dual formulation of the robust primal problem (3.6). Let now \(\mathcal M\) be the family of positive measures \(Q = Q_{\theta , \mu }\) of the form

$$\begin{aligned} dQ_{\theta , \mu }(\omega ) = G_{\theta ,\mu }(t) dP(\omega ) \text { on } \mathcal {F}_t, \end{aligned}$$
(3.12)

where \(G(t) = G_{\theta ,\mu }^y(t)\) is given by

(3.13)

and \((\theta , \mu )\) is such that \(\mu \in \mathbb M\) and \(\theta = (\theta _0, \theta _1)\) is a predictable processes satisfying (2.9) and

$$\begin{aligned} b(t) + \mu (t) + \sigma (t)\theta _0(t) + \displaystyle \int _\mathbb {R}\gamma (t,\zeta ) \theta _1(t,\zeta ) \nu (d\zeta ) = 0 \; ; \; t \in [0,T]. \end{aligned}$$
(3.14)

We let \( \Lambda \) denote the set of such processes \((\theta , \mu )\). If \(y=1\), then the measure \(Q_{\theta , \mu }\) is an ELMM for the perturbed price process \(S_\mu \) in (3.3).

Definition 3.3

The robust dual problem is for given \(y >0\), to find \(( \tilde{\theta },\tilde{\mu }) \in \Lambda \) such that

$$\begin{aligned} \sup _{(\theta , \mu ) \in \Lambda } J(\theta ,\mu ) = J(\tilde{\theta }, \tilde{\mu }) \end{aligned}$$
(3.15)

where

$$\begin{aligned} J(\theta ,\mu ) = E \left[ - V(G_{\theta , \mu }^y(T)) - \int _0^T \rho (\mu (t))dt \right] , \end{aligned}$$
(3.16)

and V is the convex conjugate function of U, as in Sect. 1.

Proposition 3.4

(Robust dual problem and its associated constrained FBSDE). A pair \((\tilde{\theta }, \tilde{\mu }) \in \Lambda \) is a solution of the robust dual problem (3.15)–(3.16) if and only if the solution \((G_{\tilde{\theta },\tilde{\mu }}^y\), \((p_2, q_2, r_2))\) of the FBSDE consisting of the FSDE (3.13) and the BSDE

(3.17)

with \(K(q_2,r_2)(t)\) defined as in (2.26), also satisfies the two equations

$$\begin{aligned}&G_{\tilde{\theta },\tilde{\mu }}^y (t) q_2(t) + \rho {^\prime }(\tilde{\mu }(t)) \sigma (t) = 0,\end{aligned}$$
(3.18)
$$\begin{aligned}&G_{\tilde{\theta },\tilde{\mu }}^y (t) r_2(t, \zeta ) + \rho {^\prime }(\tilde{\mu }(t)) \gamma (t, \zeta ) = 0, \end{aligned}$$
(3.19)

where \((p_2, q_2, r_2) \in S^2 \times H^2 \times H_{\nu }^2.\)

Proof

We proceed as in the proof of Proposition 2.6: The Hamiltonian for the constrained stochastic control problem (3.15) is

$$\begin{aligned} H_2^L(t,g,\theta _0, \theta _1,\mu ,p,q,r)&:= -\rho (\mu )+g \theta _0 q + g \int _\mathbb {R}\theta _1(\zeta ) r(\zeta ) \nu (d\zeta )\nonumber \\&\quad + L(t) \left( b(t) +\mu (t) + \sigma (t) \theta _0 + \int _\mathbb {R}\gamma (t,\zeta ) \theta _1(\zeta ) \nu (d\zeta )\right) , \end{aligned}$$
(3.20)

where L(t) is the Lagrange multiplier process.

The first order conditions for a maximum point \((\tilde{\theta }, \tilde{\mu })\) for \(H_2^L\) are \(\nabla _{\theta } H_2^L = 0\) and \( \left( \frac{\partial H_2^L}{\partial \mu } \right) = 0\) which reduce to (3.18)–(3.19). Then, we see that the corresponding BSDE for the adjoint processes \((p_2,q_2,r_2)\) is given by (3.17).

Since \(H_2\) is concave w.r.t. \(\mu \) and \(\theta \), these necessary optimality conditions are also sufficient. \(\square \)

3.3 Relations Between Robust Primal and Robust Dual Problems

We now use the characterizations above of the solutions \((\hat{\varphi }, \hat{\mu })\in \mathcal {A}\times \mathbb M\) and \((\tilde{\theta }, \tilde{\mu }) \in \Lambda \) of the robust primal and the robust dual problem, respectively, to find the relations between them.

Theorem 3.5

(i) From robust primal to robust dual.

Assume \((\hat{\varphi }, \hat{\mu }) \in \mathcal {A}\times \mathbb M\) is a solution of the robust primal problem and let \((p_1, q_1, r_1)\) be the associated adjoint processes solution of the FBSDE (3.4) & (3.8) and satisfying (3.9)–(3.10). Define

$$\begin{aligned}&\tilde{\mu }:= \hat{\mu } \end{aligned}$$
(3.21)
$$\begin{aligned}&\tilde{\theta }_0(t) := \frac{q_1(t)}{p_1(t)} \text { ; } \; \tilde{\theta }_1(t,\zeta ) = \frac{r_1(t,\zeta )}{p_1(t)} \end{aligned}$$
(3.22)

and suppose they satisfy (2.9). Then, they are optimal for the dual problem with initial value \(y = p_1(0).\) Moreover

$$\begin{aligned} p_1(t) = G_{\tilde{\theta }, \tilde{\mu }} (t) \; ; \; t \in [0,T]. \end{aligned}$$
(3.23)

In particular,

$$\begin{aligned} U{^\prime }(X_{\hat{\varphi }, \hat{\mu }}(T)) = G_{\tilde{\theta }, \tilde{\mu }} (T). \end{aligned}$$
(3.24)

(ii) From robust dual to robust primal Let \((\tilde{\theta }, \tilde{\mu }) \in \Lambda \) be optimal for the robust dual problem (3.15)–(3.16) and let \((p_2, q_2, r_2)\) be the associated adjoint processes satisfying (3.17) with the constraints (3.19) and (3.18). Define

$$\begin{aligned}&\hat{\mu }:= \tilde{\mu } \end{aligned}$$
(3.25)
$$\begin{aligned}&\hat{\varphi }(t ):= \frac{K(q_2,r_2)(t)}{ S_{\hat{\mu }}(t^-)} \; ; \; t \in [0,T]. \end{aligned}$$
(3.26)

Assume that \(\hat{\varphi }\in \mathcal {A}\). Then \((\hat{\mu }, \hat{\varphi })\) are optimal for primal problem with initial value \(x = p_2(0).\) Moreover,

$$\begin{aligned} p_2(t) = X_{\hat{\varphi },\hat{\mu }}(t) \quad t \in [0,T]. \end{aligned}$$
(3.27)

In particular

$$\begin{aligned} - V{^\prime }( G_{\tilde{\theta }}(T)) = X_{\hat{\varphi },\hat{\mu }}(T). \end{aligned}$$
(3.28)

Proof

(i) Let \((\hat{\varphi }, \hat{\mu }) \in \mathcal {A}\times \mathbb M\) is a solution of the robust primal problem and let \((p_1, q_1, r_1)\) be as in Proposition 3.2, i.e. assume that \((p_1, q_1, r_1)\) solves the FBSDE (3.4) and (3.8) and satisfies (3.9)–(3.10).

We want to find the solution \((\tilde{\theta }, \tilde{\mu }) \in \Lambda \) of the robust dual problem. By Proposition 3.4 this means that we must find a solution \((p_2, q_2, r_2)\) of the FBSDE (3.13) and (3.17) which satisfies (3.19)–(3.18). To this end, choose \(\tilde{\mu }, \tilde{\theta }_0, \tilde{\theta }_1\) given in (3.21)–(3.22). Then by (3.9) we have

$$\begin{aligned} b(t) + \tilde{\mu }(t)+ \sigma (t) \tilde{\theta }_0(t) + \int _\mathbb {R}\gamma (t,\zeta ) \tilde{\theta }_1(t,\zeta ) \nu (d\zeta ) = 0. \end{aligned}$$
(3.29)

Assume that (2.9) holds. Then \((\tilde{\mu },\tilde{\theta }) \in \Lambda \). Substituting (3.22) into (3.8), we obtain

(3.30)

Comparing with (3.13) we see that

$$\begin{aligned} \frac{dG_{\tilde{\theta }, \tilde{\mu }} (t)}{G_{\tilde{\theta }, \tilde{\mu }} (t)} = \frac{dp_1(t)}{p_1(t)} \end{aligned}$$

and hence, for \(y = G_{\tilde{\theta }, \tilde{\mu }} (0) =p_1(0)> 0\) we get (3.23) and (3.24). Define

$$\begin{aligned} p_2(t) := X_{\hat{\varphi }, \hat{\mu }}(t), q_2(t) := \hat{\varphi }(t) \sigma (t) S_{\hat{\mu }}(t^-), r_2(t, \zeta ) := \hat{\varphi }(t) \gamma (t,\zeta )S_{\hat{\mu }}(t^-). \end{aligned}$$
(3.31)

Then by (3.4) and (3.24), combined with (1.7),

(3.32)

Hence \((p_2, q_2, r_2)\) solves the BSDE (3.17), as requested. It remains to verify that (3.19) and (3.18) hold: By (3.31) we have

$$\begin{aligned} - q_2(t) \gamma (t, \zeta ) + \sigma (t) r_2(t,\zeta ) = \sigma (t)[- \hat{\varphi }(t) S_{\hat{\mu }}(t^-)\gamma (t, \zeta ) + \hat{\varphi }(t) S_{\hat{\mu }}(t^-)\gamma (t,\zeta )] = 0, \end{aligned}$$

which is (3.19). By (3.21), (3.23), (3.31) and (3.10),

$$\begin{aligned} \rho {^\prime }(\tilde{\mu }) + G_{\tilde{\theta },\tilde{\mu }}(t) q_2(t) = \rho {^\prime }(\hat{\mu }) + p_1(t) \hat{\varphi }(t) \sigma (t) S_{\hat{\mu }}(t^-)= 0, \end{aligned}$$

which is (3.18).

(ii) Next, assume that \((\tilde{\theta }, \tilde{\mu }) \in \Lambda \) is optimal for the robust dual problem (3.15)–(3.16) and let \((p_2, q_2, r_2)\) be as in Proposition 3.4. We will find \((\hat{\varphi }, \hat{\mu }) \in \mathcal {A}\times \mathbb M\) and \((p_1, q_1, r_1)\) satisfying Proposition 3.2. Choose \(\hat{\mu }\) and \(\hat{\varphi }\) given in (3.25)–(3.26) and assume that \(\hat{\varphi }\) is admissible. Then by (3.17) and (3.19)

Hence, with \(x = p_2(0) >0\), (3.27) holds.

In particular

$$\begin{aligned} X_{\hat{\varphi }, \hat{\mu }}(T) = p_2(T) = - V{^\prime }(G_{\tilde{\theta },\tilde{\mu }} (T)), \text { i.e. } G_{\tilde{\theta },\tilde{\mu }} (T) = U{^\prime } (X_{\hat{\varphi }, \hat{\mu }}(T)). \end{aligned}$$
(3.33)

We now verify that with \(\varphi = \hat{\varphi }, \mu = \hat{\mu }\), and \(p_1, q_1, r_1\) defined by

$$\begin{aligned} p_1(t) := G_{\tilde{\theta },\tilde{\mu }} (t), \; q_1(t) := G_{\tilde{\theta },\tilde{\mu }} (t) \tilde{\theta }_0(t), \; r_1(t,\zeta ) := G_{\tilde{\theta },\tilde{\mu }} (t) \tilde{\theta }_1(t,\zeta ), \end{aligned}$$
(3.34)

all the conditions of Proposition 3.2 hold: By (3.13) and (3.33),

$$\begin{aligned} \left\{ \begin{array}{ll} dp_1(t) = dG_{\tilde{\theta }, \tilde{\mu }} (t) = G_{\tilde{\theta }, \tilde{\mu }} (t^-) \big ( \tilde{\theta }_0(t) dB(t) + \displaystyle \int _\mathbb {R}\tilde{\theta }_1(t,\zeta ) \tilde{N}(dt,d\zeta ) \big )\\ \qquad = \displaystyle q_1(t)dB(t) + \displaystyle \int _\mathbb {R}r_1(t,\zeta ) \tilde{N}(dt,d\zeta ) \quad \; ; \; 0 \le t \le T \\ p_1(T) = G_{\tilde{\theta },\tilde{\mu }} (T) = U{^\prime }(X_{\hat{\varphi }, \hat{\mu }}(T)). \end{array}\right. \end{aligned}$$
(3.35)

Hence (3.8) holds. It remains to verify (3.9) and (3.10). By (3.34) and (3.14) for \(\theta = \tilde{\theta }\), we get

$$\begin{aligned} (b(t) + \hat{\mu }(t)&)p_1(t) + \sigma (t) q_1(t) + \int _\mathbb {R}\gamma (t,\zeta ) r_1 (t,\zeta ) \nu (d\zeta ) \\&= G_{\tilde{\theta },\tilde{\mu }} (t) \left[ b(t) + \hat{\mu }(t) + \sigma (t) \tilde{\theta }_0(t) + \int _\mathbb {R}\gamma (t,\zeta ) \tilde{\theta }_1(t,\zeta ) \nu (d\zeta ) \right] = 0, \end{aligned}$$

which is (3.9). By (3.25), (3.26), (3.34) and (3.18) we get

$$\begin{aligned} \rho {^\prime }(\hat{\mu }(t)) + \hat{\varphi }(t) S_{\hat{\mu }}(t^-) \sigma (t) p_1(t) = \rho {^\prime }(\tilde{\mu }(t)) + q_2(t) G_{\tilde{\theta },\tilde{\mu }} (t) = 0, \end{aligned}$$

which is (3.10). \(\square \)

3.4 Illustrating Examples

Example 3.6

We consider a robust version of the classical Merton type optimal portfolio problem:

We assume that there exists a constant \(C > 0\) such that

$$\begin{aligned} \frac{|b(t)|}{|\sigma (t)|} \le C \; ; \; 0 \le t \le T. \end{aligned}$$
(3.36)

We want to study

$$\begin{aligned} \inf _{\mu \in \mathbb M} \sup _{\varphi \in \mathcal {A}} E \left[ U(X_{\varphi , \mu }(T)) + \int _0^T \rho (\mu (t))dt \right] \end{aligned}$$
(3.37)

in the case with no jumps (\(N = \gamma = 0, \sigma \ne 0\)). Then there is only one ELMM for the price process \(S_\mu (t)\) for each given \(\mu (t)\). So \(\theta = \theta _0 = - \frac{b(t)+\mu (t)}{\sigma (t)}\) and the corresponding robust dual problem simplifies to

$$\begin{aligned} \sup _{\mu \in \mathbb M} E \left[ - V(G_\mu (T)) - \int _0^T \rho (\mu (t))dt \right] , \end{aligned}$$
(3.38)

where

$$\begin{aligned} \displaystyle dG_{\mu }(t) = - G_\mu (t^-) \frac{b(t)+\mu (t)}{\sigma (t)} dB_t \; ; \; 0 \le t \le T; \; \; G_\mu (0) = y > 0. \end{aligned}$$
(3.39)

The first order conditions for the Hamiltonian reduce to:

$$\begin{aligned} \tilde{\mu }(t) =({ \rho {^\prime }})^{-1} \left( - \frac{G_{\tilde{\mu }} (t) q_2(t)}{\sigma (t)}\right) \end{aligned}$$
(3.40)

which substituted into the adjoint BSDE equation gives:

$$\begin{aligned} \left\{ \begin{array}{ll} dp_2(t )= \frac{q_2(t)}{\sigma (t)} \left[ b(t) + ({ \rho {^\prime }})^{-1} \left( - \frac{G_{\tilde{\mu }} (t) q_2(t)}{\sigma (t)}\right) \right] dt + q_2(t )dB_t; \; ; \; t \in [0,T] \\ p_2(T) = - V{^\prime }(G_{\tilde{\mu }}(T)). \end{array}\right. \end{aligned}$$
(3.41)

We get that \(\tilde{\mu }\) is optimal for the robust dual problem if and only if there is a solution \((p_2,q_2,G_{\tilde{\mu }})\) of the FBSDE consisting of (3.41) and (3.39) with the constraint (3.40). Hence, by Theorem 3.5(ii), the optimal \(\hat{\mu }\) for the primal robust problem is given by \(\hat{\mu }:=\tilde{\mu }\), and the optimal portfolio is

$$\begin{aligned} \hat{\varphi }(t) = \frac{K(q_2,r_2)(t)}{ S_{\hat{\mu }}(t^-)} = \frac{q_2(t)}{\sigma (t) S_{\tilde{\mu }}(t^-)} \; ; \; t \in [0,T]. \end{aligned}$$
(3.42)

Now assume that

$$\begin{aligned} U(x) = \ln x \;\; \text { and } \; \rho (x) = \frac{1}{2} x^2. \end{aligned}$$
(3.43)

Then \(V(y) = - \ln y - 1.\)

If b(t) and \(\sigma (t)\) are deterministic, we can solve (3.38) by dynamic programming, and we get

$$\begin{aligned} \tilde{\mu }(t) = - \frac{b(t)}{2} ; \; t \in [0, T]. \end{aligned}$$
(3.44)

In view of this, it is natural to guess that (3.44) is the optimal choice of \(\mu \) also when b(t) and \(\sigma (t)\) are \(\mathcal{F}_t\)-adapted processes. To verify this we have to show that the system (3.39)–(3.41) is consistent. This system is now the following

$$\begin{aligned} G_{\tilde{\mu }}(t)&= y \exp \left( - \int _0^t \frac{b(s)}{2\sigma (s)} dB(s) - \frac{1}{2} \left( \frac{b(s)}{2\sigma (s)}\right) ^2 ds \right) \end{aligned}$$
(3.45)
$$\begin{aligned} q_2(t)&= \frac{1}{G_{\tilde{\mu }}(t)} . \frac{b(t)}{2\sigma (t)} \end{aligned}$$
(3.46)
$$\begin{aligned} dp_2(t)&= \frac{1}{G_{\tilde{\mu }}(t)} \left[ \frac{b(t)}{2\sigma (t)} dB(t) + \left( \frac{b(t)}{2\sigma (t)}\right) ^2 dt \right] \; ; \; p_2(T) = \frac{1}{G_{\tilde{\mu }}(T)} \end{aligned}$$
(3.47)

which gives

$$\begin{aligned} \frac{1}{G_{\tilde{\mu }}(t)} = \frac{1}{y} \exp \left( \int _0^t \frac{b(s)}{2\sigma (s)} dB(s) + \frac{1}{2} \left( \frac{b(s)}{2\sigma (s)}\right) ^2 ds\right) \end{aligned}$$
(3.48)

i.e.

$$\begin{aligned} d \left( \frac{1}{G_{\tilde{\mu }}(t)}\right) = \frac{1}{G_{\tilde{\mu }}(t)} \left[ \frac{b(t)}{2\sigma (t)} dB(t) + \left( \frac{b(t)}{2\sigma (t)}\right) ^2 dt \right] . \end{aligned}$$
(3.49)

We see that (3.47) is in agreement with (3.49) with \(p_2(t) = \frac{1}{G_{\tilde{\mu }}(t)}\), and this proves that \(\tilde{\mu }(t)\) given by (3.44) is indeed optimal also when b and \(\sigma \) are stochastic. The corresponding optimal portfolio for the robust utility maximization problem with initial value \(x = \frac{1}{y} \), is, by (3.26),

$$\begin{aligned} \hat{\varphi }(t) = \frac{b(t)}{ G_{\tilde{\mu }}(t) 2\sigma ^2(t) S_{\tilde{\mu }}(t)} ; \;\; t \in [0, T]. \end{aligned}$$
(3.50)

which means that the optimal fraction of wealth to be placed in the risky asset is

$$\begin{aligned} \hat{\pi }(t )= \frac{\hat{\varphi }(t)S_{\tilde{\mu }}(t^-)}{\hat{X}(t)} = \frac{b(t)}{ 2 \sigma ^2(t)} \end{aligned}$$
(3.51)

We have thus proved:

Proposition 3.7

Suppose (3.43) holds. Then the optimal scenario \(\hat{\mu }=\tilde{\mu }\) and optimal portfolio \(\hat{\varphi }\) for the robust primal problem (3.37) are given by (3.44) and (3.50), respectively, with \(G_{\tilde{\mu }}(t)\) as in (3.45).

Remark 3.8

Comparing (3.51) with the solution of the Merton problem in the classical, non-robust case, we see that the optimal fraction to be placed in the risky asset in the robust case is just half of the optimal fraction in the non-robust case.

Example 3.9

We consider a robust version of Example 2.10. In this case the perturbed price \(S=S_\mu \) is

$$\begin{aligned} dS(t) = S(t^-)[(b(t) + \mu (t)) dt + \gamma (t,1) d \tilde{N}(t)] \; ; \; S(0) > 0 \end{aligned}$$
(3.52)

and the wealth process \(X = X_{\varphi ,\mu }^x\) associated to a portfolio \(\varphi \) is

$$\begin{aligned} dX(t) = \varphi (t) S(t^-) [(b(t) + \mu (t)) dt + \gamma (t,1) d \tilde{N}(t)] \; ; \; X(0) = x > 0. \end{aligned}$$
(3.53)

We again choose the logarithmic utility function \(U(x) = \ln (x)\) and the quadratic penalty function \(\rho (x) = \frac{1}{2} x^2\).

Thus the robust primal problem is to find \((\hat{\varphi }, \hat{\mu }) \in \mathcal {A}\times \mathbb M \) such that

$$\begin{aligned} \inf _{\mu \in \mathbb M} \sup _{\varphi \in \mathcal {A}} E \left[ \ln X_{\varphi , \mu }^x(T) + \frac{1}{2} \int _0^T \mu ^2(t)dt \right] = E \left[ \ln X_{\hat{\varphi }, \hat{\mu }}^x(T) + \frac{1}{2} \int _0^T \hat{\mu }^2(t) dt \right] . \end{aligned}$$
(3.54)

The corresponding dual problem is to find \(( \tilde{\theta },\tilde{\mu }) \in \Lambda \) such that

$$\begin{aligned} \sup _{(\theta , \mu ) \in \Lambda } E \left[ \ln G_{\theta , \mu }^y(T) - \frac{1}{2} \int _0^T \mu ^2(t)dt \right] = E \left[ \ln G_{\tilde{\theta }, \tilde{\mu }}^y(T) - \frac{1}{2} \int _0^T \tilde{\mu }^2(t)dt \right] . \end{aligned}$$
(3.55)

First note from (3.14) that for each \(\mu \) there is only one admissible element process \(\theta \) given by

$$\begin{aligned} \theta _1(t,1) = \tilde{\theta }_1(t,1) = - \frac{b(t) + \mu (t)}{\lambda \gamma (t,1)}. \end{aligned}$$
(3.56)

Assume that (see (2.9))

$$\begin{aligned} \frac{b(t) + \mu (t)}{\lambda \gamma (t,1)} < 1 ; \; t \in [0, T] . \end{aligned}$$
(3.57)

Then we get

$$\begin{aligned} G^y_{\tilde{\theta }_1, \tilde{\mu }}(t)&= y \exp \Big ( \int _0^t \ln (1 - \frac{b(s) + \tilde{\mu }(s)}{\lambda \gamma (s,1)}) d \tilde{N}(s) \nonumber \\&\quad + \lambda \int _0^t \left\{ \ln ( 1 - \frac{b(s)+ \tilde{\mu }(s)}{\lambda \gamma (s,1)}) + \frac{b(s)+ \tilde{\mu }(s)}{\lambda \gamma (s,1)}\right\} ds \Big ) \; ; \; 0 \le t \le T. \end{aligned}$$
(3.58)

In this case, \(K(q_2, r_2)(t) = \frac{r_2(t,1)}{\gamma (t,1)}\) and the BSDE (3.17) becomes:

(3.59)

To solve this equation, we proceed as in Example 2.10. We then get:

$$\begin{aligned} \hat{p}_2 = \frac{1}{G_{\tilde{\theta },\tilde{\mu }}^y}\; ; \; \hat{r}_2 = \hat{p}_2 \psi \end{aligned}$$

with \(\psi \) given by

$$\begin{aligned} \psi (t) = \frac{b(t) + \tilde{\mu }(t)}{\lambda \gamma (t,1) - ( b(t) + \tilde{\mu }(t))}, \; t \in [0,T]. \end{aligned}$$
(3.60)

From (3.19), we get the equation:

$$\begin{aligned} \tilde{\mu }(t) = - \frac{G_{\tilde{\theta },\tilde{\mu }}^y(t) \hat{p}_2 (t) \psi (t)}{\gamma (t,1)} = - \frac{b(t) + \tilde{\mu }(t) }{\gamma (t,1) (\lambda \gamma (t,1) - (b(t) + \tilde{\mu }(t)))}, \end{aligned}$$
(3.61)

i.e.

$$\begin{aligned} \gamma (t,1) \tilde{\mu }^2(t) + (\gamma (t,1) b(t) - \lambda \gamma ^2(t,1) - 1) \tilde{\mu }(t) - b(t) = 0. \end{aligned}$$
(3.62)

The root of this quadratic equation which satisfies (3.57) is

$$\begin{aligned} \tilde{\mu }(t) = \frac{1}{2 \gamma (t,1)} ( - \gamma (t,1) b(t) + \lambda \gamma ^2(t,1) + 1 - \sqrt{\Delta }). \end{aligned}$$
(3.63)

with

$$\begin{aligned} \Delta&= (\gamma (t,1) b(t) - \lambda \gamma ^2(t,1) - 1)^2 + 4 \gamma (t,1) b(t) \\&= (\gamma (t,1) b(t) - \lambda \gamma ^2(t,1) + 1)^2 + 4 \lambda \gamma ^2(t,1). \end{aligned}$$

From Theorem 3.5 we conclude that the solution of the robust primal problem (3.54) is \(\hat{\mu }(t) = \tilde{\mu }(t)\) given by (3.63), and

$$\begin{aligned} \hat{\varphi }(t) = \frac{\hat{r}_2 (t)}{\gamma (t,1) S(t^-)} = \frac{\hat{p}_2(t) \psi (t) }{\gamma (t,1) S(t^-)} = \frac{X_{\hat{\varphi }, \hat{\mu }}^x(t) \psi (t) }{\gamma (t,1) S(t^-)} \end{aligned}$$

with \(x = \frac{1}{y}\). The optimal fraction of the wealth invested in the risky asset is

$$\begin{aligned} \hat{\pi }(t) = \frac{\hat{\varphi }(t) S(t^-)}{X_{\hat{\varphi }, \hat{\mu }}^x(t) } = \frac{\psi (t)}{\gamma (t,1)} = \frac{ b(t) + \hat{\mu }(t)}{\gamma (t,1)(\lambda \gamma (t,1) - (b(t) + \hat{\mu }(t)))} = - \hat{\mu }(t). \end{aligned}$$

We summarize this as follows:

Proposition 3.10

The optimal pair \((\hat{\mu }, \hat{\varphi }) \in \mathcal {A}\times \mathbb M\) for Problem 3.6 is given by \( \displaystyle \hat{\varphi }(t) = \frac{\hat{\pi }(t) X_{\hat{\varphi }, \tilde{\mu }}^x(t) }{S(t^-)}\) with \(\hat{\pi }(t) = - \tilde{\mu }(t)\) and \(\tilde{\mu }(t)\) given by (3.63).