2.2. Probability space

Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space. For a stochastic process $\{X_n\}_{n \in \mathbb{N}}$ , we put $\Delta X_n = X_n - X_{n-1}$ . Let $\{X_n\}$ be a d-dimensional stochastic process with $\Delta X_n$ taking values in $\{v_0,v_1,\ldots,v_d\}$ for all $n \geq 1$ and $X_0 = 0$ . By (2.1), $X_n$ takes values in L for all $n \in \mathbb{N}$ . Let $\mathscr{F}_n = \sigma(X_0,\ldots,X_n)$ , $ n\in \mathbb{N}$ , be the natural filtration generated by $\{X_n\}$ , and put $P_n = (P_{n,0},\ldots, P_{n,d})^\top$ for $n\geq 1$ , where

\begin{equation*} P_{n,j} = \mathbb{P}(\Delta X_n = v_j \mid \mathscr{F}_{n-1}), \quad j=0,\ldots, d.\end{equation*}

Note that $\{P_n\}$ is a $\Delta_d$ -valued predictable process, where

\begin{equation*} \Delta_d = \bigl\{p \in \mathbb{R}^{d+1}\ ; \ \mathbf{1}^\top p = 1, \ p \geq 0\bigr\}.\end{equation*}

We assume $P_{n,j}$ is positive for all $n \geq 1$ and $j = 0,\ldots, d$ . Let $N \in \mathbb{N}\setminus\{0\}$ . The following lemma will be of repeated use in this paper.

Lemma 2.3. For any $\Delta_d$ -valued predictable process $\{\hat{P}_n\}$ , there exists a probability measure $\hat{\mathbb{P}}$ on $(\Omega,\mathscr{F}_N)$ such that $\hat{P}_n = (\hat{P}_{n,0},\ldots, \hat{P}_{n,d})^\top$ , where

(2.4) \begin{equation} \hat{P}_{n,j} = \hat{\mathbb{P}}(\Delta X_n = v_j \mid \mathscr{F}_{n-1}), \quad j=0,\ldots, d, \ n=1,\ldots, N.\end{equation}

Proof. Define $\hat{\mathbb{P}}$ by

(2.5) \begin{equation} \hat{\mathbb{P}}(A) = \mathbb{E}[L_N1_A],\quad L_n = \prod_{k=1}^n \Biggl(\sum_{j=0}^d\frac{\hat{P}_{k,j}}{P_{k,j}}1_{\{\Delta X_k = v_j \}} \Biggr) \end{equation}

for $A \in \mathscr{F}_N$ . The measure $\hat{\mathbb{P}}$ is a probability measure because

\begin{equation*} \mathbb{E}\Biggl[ \sum_{j=0}^d\frac{\hat{P}_{n,j}}{P_{n,j}}1_{\{\Delta X_n = v_j \}} \bigg| \mathscr{F}_{n-1}\Biggr] = \sum_{j=0}^d\frac{\hat{P}_{n,j}}{P_{n,j}} P_{n,j} = 1\end{equation*}

and so $L_n$ is a martingale with $L_0 = 1$ . Using Bayes’ formula, we derive

\begin{equation*} \hat{\mathbb{P}}(\Delta X_n =v_j \mid \mathscr{F}_{n-1}) = \frac{\mathbb{E}[L_N1_{\{\Delta X_n = v_j\}}\mid \mathscr{F}_{n-1}]}{\mathbb{E}[L_N \mid \mathscr{F}_{n-1}]} = \mathbb{E}\biggl[ \frac{\hat{P}_{n,j}}{P_{n,j}}1_{\{\Delta X_n = v_j \}} \bigg| \mathscr{F}_{n-1}\biggr] = \hat{P}_{n,j},\end{equation*}

by the martingale property of $L_n$ .

Define a measure $\mathbb{Q}$ on $\mathscr{F}_N$ by

\begin{equation*} \mathbb{Q}(A) = \mathbb{E}[L_N1_A],\quad L_N = \prod_{n=1}^N \Biggl( \frac{1}{d+1} \sum_{j=0}^d\frac{1}{P_{n,j}}1_{\{\Delta X_n = v_j \}} \Biggr). \end{equation*}

Let $\mathbb{E}_\mathbb{Q}$ denote the integration under $\mathbb{Q}$ .

Lemma 2.4. The measure $\mathbb{Q}$ is the unique probability measure on $\mathscr{F}_N$ under which $\{X_n\}$ is a martingale. Under $\mathbb{Q}$ , $\{\Delta X_n\}$ is i.i.d. with

\begin{equation*} \mathbb{Q}(\Delta X_n = v_j ) = \mathbb{Q}(\Delta X_n =v_j \mid \mathscr{F}_{n-1}) = \frac{1}{d+1}\end{equation*}

for all $n =1,\ldots, N$ and $j=0,\ldots, d$ . We also have

(2.6) \begin{equation}\mathbb{E}_\mathbb{Q}[\Delta X_n \mid\mathscr{F}_{n-1}] = 0,\quad \mathbb{E}_\mathbb{Q}[\Delta X_n (\Delta X_n)^\top \mid\mathscr{F}_{n-1}] = \frac{1}{d+1} \mathbf{v}\mathbf{v}^\top.\end{equation}

Proof. By Lemma 2.3, $\mathbb{Q}$ is a probability measure with $ \mathbb{Q}(\Delta X_n =v_j \mid \mathscr{F}_{n-1}) = 1/(d+1)$ , which implies

\begin{equation*} \mathbb{E}_\mathbb{Q}[\Delta X_n \mid\mathscr{F}_{n-1}] = \frac{1}{d+1} \mathbf{v}\mathbf{1} = 0.\end{equation*}

Therefore $\{X_n\}$ is a martingale with (2.6). There is no other such measure because

\begin{equation*} \sum_{j=0}^d \alpha_j = 1, \quad \sum_{j=0}^d \alpha_j v_j = 0\end{equation*}

implies $\alpha_j = 1/(d+1)$ as in the proof of Lemma 2.1. Since the conditional law of $\Delta X_n$ given $\mathscr{F}_{n-1}$ is deterministic for every n, $\{\Delta X_n\}$ is i.i.d.

2.3. Existence, uniqueness and representation

Here we introduce our BS $\Delta$ E. Let $\mathcal{A}$ denote the set of the sequences $g = \{g_n\}_{n=1}^N$ of $\mathscr{F}_{n-1}\otimes \mathscr{B}(\mathbb{R}^d)$ measurable functions $g_n\colon \Omega \times \mathbb{R}^d \to \mathbb{R}$ . Now we state an elementary but fundamental result.

Theorem 2.1. Let $Y_N$ be an $\mathscr{F}_N$ -measurable random variable, and let $g = \{g_n\} \in \mathcal{A}$ . Then there exist uniquely an adapted process $\{Y_n\}_{n=0,\ldots,N-1}$ and an $\mathbb{R}^d$ -valued predictable process $\{Z_n\}_{n=1,\ldots,N}$ such that

(2.7) \begin{equation} \Delta Y_n = -g_n(Z_n) + Z_n^\top \Delta X_n,\quad n=1,\ldots, N.\end{equation}

Further, they admit the following representation:

\begin{align*}Y_{n-1}& = \mathbb{E}_\mathbb{Q}[Y_n\mid\mathscr{F}_{n-1}] + g_n(Z_n),\\*Z_n& = (d+1) (\mathbf{v}\mathbf{v}^\top)^{-1} \mathbb{E}_\mathbb{Q}[Y_n \Delta X_n\mid \mathscr{F}_{n-1}]= (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbf{v}\bar{Y}_n,\end{align*}

where $\bar{Y}_n = (\bar{Y}_{n,0}, \ldots, \bar{Y}_{n,d})^\top$ and

\begin{equation*} \bar{Y}_{n,j} = \mathbb{E}_\mathbb{Q}[Y_n\mid\mathscr{F}_{n-1}, \Delta X_n = v_j] = (d+1) \mathbb{E}_\mathbb{Q}[Y_n1_{\{\Delta X_n = v_j\}}\mid \mathscr{F}_{n-1}].\end{equation*}

Proof. Since $Y_N$ is $\mathscr{F}_N$ -measurable, there exists a function $f\colon L^N \to \mathbb{R}$ such that $Y_N = f(X_1,\ldots,X_N)$ . Since $\mathbb{P}_{N,j}$ are positive by the assumption, (2.7) for $n=N$ is equivalent to the system of equations for $\mathscr{F}_{N-1}$ -measurable random variables

\begin{equation*}Y\,:\!=\,\begin{bmatrix}f(X_1,\ldots,X_{N-1},X_{N-1} + v_0) \\ \vdots\\f(X_1,\ldots,X_{N-1},X_{N-1} + v_d)\end{bmatrix}= (\mathbf{1},\mathbf{v}^\top)\begin{bmatrix} Y_{N-1} - g_N(Z_N) \\ Z_N\end{bmatrix}\!.\end{equation*}

Applying Lemma 2.2, we obtain (2.7) for $n=N$ with

\begin{equation*} Z_N = b(Y), \quad Y_{N-1} = a(Y)+g_N(Z_N),\end{equation*}

where a(y) and b(y) are defined by (2.3). It is clear that both $Z_N$ and $Y_{N-1}$ are $\mathscr{F}_{N-1}$ -measurable. By backward induction, we obtain $\{Y_n\}$ and $\{Z_n\}$ . The representation follows from (2.7) and (2.6) by taking the conditional expectation under $\mathbb{Q}$ .

For $g = \{g_n\} \in \mathcal{A}$ fixed, the $\mathscr{F}_n$ -measurable random variable $Y_n$ given by Theorem 2.1 is uniquely determined by the $\mathscr{F}_N$ -measurable random variable $Y_N$ . We write this mapping as $Y_n = \mathcal{E}^g_n(Y_N)$ and call it the g-expectation of $Y_N$ (with respect to $\mathscr{F}_n$ ). The stochastic process $\{(Y_n,Z_n)\}$ given by Theorem 2.1 is called the solution of the BS $\Delta$ E (2.7).

Remark 2.2. In the literature, say, in [Reference Cohen and Elliot15], BS $\Delta$ E is formulated by decomposing $\Delta Y_n$ into a predictable part and a martingale difference part. In our formulation (2.7), $\Delta X_n$ is not necessarily a martingale difference. It is a minor reparametrization because (2.7) can be rewritten as

\begin{equation*} \Delta Y_n = - \hat{g}_n(Z_n) + Z_n^\top(\Delta X_n - A_n)\end{equation*}

with $\hat{g}_n(z) = g_n(z) - z^\top A_n$ , $A_n = \mathbb{E}[\Delta X_n \mid \mathscr{F}_{n-1}]$ .

Example 2.1. Let $\gamma \gt 0$ , $\{(\hat{P}_{n,0},\ldots,\hat{P}_{n,d})^\top\}$ be a $\Delta_d$ -valued predictable process, and

(2.8) \begin{equation} g_n(z) = -\frac{1}{\gamma} \log \Biggl(\sum_{j=0}^{d} \,{\mathrm{e}}^{-\gamma z^\top v_j} {\hat{P}}_{n,j}\Biggr).\end{equation}

Then

(2.9) \begin{equation} \mathcal{E}^g_n(Y) = -\frac{1}{\gamma} \log \hat{\mathbb{E}}\big[{\mathrm{e}}^{-\gamma Y}\mid \mathscr{F}_n\big], \quad n=0,1,\ldots, N ,\end{equation}

for any $\mathscr{F}_N$ -measurable random variable Y, where $\hat{\mathbb{E}}$ is the expectation under the measure $\hat{\mathbb{P}}$ on $\mathscr{F}_N$ defined by (2.5). To see this, note that by Lemma 2.3,

\begin{equation*}g_n(z) = -\frac{1}{\gamma} \log \hat{\mathbb{E}}\bigl[{\mathrm{e}}^{-\gamma z^\top \Delta X_n}\mid \mathscr{F}_{n-1}\bigr]. \end{equation*}

Substituting $Y_n= Y_{n-1}- g_n (Z_n) + Z_n ^\top \Delta X_n $ , we have

\begin{equation*} -\frac{1}{\gamma} \log \hat{\mathbb{E}}\big[{\mathrm{e}}^{-\gamma Y_n}\mid \mathscr{F}_{n-1}\big]= Y_{n-1} - g_n(Z_n)-\frac{1}{\gamma} \log \hat{\mathbb{E}}\bigl[{\mathrm{e}}^{-\gamma Z_n^\top \Delta X_n }\mid \mathscr{F}_{n-1}\bigr] =Y_{n-1}\end{equation*}

which implies (2.9) for $n=N-1$ . The general case follows by backward induction.

Next we give a discrete analog of the nonlinear Feynman–Kac formula, which is computationally efficient when dealing with large N.

Proposition 2.1. Let $f_n \colon L \times \mathbb{R}^d \to \mathbb{R}$ , $n=1,\ldots, N$ and $h \colon L \to \mathbb{R}$ . Define $u_n \colon L \to \mathbb{R}$ , $n=0,1,\ldots, N$ backward inductively by

\begin{equation*} u_{n-1}(x) = u_n(x) + \mathcal{L} u_n(x) + f_n(x, (\mathbf{vv}^\top)^{-1}\mathbf{v}\mathcal{N} u_n(x))\end{equation*}

with $u_N = h$ , where

\begin{align*} \mathcal{L} u_n(x) & = \frac{1}{d+1} \sum_{j=0}^d (u_n(x+v_j)-u_n(x)), \\* \mathcal{N} u_n(x) & = (u_n(x+v_0)-u_n(x), \ldots, u_n(x+v_d)-u_n(x))^\top.\end{align*}

Then the unique solution to (2.7) with $g_n = f_n(X_{n-1},\cdot)$ and $Y_N = h(X_N)$ is given by

\begin{equation*} Y_{n-1} = u_{n-1}(X_{n-1}), \quad Z_{n} = (\mathbf{vv}^\top)^{-1}\mathbf{v}\mathcal{N} u_{n}(X_{n-1}), \quad n= 1,\ldots, N. \end{equation*}

Proof. By definition, $Y_N = h(X_N) = u_N(X_N)$ . Suppose $Y_n = u_n(X_n)$ . Then, by Theorem 2.1, $Z_n = (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbf{v}\bar{Y}_n$ , where

\begin{equation*}\bar{Y}_{n,j} = \mathbb{E}_\mathbb{Q}[Y_n\mid \mathscr{F}_{n-1}, \Delta X_n = v_j] = u_n(X_{n-1} + v_j).\end{equation*}

Using $\mathbf{v1} = 0$ , we conclude $Z_n = (\mathbf{vv}^\top)^{-1}\mathbf{v}\mathcal{N} u_n(X_{n-1})$ . Further, again by Theorem 2.1,

\begin{equation*} Y_{n-1} =\mathbb{E}_\mathbb{Q}[Y_n\mid \mathscr{F}_{n-1}] + g_n(Z_n)= u_n(X_{n-1}) + \mathcal{L} u_n(X_{n-1})+ g_n(Z_n) = u_{n-1}(X_{n-1}),\end{equation*}

which concludes the proof.

2.4. A gradient constraint

Let $\Theta$ be the closed convex hull spanned by $\{v_0,v_1,\ldots,v_d\}$ , or equivalently

\begin{equation*} \Theta = \{\mathbf{v}p \ ; \ p \in \Delta_d\}.\end{equation*}

In this section we study BS $\Delta$ Es with the gradient of g being constrained in $\Theta$ .

Example 2.2. The triangular lattice of $\mathbb{R}^2$ is generated by

\begin{equation*} v_1 =\frac{1}{\sqrt{6}} \begin{pmatrix} 0\\-2 \end{pmatrix}\!, \quad v_2 = \frac{1}{\sqrt{6}} \begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}\!.\end{equation*}

In this case, $\mathbf{v}\mathbf{v}^\top = I$ and $\Theta$ is an equilateral triangle.

Proposition 2.2. Let $g_n(z) = A_n^\top z + B_n$ for a $\Theta$ -valued predictable process $\{A_n\}$ and a predictable process $\{B_n\}$ , $n=1,\ldots,N$ . Then

\begin{equation*} \mathcal{E}^g_n (Y) = \hat{\mathbb{E}}\Biggl[Y +\sum_{i=n+1}^N B_i \bigg|\mathscr{F}_n\Biggr], \quad n=0,1,\ldots,N, \end{equation*}

for any $\mathcal{F}_N$ -measurable random variable Y, where $\hat{\mathbb{E}}$ is the expectation under the measure $\hat{\mathbb{P}}$ on $\mathscr{F}_N$ defined by (2.5) with $\hat{P}_n = (\hat{P}_{n,0},\ldots, \hat{P}_{n,d})^\top$ such that $A_n = \mathbf{v}\hat{P}_n$ .

Proof. By Lemma 2.3,

\begin{equation*} \hat{\mathbb{E}}[\Delta X_n \mid \mathcal{F}_{n-1}] = \frac{\mathbb{E}[L_N \Delta X_n \mid \mathcal{F}_{n-1}]}{\mathbb{E}[L_N \mid \mathcal{F}_{n-1}]}= \sum_{j=0}^d v_j \hat{P}_{n,j}= \mathbf{v}\hat{P}_n = A_n\end{equation*}

for all n. On the other hand, from (2.7), we have

\begin{equation*} Y= Y_N = Y_n + \sum_{i=n+1}^N \big({-}g_i(Z_i) + Z_i^\top \Delta X_i\big) = Y_n + \sum_{i=n+1}^N \big({-}B_i + Z_i^{\top}(\Delta X_i - A_i)\big).\end{equation*}

Taking the conditional expectation under $\hat{\mathbb{P}}$ , we get the conclusion.

Example 2.3. Let $N=1$ , $d=1$ , $\Omega = \{+,-\}$ , $v_0 = -1, v_1 = 1$ , and $\Delta X_1(\pm) = \pm 1$ . Then $L = \mathbb{Z}$ , $\mathbf{v} = ({-}1,1)$ , $\Theta = [-1,1]$ and $\mathbf{vv}^\top = 2$ . Following the proof of Theorem 2.1, the solution of the linear BS $\Delta$ E $\Delta Y_1 = -AZ_1 + Z_1\Delta X_1$ can be constructed as

\begin{equation*} Y_0 = \frac{Y_1(+) + Y_1({-})}{2} + A \frac{Y_1(+)-Y_1({-})}{2} = \frac{1+A}{2} Y_1(+) + \frac{1-A}{2} Y_1({-})\end{equation*}

for any $A \in \mathbb{R}$ . The expression given by Proposition 2.2 is $Y_0 = \hat{E}[Y_1]$ , which can be directly seen with $\hat{P}_1 = ((1-A)/2, (1+A)/2)^\top$ for $A \in \Theta = [-1,1]$ . For $A \notin \Theta$ , we observe that $Y_0$ is not increasing in either $Y_1(+)$ or $Y_1({-})$ . In particular, $Y_0$ cannot be represented as an expectation in this case.

The set $\Theta$ plays a key role also for a comparison theorem. Let $\mathcal{B}$ denote the set of the sequence $g = \{g_n\}_{n=1}^N \in \mathcal{A}$ with

(2.10) \begin{equation} g_n(z_2) - g_n(z_1) \geq \min_{\theta\in\Theta}\theta^\top(z_2-z_1) \end{equation}

for all $z_1, z_2 \in \mathbb{R}^d$ .

Proposition 2.3. (Comparison theorem.) For $i=1,2$ , let $Y^{(i)}$ be $\mathscr{F}_N$ -measurable random variables with $Y^{(1)} \geq Y^{(2)}$ , and $g^{(i)} = \{g^{(i)}_n\} \in \mathcal{A}$ with $g^{(1)}_n \geq g^{(2)}_n$ . Let $\mathcal{E}^{(i)}_n(Y^{(i)})$ denote $\mathcal{E}^g_n(Y^{(i)})$ for $g = g^{(i)}$ , $i=1,2$ respectively. Assume also $g^{(i)} \in \mathcal{B}$ for either $i=1$ or $i=2$ . Then

\begin{equation*} \mathcal{E}^{(1)}_n(Y^{(1)}) \geq \mathcal{E}^{(2)}_n(Y^{(2)}), \quad n=0,1,\ldots, N.\end{equation*}

Proof. Note first that

(2.11) \begin{equation} \min_{\omega \in \Omega} z^\top \Delta X_n(\omega) = \min_{p \in \Delta_d} z^\top \mathbf{v}p = \min_{\theta \in \Theta}z^\top \theta\end{equation}

for any $z \in \mathbb{R}^d$ and n. Therefore, under (2.10) for $g= g^{(i)}$ , $g^{(i)}$ is balanced in the terminology of [Reference Cohen and Elliot15], so the result follows from Theorem 3.2 of [Reference Cohen and Elliot15]. Here we repeat essentially the same proof for the readers’ convenience. Let $\bigl\{\bigl(Y^{(i)}_n,Z^{(i)}_n\bigr)\bigr\}$ be the solution of (2.7) with $g = g^{(i)}$ and $Y_N = Y^{(i)}$ . We have $Y^{(1)}_N \geq Y^{(2)}_N$ by assumption. Suppose $Y^{(1)}_k \geq Y^{(2)}_k$ for some k. Then

\begin{equation*}0 \leq Y^{(1)}_k - Y^{(2)}_k = Y^{(1)}_{k-1} - Y^{(2)}_{k-1}- g^{(1)}_k\bigl(Z^{(1)}_k\bigr) + g^{(2)}_k\bigl(Z^{(2)}_k\bigr) + \bigl(Z^{(1)}_k- Z^{(2)}_k\bigr)^\top \Delta X_k\end{equation*}

and so

\begin{equation*} \bigl(Z^{(1)}_k- Z^{(2)}_k\bigr)^\top \Delta X_k \geq - Y^{(1)}_{k-1} + Y^{(2)}_{k-1} + g^{(1)}_k\bigl(Z^{(1)}_k\bigr) - g^{(2)}_k\bigl(Z^{(2)}_k\bigr).\end{equation*}

Since the right-hand side is $\mathscr{F}_{k-1}$ -measurable, this implies further

\begin{equation*} \min_{\theta \in \Theta} \theta^\top \bigl(Z^{(1)}_k- Z^{(2)}_k\bigr) \geq - Y^{(1)}_{k-1} + Y^{(2)}_{k-1} + g^{(1)}_k\bigl(Z^{(1)}_k\bigr) - g^{(2)}_k\bigl(Z^{(2)}_k\bigr)\end{equation*}

by (2.11). Therefore

\begin{align*} Y^{(1)}_{k-1} - Y^{(2)}_{k-1} & \geq g^{(1)}_k\bigl(Z^{(1)}_k\bigr) - g^{(2)}_k\bigl(Z^{(2)}_k\bigr) - \min_{\theta \in \Theta} \theta^\top \bigl(Z^{(1)}_k- Z^{(2)}_k\bigr) \\* & = g^{(1)}_k\bigl(Z^{(2)}_k\bigr) - g^{(2)}_k\bigl(Z^{(2)}_k\bigr) +g^{(1)}_k\bigl(Z^{(1)}_k\bigr) - g^{(1)}_k\bigl(Z^{(2)}_k\bigr) - \min_{\theta \in \Theta} \theta^\top \bigl(Z^{(1)}_k- Z^{(2)}_k\bigr) \\* & \geq 0 \end{align*}

under (2.10) for $g_n = g^{(1)}_n$ , and also

\begin{align*} Y^{(1)}_{k-1} - Y^{(2)}_{k-1} & \geq g^{(1)}_k\bigl(Z^{(1)}_k\bigr) - g^{(2)}_k\bigl(Z^{(2)}_k\bigr) - \min_{\theta \in \Theta} \theta^\top \bigl(Z^{(1)}_k- Z^{(2)}_k\bigr) \\* & = g^{(1)}_k\bigl(Z^{(1)}_k\bigr) - g^{(2)}_k\bigl(Z^{(1)}_k\bigr) +g^{(2)}_k\bigl(Z^{(1)}_k\bigr) - g^{(2)}_k\bigl(Z^{(2)}_k\bigr) - \min_{\theta \in \Theta} \theta^\top \bigl(Z^{(1)}_k- Z^{(2)}_k\bigr) \\* & \geq 0 \end{align*}

under (2.10) for $g_n = g^{(2)}_n$ . The result then follows by induction.

Remark 2.3. A sufficient condition for $g_n$ to meet (2.10) is that $g_n(z)$ is continuously differentiable in z with $\nabla g_n(z)$ taking values in $\Theta$ . Indeed, by Taylor’s theorem,

\begin{equation*} g_n(z_1) - g_n(z_2) = A_n^\top (z_1-z_2),\quad A_n = \int_0^1 \nabla g_n(z_2 + t(z_1 -z_2))\,\mathrm{d}t,\end{equation*}

and then notice that $A_n$ is $\Theta$ -valued because $\Theta$ is a convex set.

Example 2.4. ( Locally entropic monetary utility.) Let $\{(\hat{P}_{n,0},\ldots,\hat{P}_{n,d})^\top\}$ be a $\Delta_d$ -valued predictable process, let $\{B_n\}$ and $\{\Gamma_n\}$ be positive predictable processes, and

(2.12) \begin{equation} g_n(z) = -\frac{1}{\Gamma_n} \log \Biggl(\sum_{j=0}^{d} \,{\mathrm{e}}^{-\Gamma_n z^\top v_j} \hat{P}_{n,j}\Biggr) - \frac{1}{\Gamma_n}\log B_n.\end{equation}

Using a similar calculation to Example 2.1, we deduce the relation

\[Y_{n-1}=-\frac{1}{\Gamma_n}\{\log \hat{\mathbb{E}}[ {\mathrm{e}}^{-\Gamma_n Y_n} \mid {\mathcal F}_{n-1}]+\log B_n\}, \quad n=N,\ldots, 1.\]

In particular, when $B_n=1$ , ${\mathcal E}_{n-1}^g$ is locally the minus of the entropic risk measure with risk-aversion parameter $\Gamma_n$ extending (2.9). In contrast to the dynamic entropic risk measure studied in [Reference Acciaio and Penner1], we have the time-consistency property $\mathcal{E}_m^g(\mathcal{E}_n^g(Y))=\mathcal{E}_m^g(Y)$ for any $m \leq n$ when $B_n = 1$ for all n even if the process $\{\Gamma_n\}$ is not constant. We allow $B_n \neq 1$ in order to include an example in Section 3. We call $\mathcal{E}^g_n$ a locally entropic monetary utility. A brief numerical study for this utility is provided in Appendix B. We have

\begin{equation*} \nabla g_n(z) = \frac{\hat{\mathbb{E}}[\Delta X_n \,{\mathrm{e}}^{-\Gamma_n z^\top \Delta X_n}\mid \mathscr{F}_{n-1}]}{\hat{\mathbb{E}}[{\mathrm{e}}^{-\Gamma_n z^\top \Delta X_n}\mid \mathscr{F}_{n-1}]} = \mathbf{v}\hat{P}_n(z), \end{equation*}

where $\hat{P}_n(z) = (\hat{P}_{n,0}(z),\ldots,\hat{P}_{n,d}(z))^\top$ and

\begin{equation*} \hat{P}_{n,j}(z)= \frac{{\mathrm{e}}^{-\Gamma_n z^\top v_j} \hat{P}_{n,j}}{\sum_{k=0}^d \,{\mathrm{e}}^{-\Gamma_n z^\top v_k} \hat{P}_{n,k}}. \end{equation*}

Since $\hat{P}_n(z)$ is continuous in z and $\Delta_d$ -valued for all n, by Remark 2.3, the assumptions of Proposition 2.3 on $g^{(i)}_n$ are satisfied.

Next, we seek a robust representation of $\mathcal{E}^g$ when g is concave. Let $\mathcal{C}$ denote the set of $g = \{g_n\} \in \mathcal{B}$ with $g_n(z)$ being concave in z for all n.

Lemma 2.5. Let $g = \{g_n\} \in \mathcal{A}$ . Then $g \in \mathcal{C}$ if and only if

(2.13) \begin{equation} g_n(z) = \min_{\theta \in \Theta} \{z^\top \theta + b_n(\theta)\}, \end{equation}

where

\begin{equation*} b_n(\theta) = \sup_{z \in \mathbb{R}^d} \{g_n(z)-z^\top \theta\}. \end{equation*}

Proof. If $g_n$ is concave, then it is continuous on the interior of its domain that is $\mathbb{R}^d$ . Therefore by a well-known fact on the Legendre transform, we have

\begin{equation*} g_n(z) = \inf_{x \in \mathbb{R}^d} \{z^\top x + b_n(x)\}. \end{equation*}

Let $x \notin \Theta$ . Since $\Theta$ is a closed convex set of $\mathbb{R}^d$ , by the Hahn–Banach theorem (or the separating hyperplane theorem), there exists $z_0 \in \mathbb{R}^d$ such that

\begin{equation*} \min_{\theta \in \Theta}z_0^\top \theta > z_0^\top x.\end{equation*}

Using (2.10), for $z = \alpha z_0$ , $\alpha \gt 0$ ,

\begin{equation*} g_n(z) - z^\top x \geq g_n(0) + \min_{\theta \in \Theta} \theta^\top z -z^\top x = g_n(0)+ \alpha \min_{\theta \in \Theta} z_0^\top(\theta -x).\end{equation*}

Since the last term is positive, letting $\alpha \to \infty$ , we conclude $b_n(x)=\infty$ . This implies

\begin{equation*} g_n(z) = \inf_{\theta \in \Theta} \{z^\top \theta + b_n(\theta)\} = \inf_{(\theta,b) \in A_n} \{z^\top \theta + b\}, \end{equation*}

where $A_n = \{(\theta,b) \in \Theta \times \mathbb{R} \, ; \, g_n(w) \leq w^\top \theta + b \text{ for all } w \in \mathbb{R}^d\}$ . Fix n and z and then take a sequence $\{(\theta_k,b_k)\} \subset A_n$ such that $z^\top \theta_k + b_k \to g_n(z)$ . Since $\Theta$ is compact, there exists a converging subsequence $\{\theta_{k_j}\}$ with limit $\theta_\ast \in \Theta$ . We have $b_{k_j} = z^\top\theta_{k_j} + b_{k_j} - z^\top \theta_{k_j} \to g_n(z) - z^\top \theta_\ast = \! : \, b_\ast$ . Also, $w^\top \theta_{k_j} + b_{k_j} \geq g_n(w)$ for all w implies $w^\top \theta_\ast + b_\ast \geq g_n(w)$ for all w, hence $(\theta_\ast,b_\ast) \in A_n$ . Thus we obtain (2.13). Conversely, if (2.13) is true, then $g_n(z)$ is concave, being the minimum of concave (affine) functions. Since

\begin{equation*} z_1^\top \theta + b_n(\theta) = z_2^\top \theta + b_n(\theta) + \theta^\top(z_1-z_2) \geq z_2^\top \theta + b_n(\theta) + \min_{\theta \in \Theta} \theta^\top(z_1-z_2), \end{equation*}

we derive (2.10) from (2.13).

Let $\mathcal{P}_N$ denote the set of the probability measures on $(\Omega,\mathscr{F}_N)$ absolutely continuous with respect to $\mathbb{P}$ . For $\hat{\mathbb{P}} \in \mathcal{P}_N$ , there corresponds a $\Delta_d$ -valued predictable process $\{\hat{P}_n\}$ by (2.4). The measure $\hat{\mathbb{P}}$ is recovered from $\{\hat{P}_n\}$ by (2.5). Let $\hat{\mathbb{E}}$ denote the expectation under $\hat{\mathbb{P}}$ and define

\begin{equation*} c^g_n(\hat{\mathbb{P}}) = \hat{\mathbb{E}}\Biggl[\sum_{i=n+1}^N b_i(\mathbf{v}\hat{P}_i)\bigg|\mathscr{F}_n\Biggr],\end{equation*}

where $b_i$ is associated with $g = \{g_n\} \in \mathcal{C}$ via (2.13). The following theorem shows the nature of the g-expectation with the gradient constraint as a nonlinear expectation, taking care of Knightian uncertainty, refining a general convex duality result in [Reference Detlefsen and Scandolo18] for an explicit representation of a penalty function.

Theorem 2.2. Let $g \in \mathcal{C}$ . Then, for any $\mathscr{F}_N$ -measurable random variable Y,

(2.14) \begin{equation} \mathcal{E}^g_n(Y) = \min_{\hat{\mathbb{P}}\in \mathcal{P}_N} \big\{ \hat{\mathbb{E}}[Y\mid \mathscr{F}_n] + c^g_n(\hat{\mathbb{P}}) \big\}, \quad n=0,1,\ldots, N.\end{equation}

Proof. By (2.13), we have

\begin{equation*} g_n(z) \leq z^\top \mathbf{v}\hat{P}_n + b_n(\mathbf{v}\hat{P}_n)\end{equation*}

for any $\hat{\mathbb{P}} \in \mathcal{P}_N$ . Therefore, by Propositions 2.2 and 2.3, we have

\begin{equation*} \mathcal{E}^g_n(Y) \leq \hat{\mathbb{E}}[Y\mid \mathscr{F}_n] + c^g_n(\hat{\mathbb{P}})\end{equation*}

for any $\hat{\mathbb{P}} \in \mathcal{P}_N$ . On the other hand, for any $Y \in \mathscr{F}_N$ , there exists the solution $\{(Y_n,Z_n)\}$ of (2.7) with $Y_N= Y$ . By (2.13), there exists $\hat{P}_n$ such that

\begin{equation*} g_n(Z_n) = Z_n^\top \mathbf{v}\hat{P}_n + b_n(\mathbf{v}\hat{P}_n)\end{equation*}

for each n. Since $\{g_n\}$ and $\{Z_n\}$ are predictable, $\{\hat{P}_n\}$ is a $\Delta_d$ -valued predictable process. Let $\hat{\mathbb{P}} \in \mathcal{P}_N$ be associated with $\{\hat{P}_n\}$ . Then $\{(Y_n,Z_n)\}$ solves the BSDE with $\hat{g}_n(z) = z^\top \mathbf{v}\hat{P}_n + b_n(\mathbf{v}\hat{P}_n)$ as well, and so by Proposition 2.2,

\begin{equation*} \mathcal{E}^g_n(Y) = Y_n= \hat{\mathbb{E}}[Y\mid \mathscr{F}_n] + c^g_n(\hat{\mathbb{P}}),\end{equation*}

which implies (2.14).

Example 2.5. Let $\Theta_n \subset \Theta$ and

\begin{equation*} g_n(z) = \inf_{\theta \in \Theta_n} z^\top \theta, \quad n= 1,\ldots, N.\end{equation*}

Here, the set $\Theta_n$ can be random in such a way that $g_n$ is $\mathscr{F}_{n-1}\otimes \mathscr{B}(\mathbb{R}^d)$ -measurable. Then we have (2.13) with $b_n$ such that $b_n(\theta) = 0$ if $\theta \in \bar{\Theta}_n$ while $b_n(\theta) = \infty$ otherwise, where $\bar{\Theta}_n$ is the closure of $\Theta_n$ . In particular when $\Theta_n = \Theta$ , by Theorem 2.2,

\begin{equation*} \mathcal{E}^g_n(Y) = \min_{\hat{\mathbb{P}} \in \mathcal{P}_N} \hat{\mathbb{E}}[Y \mid \mathscr{F}_n],\quad n= 1,\ldots, N,\end{equation*}

for any $\mathscr{F}_N$ -measurable random variable Y. Note also that

\begin{equation*} \mathcal{E}^g_0(Y) = \min_{\hat{\mathbb{P}} \in \mathcal{P}_N} \hat{\mathbb{E}}[Y] = \min_{\omega \in \Omega} Y(\omega).\end{equation*}

When $\Theta_n = \{\mathbf{v}P_n^{(1)}, \ldots, \mathbf{v}P^{(m)}_n\}$ for a $\Delta_d$ -valued predictable process $\{P^{(i)}_n\}$ , letting $\mathbb{E}^{(x)}$ denote the expectation under the measure determined by $\{P^{(x_n)}_n\}$ for $x = (x_1,\ldots,x_N) \in \{1,\ldots,m\}^N$ , by Theorem 2.2, we have

\begin{equation*} \mathcal{E}^g_n(Y) = \min_x \mathbb{E}^{(x)}[Y \mid \mathscr{F}_n],\quad n= 1,\ldots, N.\end{equation*}

2.5. Filtration consistent nonlinear expectations

Inspired by Coquet et al. [Reference Coquet, Hu, Memin and Peng16], we call $\mathcal{E}\colon L^0(\Omega,\mathscr{F}_N,\mathbb{P}) \to \mathbb{R}$ a filtration consistent nonlinear expectation if:

1. (i) $Y \geq Y^\prime \Rightarrow \mathcal{E}(Y) \geq \mathcal{E}(Y^\prime)$ ,

2. (ii) $Y \geq Y^\prime$ and $ \mathcal{E}(Y) = \mathcal{E}(Y^\prime) \Rightarrow Y = Y^\prime$ ,

3. (iii) $\mathcal{E}(c) = c$ for any constant $c \in \mathbb{R}$ , and

4. (iv) for any $n=1,\ldots,N$ and Y, there exists an $\mathscr{F}_n$ -measurable $\eta$ such that $\mathcal{E}(Y1_A) = \mathcal{E}(\eta 1_A)$ for any $A \in \mathscr{F}_n$ .

Further, $\eta$ is uniquely determined as shown in [Reference Coquet, Hu, Memin and Peng16]. Let it be denoted by $\mathcal{E}_n(Y)$ . It follows that

(2.15) \begin{equation} \mathcal{E}_n(\mathcal{E}_m(Y)) = \mathcal{E}_n(Y)\end{equation}

for any $m \geq n$ , and

(2.16) \begin{equation} \mathcal{E}_n(Y)1_A = \mathcal{E}_n(Y1_A)\end{equation}

for any $A \in \mathscr{F}_n$ .

Proposition 2.4. Let $g = \{g_n \} \in \mathcal{A}$ , and assume that for $n = 1,\ldots, N$ ,

1. (i) $g_n(0) = 0$ and

2. (ii) for any $z_1, z_2 \in \mathbb{R}^d$

   \begin{equation*} g_n(z_1) -g_n(z_2) \geq \min_{\theta \in \Theta} \theta^\top (z_1-z_2),\end{equation*}
   with equality holding only if $z_1 = z_2$ .

Then $\mathcal{E}^g_0$ is a filtration consistent nonlinear expectation with $\mathcal{E}_n = \mathcal{E}^g_n$ and a translation invariance property,

(2.17) \begin{equation} \mathcal{E}_n(Y + \eta) = \mathcal{E}_n(Y) + \eta,\end{equation}

for any $\mathscr{F}_N$ -measurable random variable Y, $\mathscr{F}_n$ -measurable random variable $\eta$ , and $n=1,\ldots, N$ .

The following theorem is a discrete analog of Theorem 7.1 of [Reference Coquet, Hu, Memin and Peng16].

Proof. We have $\mathcal{E}_N(Y) = Y$ for any Y, so the claim is true for $n=N$ . Assume $\mathcal{E}_k(Y) = \mathcal{E}^g_k(Y)$ for $k \geq n$ . Then, by (2.2), there exists $\mathscr{F}_{n-1}$ -measurable $A_n$ and $Z_n$ such that

\begin{equation*} \mathcal{E}_n(Y) = A_n + Z_n^\top \Delta X_n.\end{equation*}

Then, by (2.15) and (2.17),

\begin{equation*} \mathcal{E}_{n-1}(Y) = A_n + \mathcal{E}_{n-1}(Z_n^\top \Delta X_n).\end{equation*}

The last term is $g_n(Z_n)$ by (2.16). Therefore

\begin{equation*}\mathcal{E}_n(Y)- \mathcal{E}_{n-1}(Y) = - g_n(Z_n) + Z_n^\top \Delta X_n,\end{equation*}

which implies that $\mathcal{E}_{n-1}(Y) = \mathcal{E}_{n-1}^g(Y)$ . The result follows by induction.

3.1. Monetary utility maximization

Now we consider $\{X_n\}$ to be a d-dimensional asset price process. The lattice L is then understood as the price grid. For any $\{\mathscr{F}_n\}$ -predictable $\mathbb{R}^d$ -valued process $Z = \{Z_n\}$ and $w \in \mathbb{R}$ ,

\begin{equation*} W_n(w,Z) \,:\!=\, w + \sum_{i=1}^n Z_i^\top \Delta X_i\end{equation*}

represents the wealth process associated with the portfolio strategy Z and the initial wealth w. Applying Theorem 2.1 with $g_n = 0$ , $n=1,\ldots, N$ , we observe that the market is complete, that is, for any $\mathscr{F}_N$ -measurable random variable Y, there exists a predictable process Z and $w \in \mathbb{R}$ such that $Y = W_N( w,Z)$ .

Consider an agent whose utility functional is $\mathcal{E}^g_0$ for $g \in \mathcal{C}$ . This utility is monetary in the sense that for all $n = 0, \ldots, N$ , $\mathcal{E}_n^g(Y+A) = \mathcal{E}_n^g(Y)+A$ for any $\mathscr{F}_N$ -measurable random variable Y and $\mathscr{F}_n$ -measurable random variable A. When assuming $g_n(0) = 0$ for all n in addition, the utility is normalized in the sense that $\mathcal{E}^g_n(0)=0$ , $n =0,\ldots, N$ , and it is time-consistent in the sense that $\mathcal{E}_m^g(\mathcal{E}_n^g(Y))=\mathcal{E}_m^g(Y)$ for any $m \leq n$ and for any $\mathscr{F}_N$ -measurable random variable Y. The simplest example is $\mathcal{E}^g_n(Y) = \mathbb{E}[Y\mid \mathscr{F}_n]$ corresponding to $g_n(z) = z^\top \mathbb{E}[\Delta X_n \mid \mathscr{F}_{n-1}]$ . More generally, $\mathcal{E}^g_n(Y)$ is a conditional expectation with respect to a probability measure when $g_n$ are linear for all n by Proposition 2.2. The driver $\{g_n\}$ should reflect the agent’s belief in the distribution of the price process $\{X_n\}$ . For example, if $g_n = 0$ for all n, then $\mathcal{E}^g_n(Y) = \mathbb{E}_\mathbb{Q}[Y\mid \mathscr{F}_n]$ irrespective of $\mathbb{P}$ . The choice of nonlinear $\{g_n\}$ accommodates a nonlinear evaluation of risk, extending the exponential utility (2.9). In light of Theorem 2.2, our problem can be interpreted as a robust utility maximization.

The agent’s objective is to maximize $\mathcal{E}^g_0(H + W_N(w,\pi))$ among the predictable process $\pi = \{\pi_n\}$ , where H is a given $\mathscr{F}_N$ -measurable random variable representing an initial endowment of the agent, i.e. an initially endowed asset or a scheduled random cashflow. Since the utility is monetary, it suffices to treat the case $w = 0$ . Let

(3.1) \begin{equation} Y^\pi_n =\mathcal{E}^g_n(H + W_N(0,\pi)- W_n(0,\pi) ) =\mathcal{E}^g_n(H + W_N(0,\pi))- W_n(0,\pi).\end{equation}

Then the problem is equivalent to maximizing $Y^\pi_0$ among $\pi$ . The following theorem characterizes the maximizer.

Theorem 3.1. Assume that there exists a predictable process $\{Z^\dagger_n\}$ such that

(3.2) \begin{equation} g_n(Z^\dagger_n) = \sup_{z \in \mathbb{R}^d} g_n(z), \quad n=1,\ldots, N.\end{equation}

Then

\begin{equation*} \max_{\pi} Y^\pi_0 = \max_{\pi}\mathcal{E}^g_0(H + W_{N} (0,\pi)) = \mathcal{E}^g_0(H + W_{N}(0,\pi^\ast)),\end{equation*}

where $\pi^\ast_n = Z^\dagger_n - Z^\ast_n$ , $ Z^\ast_n = Z^H_n + Z^g_n$ ,

(3.3) \begin{align} Z^H_n& =(d+1) (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbb{E}_\mathbb{Q}[H\Delta X_n \mid \mathscr{F}_{n-1}],\nonumber\\Z^g_n & =(d+1) (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbb{E}_\mathbb{Q}[G_n \Delta X_n \mid \mathscr{F}_{n-1}],\end{align}

and

\begin{equation*} G_n = \sum_{i=n+1}^N \mathbb{E}_\mathbb{Q}[ g_i(Z_i^\dagger) \mid \mathscr{F}_n]\end{equation*}

for $n=1,\ldots, N$ . Moreover,

(3.4) \begin{equation}Y^{\pi^\ast}_n = \mathbb{E}_\mathbb{Q}[ H \mid \mathscr{F}_n]+ G_n\end{equation}

for $n=0,1,\ldots, N$ . If in addition $Z^\dagger_n$ is unique, then $\pi^\ast_n$ is the unique maximizer.

Proof. Let $Y^\ast_n$ denote the right-hand side of (3.4). Then $\{(Y^\ast_n,Z^\ast_n)\}$ is the solution of the BS $\Delta$ E

(3.5) \begin{equation} \Delta Y^\ast_n = - g_n(Z_n^\dagger) + (Z^\ast_n)^\top \Delta X_n, \quad Y^\ast_N = H ,\end{equation}

by Theorem 2.1. Note that

\begin{equation*} \Delta Y^\pi_n = -g_n(Z_n) + Z_n^\top \Delta X_n - \pi_n^\top \Delta X_n,\quad Y^\pi_N = H ,\end{equation*}

for a predictable process $\{Z_n\}$ by (3.1). This means $\{(Y^\pi_n,Z^\pi_n)\}$ , $Z^\pi_n = Z_n -\pi_n$ solves the BS $\Delta$ E

\begin{equation*}\Delta Y^\pi_n = -h^\pi_n(Z^\pi_n) + (Z_n^\pi)^\top \Delta X_n,\quad Y^\pi_N = H,\end{equation*}

where $h^\pi_n(z) = g_n(z+ \pi_n)$ . For any predictable process $\pi$ , we have $h^\pi_n \leq g_n(Z^\dagger_n)$ . Therefore $Y^\pi_n \leq Y^\ast_n$ by Proposition 2.3. Notice also that by choosing $\pi^\ast_n = Z^\dagger_n -Z^\ast_n$ we have $g_n(Z^\dagger) = h_n^{\pi^\ast}(Z^\ast_n)$ , so that $\{(Y^\ast_n,Z^\ast_n)\}$ satisfies the same BS $\Delta$ E as $\{(Y^{\pi^\ast}_n,Z^{\pi^\ast}_n)\}$ . Hence $Y^\ast_n = Y^{\pi^\ast}_n$ .

Remark 3.1. Applying Theorem 2.1 to $g_n = 0$ and $Y_N = H$ , we have

\begin{equation*} H = \mathbb{E}_\mathbb{Q}[H] + \sum_{n=1}^N Z^H_n \Delta X_n \end{equation*}

with $\{Z^H_n\}$ defined by (3.3). Therefore the optimal strategy $\pi^\ast = Z^\dagger_n - Z^\ast_n = -Z^H_n + Z^\dagger_n - Z^g_n$ of Theorem 3.1 is decomposed into the hedging part $-Z^H_n$ and the optimal investment part $Z^\dagger_n - Z^g_n$ .

Example 3.1. (Locally entropic monetary utility.) Consider (2.12). Let $\Delta_d^\circ$ denote the interior of $\Delta_d$ and assume $\{(\hat{P}_{n,1},\ldots, \hat{P}_{n,d})^\top\}$ to be $\Delta_d^\circ$ -valued. Then the map $z \mapsto g_n(z)$ is strictly concave and its unique maximizer is given by $Z^\dagger_n = b(y)$ of (2.3) for

\begin{equation*} y = \frac{1}{\Gamma_n} \log \hat{P}_n \,:\!=\, \frac{1}{\Gamma_n} \big( \log \hat{P}_{n,1}, \ldots, \log \hat{P}_{n,d}\big)^\top, \end{equation*}

or equivalently

(3.6) \begin{equation} Z^\dagger_n = \frac{1}{\Gamma_n} (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbf{v} \log \hat{P}_n. \end{equation}

Indeed, $v_j^\top Z^\dagger_n = \Gamma_n^{-1}\log \hat{P}_{n,j} - a(y)$ implies

\begin{equation*} \sum_{j=0}^d v_j \,{\mathrm{e}}^{-\Gamma_n v_j^\top Z^\dagger_n} \hat{P}_{n,j}= 0, \end{equation*}

and thus $\nabla g_n(Z^\dagger_n) =0$ for all n. We also have

(3.7) \begin{align} \frac{1}{\Gamma_n}\log B_n + g_n(Z^\dagger_n) &= -\frac{1}{\Gamma_n} \log \Biggl( \sum_{j=0}^d \,{\mathrm{e}}^{-\Gamma_n v_j^\top Z^\dagger_n} \hat{P}_{n,j}\Biggr)\notag \\*& =-\frac{1}{\Gamma_n} \log \Bigl((d+1) \,{\mathrm{e}}^{\frac{1}{d+1}\mathbf{1}^\top \log \hat{P}_n }\Bigr)\notag \\ &= \frac{1}{\Gamma_n} \frac{1}{d+1}\sum_{j=0}^d \log \frac{1}{(d+1)\hat{P}_{n,j}} \notag \\*&= \frac{1}{\Gamma_n}D_{\mathrm{KL}}(Q_n ||\hat{P}_n) \notag \\*&\geq 0, \end{align}

where $D_{\mathrm{KL}}$ denotes the Kullback–Leibler divergence on $\Delta_d$ and $Q_n = \mathbf{1}/(d+1)$ . By (3.3) and (3.6),

\begin{equation*} \pi^\ast_n = Z^\dagger_n - Z^g_n - Z^H_n= - Z^H_n + (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbf{v} \biggl( \frac{\log \hat{P}_n}{\Gamma_n} - \hat{Y}_n \biggr), \end{equation*}

where $\hat{Y}_n = (\hat{Y}_{n,0},\ldots, \hat{Y}_{n,d})^\top$ and $ \hat{Y}_{n,j} = \mathbb{E}_\mathbb{Q}[G_n \mid \mathscr{F}_{n-1}, \Delta X_n = v_j]$ . The first term $-Z^H_n$ is the hedging term as noted in Remark 3.1. The term

\begin{equation*} Z^\dagger_n = (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbf{v} \frac{\log \hat{P}_n}{\Gamma_n} \approx (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbf{v} \frac{\hat{P}_n - \mathbf{1}}{\Gamma_n} = \frac{1}{\Gamma_n} (\mathbf{v}\mathbf{v}^\top)^{-1}\hat{\mathbb{E}}[\Delta X_n \mid \mathscr{F}_{n-1}] \end{equation*}

can be interpreted as the discrete counterpart of the Merton portfolio (see e.g. Remark 8.9 of [Reference Karatzas and Shreve27]). The term $\hat{Y}_n$ adjusts the expected return depending on the stochastic dynamics of $D_{\mathrm{KL}}(Q_i ||\hat{P}_i)$ for $i \geq n+1$ . Indeed, when $B_n = 1$ and $\Gamma_n = \gamma$ for all n and a constant $\gamma \gt 0$ as in (2.8), we have

\begin{align*} G_n &= \sum_{i=n+1}^n \mathbb{E}_\mathbb{Q}[g_i(Z^\dagger_i) \mid \mathscr{F}_n]\\ &= \frac{1}{\gamma}\sum_{i=n+1}^n \mathbb{E}_\mathbb{Q}[D_{\mathrm{KL}}(Q_i||\hat{P}_i) | \mathscr{F}_n]\\ &= \frac{1}{\gamma} \mathbb{E}_\mathbb{Q}\biggl[ \log \frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\hat{\mathbb{P}}} - \log \mathbb{E}_\mathbb{Q}\biggl[ \frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\hat{\mathbb{P}}}\,\bigg|\, \mathscr{F}_n\biggr]\, \bigg|\, \mathscr{F}_n\biggr]\end{align*}

by (3.7), where $\hat{\mathbb{P}}$ is associated with $\{\hat{P}_n\}$ by (2.5). Note also that $Z^g_n = 0$ if $G_n$ is deterministic, which is the case when $\{\hat{P}_n\}$ , $\{\Gamma_n\}$ , and $\{B_n\}$ are deterministic.

Example 3.2. Let $d=2$ and

\begin{equation*} \mathbf{v} = \begin{pmatrix} 0 & \quad 1 & \quad -1 \\ -2c & \quad c & \quad c \end{pmatrix}\!, \end{equation*}

where $c \in \mathbb{R}$ . Then we have

\begin{equation*} X_{n,2} = c\Biggl(3 \sum_{k=1}^n |\Delta X_{k,1}|^2 - 2n\Biggr).\end{equation*}

Indeed, if $X_n = (n-a-b)v_0 + av_1 + bv_2$ for $(a,b) \in \mathbb{N}^2$ , then

\begin{equation*} a + b = \sum_{k=1}^n |\Delta X_{k,1}|^2, \quad X_{n,2} = -2c(n-a-b) + c(a+b) = c(3(a+b)-2n).\end{equation*}

Regarding $\{X_{n,1}\}$ as a price process of an asset, the above identity allows us to interpret $X_{N,2}$ as an affine transform of the variance swap payoff of the asset. Further, regarding $\mathbb{Q}$ as the pricing measure, $\{X_{n,2}\}$ corresponds to the price process of the variance swap payoff. Note that $\{X_{n,1}\}$ describes a trinomial model for the asset, which is not complete. The variance swap trading makes the two-dimensional market $\{X_n\}$ complete.

3.2. General equilibrium

Consider m agents who maximize respective utilities $\mathcal{E}_0^{(i)}(H^{(i)} + W_N(0,\pi^{(i)}))$ , $i=1,\ldots, m$ through trading strategies $\pi^{(i)}$ of the d-dimensional asset $\{X_n\}$ , where $\mathcal{E}^{(i)}$ is the solution map of the BS $\Delta$ E (2.7) with $g = g^{(i)} \in \mathcal{C}$ , and $H^{(i)}$ is an $\mathscr{F}_N$ -measurable random variable representing an endowment for the agent i, $i=1,\ldots,m$ . Let $H_n$ denote the total supply vector of the asset vector $X_n$ and assume $\{H_n\}$ to be an $\mathbb{R}^d$ -valued predictable process. We say the market is in general equilibrium if there exist predictable processes $\pi^{(i)} = \{\pi^{(i)}_n\}$ , $i=1,\ldots, m$ such that

1. (i) $\mathcal{E}_0^{(i)}(H^{(i)} + W_N(0,\pi^{(i)})) = \max_\pi \mathcal{E}_0^{(i)}(H^{(i)} + W_N(0,\pi))$ for all $i=1,\ldots, m$ , and

2. (ii) $\sum_{i=1}^m \pi^{(i)}_n = H_n $ for all $n=1,\ldots, N$ ,

where the maximum is among all predictable processes $\pi$ .

Proposition 3.1. Let $H^{(i)}$ and $\tilde{H}^{(i)}$ , $i=1,\ldots,m$ be $\mathscr{F}_N$ -measurable random variables, and let $\{H_n\}$ and $\{\tilde{H}_n\}$ be $\mathbb{R}^d$ -valued predictable processes satisfying

\begin{equation*} \sum_{i=1}^m H^{(i)} + \sum_{n=1}^N H_n^\top \Delta X_n = \sum_{i=1}^m \tilde{H}^{(i)} + \sum_{n=1}^N \tilde{H}_n^\top \Delta X_n. \end{equation*}

Then the market with the endowments $H^{(i)}$ and the total supply $\{H_n\}$ is in general equilibrium if and only if the market with the endowments $\tilde{H}^{(i)}$ and total supply $\{\tilde{H}_n\}$ is in general equilibrium.

Proof. Let $Z^{\dagger(i)}$ , $Z^{H(i)}$ , and $Z^{g(i)}$ , respectively, denote $Z^{\dagger}$ , $Z^{H}$ , and $Z^{g}$ in Theorem 3.1 with $g = g^{(i)}$ and $H = H^{(i)}$ . Then, by Theorem 3.1,

\begin{equation*}-H_n + \sum_{i=1}^m \pi^{(i)}_n = -H_n - \sum_{i=1}^m Z^{H(i)} + \sum_{i=1}^m \big(Z^{\dagger(i)} - Z^{g(i)}\big) =- Z^H + \sum_{i=1}^m \big(Z^{\dagger(i)} - Z^{g(i)}\big),\end{equation*}

where $Z^H$ is defined by (3.3) with

\begin{equation*} H = \sum_{i=1}^m H^{(i)} + \sum_{n=1}^N H_n^\top \Delta X_n.\end{equation*}

Therefore, whether or not $\sum_{i=1}^m \pi^{(i)}_n =H_n$ depends on $H^{(i)}$ and $\{H_n\}$ only through H.

We are interested in conditions on $g^{(i)}$ for the market to be in general equilibrium. In light of Proposition 3.1, we assume hereafter $H^{(i)}=0$ ( $i \geq 2)$ and $H_n =0$ ( $n\geq 1$ ), without loss of generality. Let H denote $H^{(1)}$ .

For functions $f^{(1)}$ and $f^{(2)}$ on $\mathbb{R}^d$ , define the sup-convolution $f^{(1)} \square f^{(2)}$ by

\begin{equation*} f^{(1)}\square \, f^{(2)}(z) = \sup_{x \in \mathbb{R}^d} \{\, f^{(1)}(x) + f^{(2)}(z-x)\}.\end{equation*}

For the drivers $g^{(i)} = \{g^{(i)}_n\}$ , $i=1,\ldots,m$ of the m agents’ utilities, let

(3.8) \begin{equation} g_n(z) = g^{(1)}_n \square \cdots \square g^{(m)}_n(z).\end{equation}

Lemma 3.1. For all n,

(3.9) \begin{equation} \sup_{z \in \mathbb{R}^d} g_n(z) = \sum_{i=1}^m \sup_{z \in \mathbb{R}^d} g_n^{(i)}(z).\end{equation}

Proof. It is trivial that $g_n(z)$ is upper-bounded by the right-hand side sum for any z, and hence its supremum is upper-bounded. Conversely, let $\bigl\{z^{(i)}_k\bigr\}$ be a sequence for which $ \lim_{k \to \infty} g_n^{(i)}\bigl(z^{(i)}_k\bigr) = \sup_{z \in \mathbb{R}^d} g_n^{(i)}(z)$ for each i. Then

\begin{equation*} \sum_{i=1}^m g_n^{(i)}\bigl(z^{(i)}_k\bigr) \leq g_n\Biggl(\sum_{i=1}^m z^{(i)}_k \Biggr) \leq \sup_{z \in \mathbb{R}^d} g_n(z)\end{equation*}

for any k, and the limit is similarly bounded.

A single agent whose utility is $\mathcal{E}^g_0$ with $g = \{g_n\}$ defined by (3.8) and whose endowment is H is called the representative agent of the market. The following theorem reduces the general equilibrium problem for the multi-agent market to the one for a single agent market. This extends the idea of the well-known Gorman aggregation theorem.

Proposition 3.2. Assume that there exists a unique predictable process $\{Z^{\dagger(i)}_n\}$ for each $i=1,\ldots,m$ such that

\begin{equation*} g^{(i)}_n(Z^{\dagger(i)}_n) = \sup_{z \in \mathbb{R}^d}g^{(i)}_n(z)\end{equation*}

for all $n = 1,\ldots, N$ . Assume further that the maximizer of the map $z \mapsto g_n(z)(\omega)$ is unique for each n and $\omega \in \Omega$ . The market for the m agents is in general equilibrium if and only the market for the representative agent is in general equilibrium.

Proof. Let $\{(Y^{\ast(i)}_n,Z^{\ast(i)}_n)\}$ be the solution of

\begin{equation*} \Delta Y^{\ast (i)}_n = -g^{(i)}_n(Z^{\dagger (i)}_n) + (Z^{\ast (i)}_n)^\top \Delta X_n,\end{equation*}

with $Y^{\ast (1)}_N = H$ and $Y^{\ast (i)}_N = 0$ for $i \geq 2$ . By Theorem 3.1, the unique optimal strategy $\pi^{(i)}$ for the agent i is given by $\pi^{(i)}_n = Z^{\dagger (i)}_n -Z^{\ast(i)}_n $ . By (3.9), we have

\begin{equation*} \sum_{i=1}^m g^{(i)}_n(Z^{\dagger(i)}_n) = \sup_{z \in \mathbb{R}^d}g_n(z) = g_n(Z^\dagger_n), \quad Z^\dagger_n \,:\!=\, \sum_{i=1}^m Z^{\dagger(i)}_n.\end{equation*}

Therefore

\begin{equation*} Y^\ast_n \,:\!=\, \sum_{i=1}^m Y^{\ast(i)}_n, \quad Z^\ast_n \,:\!=\, \sum_{i=1}^m Z^{\ast (i)}_n\end{equation*}

solves

\begin{equation*} \Delta Y^\ast_n = -g_n(Z^\dagger_n) + (Z^\ast_n)^\top \Delta X_n, \quad Y^\ast_N = H.\end{equation*}

Since $Z^\dagger_n$ is the unique maximizer of $g_n$ , the unique optimal strategy for the representative agent is $\pi^\ast_n = Z^\dagger_n - Z^\ast_n$ , again by Theorem 3.1. Hence

\begin{equation*} \sum_{i=1}^m \pi^i_n = \sum_{i=1}^m Z^{\ast (i)}_n - \sum_{i=1}^m Z^{\dagger (i)}_n = Z^\ast_n - Z^\dagger_n = \pi^\ast_n.\end{equation*}

Therefore $\sum_{i=1}^m \pi^i_n = 0$ if and only if $\pi^\ast_n = 0$ .

3.3. Equilibrium in a single agent market

By Proposition 3.2, it suffices to consider the case $m=1$ in order to characterize the general equilibrium. Let $m=1$ , and put $g_n = g^{(1)}_n$ and $\pi_n = \pi^{(1)}_n$ . The market is in general equilibrium if and only if $\mathcal{E}_0^g (H) = \max_\pi \mathcal{E}_0^{g}(H + W_N(0,\pi))$ , that is, $\pi^\ast_n \equiv 0$ is the maximizer. The following theorems characterize the general equilibrium by a backward recurrence relation for the BS $\Delta$ E driver $\{g_n\}$ .

Theorem 3.2. The market is in general equilibrium if (3.2) holds with

(3.10) \begin{equation}Z^\dagger_n = (d+1)(\mathbf{v}\mathbf{v}^\top)^{-1} \mathbb{E}_\mathbb{Q}[(H+G_n)\Delta X_n \mid \mathscr{F}_{n-1}],\quad n=1,\ldots, N,\end{equation}

where

\begin{equation*} G_n =\mathbb{E}_\mathbb{Q}\Biggl[ \sum_{i=n+1}^N \sup_{z\in \mathbb{R}^d} g_i(z) \bigg| \mathscr{F}_n\Biggr].\end{equation*}

Conversely, if the market is in general equilibrium and there exist sequences of maximizers $\{Z^\dagger_n\}$ of $\{g_n\}$ , then (3.10) is one of them.

Example 3.3. ( Locally entropic monetary utility.) We consider (2.12) again with $\{\hat{P}_n\}$ , $\hat{P}_n = (\hat{P}_{n,1},\ldots, \hat{P}_{n,d})^\top$ , being $\Delta_d^\circ$ -valued. The driver $g_n$ is of the form $g_n(z) = (f(\Gamma_nz,\hat{P}_n) -\log B_n)/\Gamma_n $ , where

(3.11) \begin{equation} f(z,p) = -\log \sum_{j=0}^d \,{\mathrm{e}}^{-z^\top v_j}p_j,\quad z \in \mathbb{R}^d, \quad p = (p_0,\ldots,p_d)^\top \in \Delta_d. \end{equation}

By (3.6), we have $\nabla f_n(Z_n,\hat{P}_n) = 0$ if and only if

\begin{equation*} Z_n = \frac{1}{\Gamma_n} (\mathbf{v}\mathbf{v}^\top)^{-1}\mathbf{v} \log \hat{P}_n. \end{equation*}

By Theorem 3.2, the market is in general equilibrium if and only if

\begin{equation*} \mathbf{v}\log \hat{P}_n = (d+1)\Gamma_n \mathbb{E}_\mathbb{Q}[(H+G_n) \Delta X_n\mid \mathscr{F}_{n-1}], \quad n=1,\ldots, N.\end{equation*}

This is a backward recurrence equation for $\{\hat{P}_n\}$ because

\begin{equation*} G_n = \mathbb{E}_\mathbb{Q}\Biggl[ \sum_{i=n+1}^N \frac{1}{\Gamma_i} ( D_{\mathrm{KL}}(Q_i||\hat{P}_i) - \log B_i )\, \bigg|\, \mathscr{F}_n \Biggr]\end{equation*}

by (3.7). Since $\mathbf{v}$ has rank d and $\mathbf{v}\mathbf{1} = 0$ , an equivalent condition is that

(3.12) \begin{equation} \hat{P}_{n,j} = \frac{{\mathrm{e}}^{\Gamma_n\bar{Y}_{n,j}}}{\sum_{k=0}^d\,{\mathrm{e}}^{\Gamma_n\bar{Y}_{n,k}} }, \quad \bar{Y}_{n,j} = \mathbb{E}_\mathbb{Q}[H+G_n \mid \mathscr{F}_{n-1},\Delta X_n = v_j], \quad j=0,\ldots, d, \end{equation}

for $n=1,\ldots, N$ . Let $ Y^\ast_n = \mathbb{E}_\mathbb{Q}[H\mid \mathscr{F}_n] + G_n$ . Then

\begin{equation*}Y_n = \mathbb{E}_\mathbb{Q}[Y_n\mid \mathscr{F}_{n-1},\Delta X_n] = \sum_{j=0}^d \bar{Y}_{n,j} 1_{\{\Delta X_n = v_j\}}, \end{equation*}

which implies

\begin{equation*} \hat{P}_{n,j}1_{\{\Delta X_n = v_j\}} = \frac{{\mathrm{e}}^{\Gamma_nY^\ast_n}}{(d+1)\mathbb{E}_\mathbb{Q}[{\mathrm{e}}^{\Gamma_nY^\ast_n}\mid \mathscr{F}_{n-1}]}1_{\{\Delta X_n=v_j\}},\quad j=0,\ldots,d, \end{equation*}

under (3.12). Therefore the market is in general equilibrium if and only if

\begin{equation*} \frac{\mathrm{d}\hat{\mathbb{P}}}{\mathrm{d}\mathbb{Q}}=\prod_{n=1}^N\frac{{\mathrm{e}}^{\Gamma_nY^\ast_n}}{\mathbb{E}_{\mathbb{Q}}[{\mathrm{e}}^{\Gamma_nY^\ast_n}\mid \mathscr{F}_{n-1}]}, \end{equation*}

where $\hat{\mathbb{P}}$ is defined by (2.5). Further, by (3.5), we have

\begin{align*} \log \hat{\mathbb{E}}[{\mathrm{e}}^{-\Gamma_n Y^\ast_n}\mid \mathscr{F}_{n-1}]& = -\Gamma_n(Y^\ast_{n-1} - g_n(Z^\dagger_n)) +\log \hat{\mathbb{E}}[{\mathrm{e}}^{-\Gamma_n(\Delta X_n)^\top Z^\dagger_n}\mid \mathscr{F}_{n-1}] \\*&=-\Gamma_n(Y^\ast_{n-1} - g_n(Z^\dagger_n)) -f_n(\Gamma_n Z^\dagger_n, \hat{P}_n) \\*& =-\Gamma_nY^\ast_{n-1} - \log B_n. \end{align*}

This implies

(3.13) \begin{equation} \frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\hat{\mathbb{P}}}=\prod_{n=1}^N\frac{{\mathrm{e}}^{-\Gamma_nY^\ast_n}}{\hat{\mathbb{E}}[{\mathrm{e}}^{-\Gamma_n Y^\ast_n}\mid \mathscr{F}_{n-1}]}= \exp\Biggl\{\sum_{n=1}^N -\Gamma_n \Delta Y^\ast_n \Biggr\} \prod_{n=1}^N B_n. \end{equation}

In particular, when $B_n = 1$ and $\Gamma_n = \gamma$ for all n and a constant $\gamma \gt 0$ as in (2.8), we have

(3.14) \begin{equation}\frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\hat{\mathbb{P}}}=\frac{{\mathrm{e}}^{-\gamma H}}{\hat{\mathbb{E}}[{\mathrm{e}}^{-\gamma H}]},\end{equation}

which characterizes the equilibrium probability measure $\hat{\mathbb{P}}$ and is consistent with the well-known computation under exponential utility.

Remark 3.2. In addition to the conditions of Proposition 3.3, if $f_n$ is of the form $f_n(z,p) = f(\Gamma_n z,p)/\Gamma_n$ for a smooth function $f\colon \mathbb{R}^d \times \Delta_d \to \mathbb{R}$ with $\nabla f(0,p) = p$ and a positive predictable process $\{\Gamma_n\}$ as in Example 3.3 with $B_n = 1$ , then we have

\begin{equation*} g_n(z) = f_n(z,\hat{P}_n) \approx z^\top \hat{P}_n \end{equation*}

considering $\Gamma_n$ to be small. This approximation implies in turn

\begin{equation*} \mathcal{E}^g_n(Y) \approx \hat{\mathbb{E}}[Y \mid \mathscr{F}_n] \end{equation*}

in light of Proposition 2.2, where $\hat{\mathbb{E}}$ is the expectation under $\hat{\mathbb{P}}$ defined by (2.5). Therefore, in this case, extending Example 3.3, we can interpret $\Gamma_n$ as a risk-aversion parameter and $\hat{\mathbb{P}}$ as the belief of the agent. If the market is in general equilibrium,

\begin{equation*} \hat{\mathbb{E}}[\Delta X_n \mid \mathscr{F}_{n-1}] = \mathbf{v}\hat{P}_n \end{equation*}

is interpreted as an equilibrium return.

Example 3.4. If $H = h(X_N)$ and $g_n(z) = f_n(X_{n-1},z)$ for deterministic functions h and $f_n$ , as in Proposition 2.1, and if f(x,z) is strictly concave and continuously differentiable in z, then the condition (3.10) follows from the deterministic identity

\begin{equation*} \nabla_z f_n(x,(\mathbf{vv}^\top)^{-1}\mathbf{v}\mathcal{N}u_n(x)) = 0,\quad x \in L, \ n=1,\ldots, N, \end{equation*}

where $\mathcal{N}u_n$ is as in Proposition 2.1. For example, for the market with no random endowments ( $H^{(i)}=0$ ) and unit total supply ( $H_n = \mathbf{1}$ ), we have $H = \mathbf{1}^\top X_N$ .

Example 3.5. Consider (2.12) with $B_n = 1$ , $\Gamma_n = \gamma_n(X_{n-1})$ and $\hat{P}_n = p_n(X_{n-1})$ for deterministic functions $\gamma_n\colon L \to (0,\infty)$ and $p_n\colon L \to \Delta_d^\circ$ . Assume also $H = h (X_N)$ as in Example 3.4. Then, from Examples 3.3 and 3.4, the market is in general equilibrium if

\begin{equation*} \mathbf{v}\log p_n(x) = \gamma_n(x) \mathbf{v} \mathcal{N}u_n(x), \quad x \in L, \ n=1,\ldots, N. \end{equation*}

The function $u_n(x)$ is computed backward inductively without using $p_n(x)$ . For a given function $\gamma_n(x)$ , there exists a unique $p_n(x) \in \Delta_d^\circ$ satisfying this equation for each $x \in L$ . For the sequence of such functions $p_n(x)$ obtained in the backward manner, the $\Delta_d$ -valued sequence $\hat{P}_n = p_n(X_{n-1})$ defines a unique equilibrium probability $\hat{\mathbb{P}}$ by (2.5) associated with the sequence $\Gamma_n = \gamma_n(X_{n-1})$ . The equilibrium return is approximated as

\begin{equation*} \hat{\mathbb{E}}[\Delta X_n \mid \mathscr{F}_{n-1}] = \mathbf{v} \hat{P}_n\approx \mathbf{v}\log \hat{P}_n = \gamma_n(X_{n-1})\mathbf{v} \mathcal{N}u_n(X_{n-1}) \end{equation*}

using $\mathbf{v1} = 0$ .

3.4. Equilibrium under heterogeneous beliefs

Here we assume $m>1$ again and give more explicit computations of the sup-convolution in special cases. First, we consider a homogeneous case, i.e. the case where all of the drivers $g^{(i)}$ have the same functional form determined by a common $\Delta_d$ -valued predictable process $\{\hat{P}_n\}$ as

(3.15) \begin{equation} g^{(i)}_n(z) = \frac{1}{\Gamma^{(i)}_n}f_n(\Gamma^{(i)}_nz,\hat{P}_n),\end{equation}

where $f_n\colon \Omega \times \mathbb{R}^d \times \Delta_d \to \mathbb{R}$ is as in Proposition 3.3 and $\{\Gamma^{(i)}_n\}$ , $i=1,\ldots,m$ are positive predictable processes quantifying each agent’s risk preference; see Remark 3.2. By induction, we can show that

\begin{equation*}g_n(z) \,:\!=\, g^{(1)}_n \square \cdots \square g^{(m)}_n(z)= \frac{1}{\Gamma_n} f_n(\Gamma_nz,\hat{P}_n),\end{equation*}

with

(3.16) \begin{equation} \frac{1}{\Gamma_n} = \sum_{i=1}^m \frac{1}{\Gamma^{(i)}_n}.\end{equation}

Now we consider a heterogeneous case. We assume locally entropic monetary utilities $g^{(i)}_n(z) = f(\Gamma^{(i)}_nz,\hat{P}^{(i)}_n)/\Gamma^{(i)}_n$ , where f is defined by (3.11), and $\{\Gamma^{(i)}_n\}$ and

\[\big\{\hat{P}^{(i)}_n\big\} =\Bigl\{\bigl(\hat{P}^{(i)}_{n,0},\ldots,\hat{P}^{(i)}_{n,d}\bigr)^\top \Bigr\} ,\]

respectively, are $(0,\infty)$ -valued and $\Delta_d^\circ$ -valued predictable processes for each $i=1,\ldots,m$ . Each sequence $\{\hat{P}^{(i)}_n\}$ determines a probability measure $\hat{P}^{(i)}$ on $\mathscr{F}_N$ by (2.5), which is interpreted as the agent i’s belief in the law of $\{X_n\}$ .

Proposition 3.5. Define $\{\Gamma_n\}$ by (3.16). Then

\begin{equation*} g^{(1)}_n \square \cdots \square g^{(m)}_n(z) = \frac{1}{\Gamma_n}f(\Gamma_nz,\tilde{P}_n) - \frac{1}{\Gamma_n}\log B_n, \end{equation*}

where $ \tilde{P}_n = ( \tilde{P}_{n,0},\ldots, \tilde{P}_{n,d})^\top$ and

\begin{equation*} \tilde{P}_{n,j} = \frac{1}{B_n}\prod_{i=1}^m \bigl(\hat{P}^{(i)}_{n,j}\bigr)^{\Gamma_n/\Gamma_n^{(i)}}, \quad B_n = \sum_{j=0}^d \prod_{i=1}^m \bigl(\hat{P}^{(i)}_{n,j}\bigr)^{\Gamma_n/\Gamma_n^{(i)}}. \end{equation*}

Proof. The case $m=2$ follows by solving the equation

\begin{equation*} \nabla f\bigl(\Gamma^{(1)}_nx,\hat{P}^{(1)}_n\bigr) = \nabla f\bigl(\Gamma^{(2)}_n(z-x),\hat{P}^{(2)}_n\bigr)\end{equation*}

in $x \in \mathbb{R}^d$ ; see Lemma A.1. The general case then follows by induction.

By Proposition 3.5, the representative agent’s market falls into Example 3.3. In particular, when $\Gamma_n = \gamma>0$ (a constant), the market is in general equilibrium if and only if $\tilde{P}_n =\hat{P}_n$ , where

\begin{equation*} \frac{\mathrm{d}\mathbb{Q}}{\mathrm{d}\hat{\mathbb{P}}}= \frac{{\mathrm{e}}^{-\gamma H} \prod_{n=1}^N B_n }{\hat{\mathbb{E}}[{\mathrm{e}}^{-\gamma H} \prod_{n=1}^N B_n ]}. \end{equation*}

To highlight the outcome of heterogeneous beliefs, let us further assume there are only two agents ( $m=2$ ) with constant risk aversion $\Gamma^{(i)}_n = \gamma_i>0$ and with no endowment ( $H=0$ ). In this case, $\nabla g^{(i)}(0,\mathbb{Q}) = 0$ , and so, if the two agents have a common belief $\hat{\mathbb{P}}$ , we need $\hat{\mathbb{P}}=\mathbb{Q}$ for the market to be in general equilibrium. The optimal strategies are simply $\pi^{(1)}_n = \pi^{(2)}_n = 0$ . On the other hand, for any $\Delta_d^\circ$ -valued deterministic sequence $\{\hat{P}^{(1)}_n\}$ , by choosing $\hat{P}^{(2)}_n$ as

\begin{equation*}\hat{P}^{(2)}_{n,j} = \frac{(\hat{P}^{(1)}_{n,j})^{-\gamma_2/\gamma_1}}{\sum_{k=0}^d (\hat{P}^{(1)}_{n,k})^{-\gamma_2/\gamma_1}},\end{equation*}

we have $\tilde{P}_{n,j} = 1/(d+1)$ for all n and j, which makes this market with heterogeneous beliefs be in general equilibrium. The individual optimal strategies $\pi^{(1)}_n = - \pi^{(2)}_n$ are non-zero; the agents bet on their beliefs.

Another observation is that the equilibrium return is mostly affected by the belief of the least risk-averse agent. Indeed, if $\Gamma^{(1)}_n \ll \Gamma^{(i)}_n$ for $i \geq 2$ , we have $\Gamma_n/\Gamma_n^{(1)} \approx 1$ , while $\Gamma_n/\Gamma_n^{(i)} \approx 0$ for $i \geq 2$ . Therefore $\tilde{P}_n \approx \hat{P}^{(1)}_n$ .