Price formation and optimal trading in intraday electricity markets

Abstract

We develop a tractable equilibrium model for price formation in intraday electricity markets in the presence of intermittent renewable generation. Using stochastic control theory, we identify the optimal strategies of agents with market impact and exhibit the Nash equilibrium in closed form for a finite number of agents as well as in the asymptotic framework of mean field games. Our model reproduces the empirical features of intraday market prices, such as increasing price volatility at the approach of the delivery date and the correlation between price and renewable infeed forecasts, and relates these features with market characteristics like liquidity, number of agents, and imbalance penalty.

Appendix

1.1 Proof of Theorem 1

Proof

Step 1. First order condition of optimality for a single agent In this step, we are going to show that for fixed $\phi ^{-i*}$, the strategy $\phi ^{i*}$ satisfies (4) if and only if ${\mathbb {E}}[\phi ^{i*}_T-X^i_T|{\mathcal {F}}_0]=0$ and there exists a square integrable ${\mathbb {F}}$-martingale $Y^i$ such that, almost surely,

$$\begin{aligned}&Y^i_t + \alpha (t){\left\{ {{\dot{\phi }}}^{i*}_t\left( 1-\frac{b_N}{N}\right) + b_N\dot{{\bar{\phi }}}^{N*}_t\right\} } \nonumber \\&\qquad + S_t + a\big ({\bar{\phi }}^{N*}_t-{\bar{\phi }}^{N*}_0\big ) - \frac{a}{N}\big (\phi ^{i*}_t-\phi ^{i*}_0\big ) = 0,\ 0\le t\le T, \nonumber \\&Y^i_T = \frac{a}{N} \big (\phi ^{i*}_T-\phi ^{i*}_0\big ) + \lambda \big (\phi ^{i*}_T - X^i_T\big )-S_T, \end{aligned}$$

(22)

with the shorthand notation $b_N = \frac{N}{N-1}\frac{b}{2}$. Assume that $\phi ^{i*}$ satisfies (4). Then, for any adapted square integrable process $(\nu _t)_{0\le t\le T}$, and for any $\delta \in {\mathcal {F}}_0$,

$$\begin{aligned} J^{N,i}\left( \phi ^{i*} +\delta + \int _0^\cdot \nu _s ds,\phi ^{-i*}\right) \le J^{N,i}\big (\phi ^{i*}, \phi ^{-i*}\big ). \end{aligned}$$

Developing the expressions, this is equivalent to

$$\begin{aligned}&{\mathbb {E}}\Big [\int _0^T \nu _t \Big \{\alpha (t){\Big [{{\dot{\phi }}}^{i*}_t\Big (1-\frac{b_N}{N}\Big ) + b_N\dot{{\bar{\phi }}}^{N*}_t\Big ] }+ S_t + a({\bar{\phi }}^{N*}_t-{\bar{\phi }}^{N*}_0) {- \frac{a}{N}\big (\phi ^{i*}_t-\phi ^{i*}_0\big )}\Big \}dt \\&\quad + \left( \frac{a}{N} \big (\phi ^{i*}_T-\phi ^{i*}_0\big ) + \lambda \big (\phi ^{i*}_T - X^i_T\big )-S_T\right) \int _0^T \nu _t dt +\lambda \big (\phi ^{i*}_T - X^i_T\big )\delta \Big ] \\&\quad +{\mathbb {E}}\Big [\frac{1}{2}\int _0^T \alpha (t)\nu _t^2 dt + \Big (\frac{a}{N} + \frac{\lambda }{2}\Big )\Big (\int _0^T \nu _t dt\Big )^2 + \frac{\lambda }{2}\delta ^2 \Big ] \ge 0,\nonumber \\ \end{aligned}$$

and since $\nu$ and $\delta$ are arbitrary, we see that optimality is equivalent to

$$\begin{aligned}&{\mathbb {E}}\Big [\int _0^T \nu _t \Big \{\alpha (t){\Big [{{\dot{\phi }}}^{i*}_t\Big (1-\frac{b_N}{N}\Big ) + b_N\dot{{\bar{\phi }}}^{N*}_t\Big ]} + S_t + a\big ({\bar{\phi }}^{N*}_t-{\bar{\phi }}^{N*}_0\big ) {- \frac{a}{N}\big (\phi ^{i*}_t-\phi ^{i*}_0\big )}\Big \}dt\nonumber \\&\quad + \Big (\frac{a}{N} \big (\phi ^{i*}_T-\phi ^{i*}_0\big )+ \lambda \big (\phi ^{i*}_T - X^i_T\big )-S_T\Big )\int _0^T \nu _t dt \Big ]=0, \end{aligned}$$

(23)

for any adapted square integrable $\nu$, together with the condition that ${\mathbb {E}}[\phi ^{i*}_T - X^i_T|{\mathcal {F}}_0] = 0$. Now, assume that $Y^i$ is a square integrable martingale satisfying (22). Then, by integration by parts, the expression in the previous line equals

$$\begin{aligned} {\mathbb {E}}\Big [-\int _0^T \nu _t Y_t dt + Y_T \int _0^T \nu _t dt\Big ] = {\mathbb {E}}\Big [\int _0^T \Big (\int _0^t \nu _s ds\Big )dY_t\Big ] = 0, \end{aligned}$$

and we see that the optimality condition is satisfied. Conversely, assume that (23) is satisfied for any adapted square integrable process $\nu$, and let $Y^i$ be a martingale such that

$$\begin{aligned} Y^i_T = \frac{a}{N} \big (\phi ^{i*}_T-\phi ^{i*}_0\big ) + \lambda \big (\phi ^{i*}_T - X^i_T\big )-S_T. \end{aligned}$$

Then, by integration by parts, (23) is equivalent to

$$\begin{aligned}&{\mathbb {E}}\Big [\int _0^T \nu _t \Big \{\alpha (t){\Big [{{\dot{\phi }}}^{i*}_t\Big (1-\frac{b_N}{N}\Big ) + b_N\dot{{\bar{\phi }}}^{N*}_t\big ] } \\&\quad + S_t + a\big ({\bar{\phi }}^{N*}_t-{\bar{\phi }}^{N*}_0\big ) {- \frac{a}{N}\big (\phi ^{i*}_t}-\phi ^{i*}_0\big )+ Y^i_t\Big \}dt\Big ] = 0, \end{aligned}$$

and since $\nu$ is arbitrary, we see that (22) is satisfied.

Step 2. Computing the average position Let $(\phi ^{i*})_{i = 1, \dots , N}$ be a Nash equilibrium. We have seen that this is equivalent to (22) together with the condition that ${\mathbb {E}}[\phi ^{i*}_T - X^i_T|{\mathcal {F}}_0] = 0$ for $i=1,\dots ,N$. Summing up these expressions for $i=1,\dots ,N$ and denoting ${\overline{Y}}^N_t = \frac{1}{N}\sum _{i=1}^N Y^i_t$, we get

$$\begin{aligned}&{\overline{Y}}^N_t + \alpha (t){\Big (1+\frac{b}{2}\Big )}\dot{{\bar{\phi }}}^{N*}_t + S_t + a{\frac{N-1}{N}}\big ({\bar{\phi }}^{N*}_t-{\bar{\phi }}^{N*}_0\big ) = 0,\quad 0\le t\le T,\\&{\overline{Y}}^N_T = \frac{a}{N} \big ({\bar{\phi }}^{N*}_T-{\bar{\phi }}^{N*}_0\big ) + \lambda \big ({\bar{\phi }}^{N*}_T - {\overline{X}}^N_T\big )-S_T,\\&{\mathbb {E}}\big [{\bar{\phi }}^{N*}_T - {\overline{X}}^N_T|{\mathcal {F}}_0\big ] = 0. \end{aligned}$$

The first equation can be solved explicitly for ${\bar{\phi }}^{N*}$:

$$\begin{aligned} {\bar{\phi }}^{N*}_t ={\bar{\phi }}^{N*}_0 -\int _0^t \eta ^{N}_{s,t} \frac{{\overline{Y}}^N_s + S_s}{\alpha (s){\left( 1+\frac{b}{2}\right) }}ds. \end{aligned}$$

(24)

Denoting ${\hat{\phi }}_t: = {\bar{\phi }}^{N*}_t + I^N_t$, we obtain simplified equations:

$$\begin{aligned} {\hat{\phi }}_t&= {\bar{\phi }}^{N*}_0-\int _0^t \eta ^{N}_{s,t} \frac{{\overline{Y}}^N_s}{\alpha (s){\left( 1+\frac{b}{2}\right) }}ds,\\ {\overline{Y}}^N_T&= \left( \frac{a}{N} +\lambda \right) \big ({\hat{\phi }}_T-I^N_T\big ) -\frac{a}{N}{\bar{\phi }}_0^{N*} - \lambda {\overline{X}}^N_T-S_T. \end{aligned}$$

Substituting ${\hat{\phi }}_T$ into the second equation and taking the expectation, we obtain another linear equation, this time for ${\overline{Y}}^N_t$:

$$\begin{aligned} {\overline{Y}}^N_T&= -\left( \frac{a}{N} +\lambda \right) \int _0^T \eta ^{N}_{s,T} \frac{{\overline{Y}}^N_s}{\alpha (s){\left( 1+\frac{b}{2}\right) }}ds -\left( \frac{a}{N} +\lambda \right) I_T - \lambda {\overline{X}}^N_T + \lambda {\bar{\phi }}^{N*}_0-S_T,\\ {\overline{Y}}^N_t&= -\left( \frac{a}{N}+\lambda \right) \int _0^t {\overline{Y}}^N_s\frac{ \eta ^{N}_{s,T}}{\alpha (s){\left( 1+\frac{b}{2}\right) }} ds - \left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{t,T}{\overline{Y}}^N_t\\&- \left( \frac{a}{N}+\lambda \right) {{\widetilde{I}}}^N_t- \lambda {\overline{X}}^N_t+ \lambda {\bar{\phi }}^{N*}_0 - {{\widetilde{S}}}_t. \end{aligned}$$

By integration by parts, this is equivalent to

$$\begin{aligned} {\overline{Y}}^N_t =&-\left( \frac{a}{N}+\lambda \right) \int _0^t \varDelta ^{N}_{s,T}d{\overline{Y}}^N_s - \left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{0,T}{\overline{Y}}^N_0 \\&- \left( \frac{a}{N}+\lambda \right) {{\widetilde{I}}}_t- \lambda {\overline{X}}^N_t+ \lambda {\bar{\phi }}^{N*}_0 - {{\widetilde{S}}}_t. \end{aligned}$$

Taking $t=0$, we get:

$$\begin{aligned} {\overline{Y}}^N_0=\frac{ - \left( \frac{a}{N}+\lambda \right) {{\widetilde{I}}}^N_0 - \lambda ({\overline{X}}^N_0- {\bar{\phi }}^{N*}_0)-{{\widetilde{S}}}_0}{1+\left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{0,T}} \end{aligned}$$

On the other hand, in differential form,

$$\begin{aligned} \left\{ 1+\left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{t,T}\right\} d{\overline{Y}}^N_t = - \left( \frac{a}{N}+\lambda \right) d{{\widetilde{I}}}^N_t - \lambda d{\overline{X}}^N_t-d{{\widetilde{S}}}_t, \end{aligned}$$

which is solved therefore explicitly by

$$\begin{aligned} {\overline{Y}}^N_t = \frac{ - \left( \frac{a}{N}+\lambda \right) {{\widetilde{I}}}^N_0 - \lambda \big ({\overline{X}}^N_0- {\bar{\phi }}^{N*}_0\big )-{{\widetilde{S}}}_0}{1+\left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{0,T}}- \int _0^t \frac{ \left( \frac{a}{N}+\lambda \right) d{{\widetilde{I}}}^N_s+ \lambda d{\overline{X}}^N_s+d{{\widetilde{S}}}_t}{1+\left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{s,T}} \end{aligned}$$

(25)

Finally

$$\begin{aligned} \begin{aligned} {\bar{\phi }}^{N*}_t&={\bar{\phi }}^{N*}_0 -I^N_t + \int _0^t {\overline{Y}}^N_s d\varDelta ^{N}_{s,t} ={\bar{\phi }}^{N*}_0-I^N_t - {\overline{Y}}^N_0 \varDelta ^{N}_{0,t} - \int _0^t \varDelta ^{N}_{s,t} d {\overline{Y}}^N_s\\&= {\bar{\phi }}^{N*}_0- I^N_t + \varDelta ^{N}_{0,t}\frac{ \left( \frac{a}{N}+\lambda \right) {{\widetilde{I}}}^N_0 + \lambda \big ({\overline{X}}^N_0-{\bar{\phi }}^{N*}_0\big )+{{\widetilde{S}}}_0}{1+\left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{0,T}} \\&\quad + \int _0^t \varDelta ^{N}_{s,t}\frac{ \left( \frac{a}{N}+\lambda \right) d{{\widetilde{I}}}^N_s+ \lambda d{\overline{X}}^N_s+ d{{\widetilde{S}}}_t}{1+\left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{s,T}}. \end{aligned} \end{aligned}$$

(26)

It remains to compute ${\bar{\phi }}^{N*}_0$ from the condition ${\mathbb {E}}[{\bar{\phi }}^{N*}_T - {\overline{X}}^N_T|{\mathcal {F}}_0] = 0$. Substituting this into the above expression, we find

$$\begin{aligned} {\bar{\phi }}^{N*}_0={\overline{X}}^N_0+\frac{{{\widetilde{I}}}^N_0 - \varDelta ^{N}_{0,T} {{\widetilde{S}}}_0}{1+\frac{a}{N} \varDelta ^{N}_{0,T}} \end{aligned}$$

so that

$$\begin{aligned} {\overline{\phi }}^{N*}_t&= {\overline{X}}^N_0+\frac{1+\frac{a}{N} \varDelta ^{N}_{0,t}}{1+\frac{a}{N} \varDelta ^{N}_{0,T}}\big ({{\widetilde{I}}}^N_0 - \varDelta ^{N}_{0,T} {{\widetilde{S}}}_0\big ) \nonumber \\&- \big (I^N_t-\varDelta ^N_{0,t}{{\widetilde{S}}}_0\big )+ \int _0^t \varDelta ^{N}_{s,t}\frac{ \left( \frac{a}{N}+\lambda \right) d{{\widetilde{I}}}^N_s+ \lambda d{\overline{X}}^N_s+ d{{\widetilde{S}}}_t}{1+\left( \frac{a}{N}+\lambda \right) \varDelta ^{N}_{s,T}}. \end{aligned}$$

(27)

Step 3: computing the position of the agent Let ${{\check{\phi }}}^{i*}_t:= \phi ^{i*}_t - {\bar{\phi }}^{N*}_t$, ${\check{X}}^i_t = X^i_t - {\overline{X}}^N_t$ and ${\check{Y}}^i_t:= Y^i_t - {\overline{Y}}^N_t$. Then, ${\check{Y}}^i$ is an ${\mathbb {F}}$-martingale and satisfies

$$\begin{aligned} {\check{Y}}^i_T = \frac{a}{N}{{\check{\phi }}}^{i*}_T + \lambda \big ({{\check{\phi }}}^{i*}_T -{\check{X}}^i_T\big ),\qquad {\check{Y}}^i_t = -\alpha (t) \dot{{{\check{\phi }}}}^{i*}_t {+ \frac{a}{N}{{\check{\phi }}}^{i*}_t.} \end{aligned}$$

together with the additional condition ${\mathbb {E}}[{{\check{\phi }}}^{i*}_T - {\check{X}}^i_T|{\mathcal {F}}_t] = 0$. Similarly to the second part, this system admits an explicit solution:

$$\begin{aligned} {\check{Y}}^i_t = -\frac{\lambda ({\check{X}}^i_0-{{\check{\phi }}}^{i*}_0)}{1+ \left( \frac{a}{N}+\lambda \right) {\widetilde{\varDelta }}^{N}_{0,T}} -\int _0^t\frac{\lambda d{\check{X}}^i_s}{1+ \left( \frac{a}{N}+\lambda \right) {\widetilde{\varDelta }}^{N}_{s,T}}. \end{aligned}$$

and

$$\begin{aligned} {{\check{\phi }}}^{i*}_t = {\check{X}}^i_0 + \int _0^t {\widetilde{\varDelta }}^{N}_{s,t} \frac{\lambda d{\check{X}}^i_s}{1+ \left( \frac{a}{N}+\lambda \right) {\widetilde{\varDelta }}^{N}_{s,T}} . \end{aligned}$$

$\square$

Remark 1

The optimal strategy is obtained by solving a series of linear equations from an equivalent characterization of optimality. Since all equations admit unique solutions, the equilibrium strategy is unique. This is a consequence of the strict concavity of the objective function in our linear quadratic setting.

1.2 Proofs of Propositions 1 and 2

Proof of Proposition 1

For all $t \in [0,T]$, we define:

$$\begin{aligned}&g^N_{s,t} =\frac{\varDelta ^{N}_{s,t}}{\big (1+\big (\frac{a}{N}+\lambda \big )\varDelta ^{N}_{s,T}\big )}, \quad g_{s,t} =\frac{\varDelta _{s,t}}{\big (1+\lambda \varDelta _{s,T}\big )}, \\&{\tilde{g}}^{N}_{s,t} =\frac{{\widetilde{\varDelta }}^{N}_{s,t}}{\big (1+\big (\frac{a}{N}+\lambda \big ){\widetilde{\varDelta }}^N_{s,T}\big )}, \quad {\tilde{g}}_{s,t} =\frac{{\widetilde{\varDelta }}_{s,t}}{\big (1+\lambda {\widetilde{\varDelta }}_{s,T}\big )}, \end{aligned}$$

so that, for some constant C depending only on the parameters a, b, $\alpha$ and $\lambda$, but not on other ingredients of the model,

$$\begin{aligned} |\varDelta ^N_{s,t}-\varDelta _{s,t}|+|g^N_{s,t}-g_{s,t}| + |{\tilde{g}}^N_{s,t} - {\tilde{g}}_{s,t}| \le \frac{C}{N}. \end{aligned}$$

for all $s,t\in [0,T]$. Now, let us consider the optimal strategies $(\phi _t^{i*})_{t\in [0,T]}$ and $(\phi _t^{MF,i*})_{t\in [0,T]}$ of the generic agent i respectively in the N-player setting and the mean field setting. Fix $t \in [0,T]$. Then,

$$\begin{aligned} \phi _t^{i*}-\phi _t^{MF,i*} =&\frac{a}{N}\frac{\varDelta ^N_{0,t}-\varDelta ^N_{0,T}}{1+\frac{a}{N}\varDelta ^N_{0,T}}\big ({{\widetilde{I}}}^N_0 - {{\widetilde{S}}}_0 \varDelta ^N_{0,T}\big ) + {{\widetilde{I}}}^N_0 - {{\widetilde{I}}}_0 + {{\widetilde{S}}}_0\big (\varDelta _{0,T}-\varDelta ^N_{0,T}\big )\\&- I^N_t+ I_t + {{\widetilde{S}}}_0\big (\varDelta ^N_{0,t}-\varDelta _{0,t}\big )+ \int _0^t \big (g^N_{s,t}-g_{s,t}\big )\\&\left\{ \left( \frac{a}{N}+\lambda \right) d{{\widetilde{I}}}^N_s+ \lambda d{\overline{X}}^N_s + d{{\widetilde{S}}}_s\right\} \\&+\lambda \int _0^t g_t(s) d\big ( {{\widetilde{I}}}^N_s-{{\widetilde{I}}}_s+ {\overline{X}}^N_s-{\overline{X}}_s\big )\\&+\int _0^t \lambda \big ({\tilde{g}}^N_{s,t}-{\tilde{g}}_{s,t}\big ) d\big (X^i_s - {\overline{X}}^N_s\big ) + \lambda \int _0^t {\tilde{g}}_{s,t} d\big ( {\overline{X}}_s- {\overline{X}}^N_s\big ) \end{aligned}$$

Therefore, for some constant C depending only on the parameters a, b, $\alpha$ and $\lambda$, but not on other ingredients of the model,

$$\begin{aligned} {\mathbb {E}}\big [\big (\phi _t^{i*}-\phi _t^{MF,i*} \big )^2\big ]\le&\frac{C}{N^2} {\mathbb {E}}\big [\big ({{\widetilde{I}}}^N_0\big )^2\big ] + \frac{C}{N^2} {\mathbb {E}}\big [\big ({{\widetilde{S}}}_0\big )^2\big ] + {\mathbb {E}}\big [\big ({{\widetilde{I}}}^N_0 - {{\widetilde{I}}}_0\big )^2\big ] + {\mathbb {E}}\big [\big (I^N_t - I_t\big )^2\big ] \\&+ \frac{C}{N^2} {\mathbb {E}}[({{\widetilde{I}}}^N_t)^2] + C{\mathbb {E}}\big [\big ({{\widetilde{I}}}^N_t - {{\widetilde{I}}}_t\big )^2\big ] +\frac{C}{N^2}{\mathbb {E}}\big [{{\widetilde{S}}}_t^2\big ]\\&+ \frac{C}{N^2} {\mathbb {E}}\big [\big ({\overline{X}}_t^N\big )^2\big ] + C{\mathbb {E}}\big [\big ({\overline{X}}^N_t - {\overline{X}}_t\big )^2\big ] + \frac{C}{N^2} {\mathbb {E}}\big [\big ({\check{X}}^i_t\big )^2\big ]\\ \le&{\mathbb {E}}\big [\big (I^N_t - I_t\big )^2\big ] + \frac{C}{N^2} {\mathbb {E}}\big [\big ( I^N_T\big )^2\big ] + C{\mathbb {E}}\big [\big (I^N_T - I_T\big )^2\big ]\\&+\frac{C}{N^2} {\mathbb {E}}[S_T^2]+\frac{C}{N^2} {\mathbb {E}}\big [\big ({\overline{X}}_t\big )^2\big ] + \frac{C}{N^2} \sum _{i=1}^N{\mathbb {E}}\big [\big ({\check{X}}_t^i\big )^2\big ]\\ \le&\frac{C}{N^2} {\mathbb {E}}\left[ \sup _{0\le s\le T} S_s^2\right] + \frac{C}{N^2} {\mathbb {E}}\big [\big ({\overline{X}}_t\big )^2\big ] + \frac{C}{N} {\mathbb {E}}\big [\big ({\check{X}}_t^i\big )^2\big ]. \end{aligned}$$

where the estimate for the first line above is obtained through Jensen’s inequality. The other estimates of the proposition are obtained in a similar way. $\square$

Proof of Proposition 2

To lighten notation, and since we now have only one strategy, we omit in this proof the superscript MF in the candidate strategy $\phi ^{MF,i*}$. The "distance to optimality" for this strategy is estimated as follows.

$$\begin{aligned}&J^{N,i}\big (\phi ^{i},\phi ^{-i*}\big ) - J^{N,i}\big (\phi ^{i*},\phi ^{-i*}\big ) \\&= J^{N,i}\big (\phi ^{i},\phi ^{-i*}\big ) - J^{MF}\big (\phi ^{i},{\bar{\phi }}^*\big )+J^{MF}\big (\phi ^{i},{\bar{\phi }}^*\big ) - J^{MF}\big (\phi ^{i*},{\bar{\phi }}^*\big )\\ {}&\quad + J^{MF}\big (\phi ^{i^*},{\bar{\phi }}^*\big ) - J^{N,i}\big (\phi ^{i*},\phi ^{-i*}\big )\\&\le J^{N,i}\big (\phi ^{i},\phi ^{-i*}\big ) - J^{MF}\big (\phi ^{i},{\bar{\phi }}^*\big ) + J^{MF}\big (\phi ^{i^*},{\bar{\phi }}^*\big ) - J^{N,i}\big (\phi ^{i*},\phi ^{-i*}\big ). \end{aligned}$$

The second difference is estimated as follows:

$$\begin{aligned}&J^{MF}\big (\phi ^{i^*},{\bar{\phi }}^*\big ) - J^{N,i}\big (\phi ^{i*},\phi ^{-i*}\big ) \nonumber \\&\quad = -{\mathbb {E}}\Big [\int _0^T {{\dot{\phi }}}^{i*}_t\Big \{ a\big ({\bar{\phi }}^*_t - {\bar{\phi }}^{N*}_t-{\bar{\phi }}^*_0 + {\bar{\phi }}^{N*}_0\big ) {+ \frac{\alpha (t)b}{2}\big (\dot{{\bar{\phi }}}^*_t-\dot{{\bar{\phi }}}^{N,-i*}_t\big )}\Big \}dt\Big ] \end{aligned}$$

(28)

Since

$$\begin{aligned} \dot{{\bar{\phi }}}^*_t-\dot{{\bar{\phi }}}^{N,-i*}_t = \dot{{\bar{\phi }}}^*_t-\dot{{\bar{\phi }}}^{N*}_t + \frac{{{\dot{\phi }}}^{i*}-\dot{{\bar{\phi }}}^{N*}}{N-1}, \end{aligned}$$

(29)

it follows from the Cauchy–Schwarz inequality and Proposition 1 that

$$\begin{aligned} \big |J^{MF}\big (\phi ^{i^*},{\bar{\phi }}^*\big ) - J^{N,i}\big (\phi ^{i*},\phi ^{-i*}\big ) \big |\le \frac{C}{\sqrt{N}}. \end{aligned}$$

(30)

The first difference admits the following estimate.

$$\begin{aligned}&J^{N,i}\big (\phi ^{i},\phi ^{-i*}\big ) - J^{MF}\big (\phi ^{i},{\bar{\phi }}^*\big ) = a{\mathbb {E}}\Big [\int _0^T {{\dot{\phi }}}^{i}_t \big ({\bar{\phi }}^*_t - {\bar{\phi }}^{N*}_t-{\bar{\phi }}^*_0 + {\bar{\phi }}^{N*}_0\big )dt\Big ]\nonumber \\&- \frac{a}{N}{\mathbb {E}}\Big [\int _0^T {{\dot{\phi }}}^{i}_t \big (\phi ^i_t-\phi ^{i*}_t-\phi ^i_0+\phi ^{i*}_0\big )dt\Big ]+ b{\mathbb {E}} \Big [\int _0^T\alpha (t){{\dot{\phi }}}^{i}_t \big (\dot{{\bar{\phi }}}^*_t-\dot{{\bar{\phi }}}^{N,-i*}_t\big )dt\Big ] \nonumber \\&\le {\mathbb {E}}\Big [\int _0^T \big ({{\dot{\phi }}}^{i}_t\big )^2 dt\Big ]^{\frac{1}{2}}\Big \{a {\mathbb {E}}\Big [\int _0^T \big ({\bar{\phi }}^*_t - {\bar{\phi }}^{N*}_t-{\bar{\phi }}^*_0 + {\bar{\phi }}^{N*}_0\big )^2 dt\Big ]^{\frac{1}{2}} \nonumber \\&\quad + \frac{a}{N} {\mathbb {E}}\Big [\int _0^T \big (\phi ^{i*}_t-\phi ^{i*}_0\big )^2 dt\Big ]^{\frac{1}{2}}+ \frac{b}{N-1} {\mathbb {E}}\Big [\int _0^T \alpha ^2(t)\big ({{\dot{\phi }}}^{i*}_t-\dot{{\bar{\phi }}}^{N*}\big )^2 dt\Big ]^{\frac{1}{2}} \nonumber \\&\quad + b {\mathbb {E}}\Big [ \int _0^T\alpha ^2(t)\big (\dot{{\bar{\phi }}}^*_t-\dot{{\bar{\phi }}}^{N,*}_t\big )^2 \Big ]^\frac{1}{2} \Big \}\le \frac{C_0}{\sqrt{N}}{\mathbb {E}}\Big [\int _0^T ({{\dot{\phi }}}^{i}_t)^2 dt\Big ]^{\frac{1}{2}} \end{aligned}$$

(31)

for some constant $C_0<\infty$, which does not depend on $\phi ^i$, in view of Proposition 1 and (29). On the other hand, the following estimate also holds true.

$$\begin{aligned} J^{N,i}\big (\phi ^i, \phi ^{-i*}\big ) =&- {\mathbb {E}}\Big [ \int _{0}^{T}\Big \{\frac{\alpha (t)}{2}\dot{\phi ^i_t}\Big ({{\dot{\phi }}}^i_t + b \dot{{\bar{\phi }}}^{N,-i*}_t\Big )+\dot{\phi ^i_{t}}\big (S_t + a\big ({\bar{\phi }}^{N*}_t - {\bar{\phi }}^{N*}_0\big ) \Big \}dt \nonumber \\&\quad +\frac{a}{N}\int _0^T{{\dot{\phi }}}^i_t\big (\phi ^i_t - \phi ^i_0 - \phi ^{i*}_t + \phi ^{i*}_0\big )dt +\phi ^i_0 S_0 \nonumber \\&\quad - \big (\phi ^i_T-X^i_T\big )S_T+\frac{\lambda }{2}\big (\phi ^i_{T}- X^i_{T}\big )^2\Big ],\nonumber \\&\le -\frac{{\bar{\alpha }}}{2}{\mathbb {E}}\Big [\int _0^T \big ({{\dot{\phi }}}^i_t\big )^2dt\Big ] - \frac{\lambda }{2}{\mathbb {E}}\big [\big (\phi ^i_T\big )^2\big ]+ C_1 {\mathbb {E}}\Big [\int _0^T \big ({{\dot{\phi }}}^i_t\big )^2dt\Big ]^{1\over 2}\nonumber \\&\quad + C_2 {\mathbb {E}}\big [\big (\phi ^i_T\big )^2\big ]^{1\over 2} + C_3, \end{aligned}$$

(32)

where $\bar{{\bar{\alpha }}} = \max _{0\le t\le T}\alpha (t)$ and

$$\begin{aligned} C_1&= \frac{b}{2} {\mathbb {E}}\Big [\int _0^T\alpha ^2(t) \big (\dot{{\bar{\phi }}}^{N,-i*}_t\big )^2dt\Big ]^{1\over 2} + a {\mathbb {E}}\Big [\int _0^T ({\bar{\phi }}^{N*}_t-{\bar{\phi }}^{N*}_0)^2dt\Big ]^{1\over 2}\\&\quad + \frac{a}{N}{\mathbb {E}}\Big [\int _0^T \big ({\bar{\phi }}^{i*}_t-{\bar{\phi }}^{i*}_0\big )^2dt\Big ]^{1\over 2} + {\mathbb {E}}\Big [\int _0^T \big (S_t-S_0\big )^2\Big ]^{1\over 2},\\ C_2&= {\mathbb {E}}\big [(S_T-S_0)^2\big ]^{1\over 2} + 2 {\mathbb {E}}\big [\big (X^i_T\big )^2\big ]^{\frac{1}{2}},\qquad C_3 = \big |{\mathbb {E}}\big [X^i_T S_T\big ]\big |. \end{aligned}$$

Thus, there exists a constant $C^*<\infty$, which does not depend on $\phi ^i$, such that if

$$\begin{aligned} {\mathbb {E}}\Big [\int _0^T \big ({{\dot{\phi }}}^i_t\big )^2dt\Big ] +{\mathbb {E}}\big [\big (\phi ^i_T\big )^2\big ]>C^*, \end{aligned}$$

then

$$\begin{aligned} J^{N,i}\big (\phi ^i, \phi ^{-i*}\big ) - J^{N,i}\big (\phi ^{i*}, \phi ^{-i*}\big )<0. \end{aligned}$$

Therefore, from (30) and (31) it follows that for any admissible strategy $\phi ^i$,

$$\begin{aligned}&J^{N,i}\big (\phi ^i, \phi ^{-i*}\big ) - J^{N,i}\big (\phi ^{i*}, \phi ^{-i*}\big ) \\&\quad \le {\mathbf {1}}_{{\mathbb {E}}[\int _0^T ({{\dot{\phi }}}^i_t)^2dt ] +{\mathbb {E}}[(\phi ^i_T)^2]\le C^*}\Big \{\frac{C}{\sqrt{N}} + \frac{C_0}{\sqrt{N}} {\mathbb {E}}\Big [\int _0^T\big ({{\dot{\phi }}}_t^i\big )^2 dt\Big ]^{1\over 2}\Big \}\\&\quad \le \frac{C}{\sqrt{N}} + \frac{C_0 \sqrt{C^*}}{\sqrt{N}}. \end{aligned}$$

$\square$