Regulation, Volatility and Efficiency in Continuous-Time Markets

# Regulation, Volatility and Efficiency in Continuous-Time Markets

Arman C. Kizilkale and Shie Mannor Arman C. Kizilkale is with the Department of Electrical and Computer Engineering at McGill University, Montreal, Canada. arman@cim.mcgill.caShie Mannor is with the Department of Electrical Engineering at the Technion, Haifa, Israel. shie@ee.technion.ac.il
###### Abstract

We analyze the efficiency of markets with friction, particularly power markets. We model the market as a dynamic system with the demand process and the supply process. Using stochastic differential equations to model the dynamics with friction, we investigate the efficiency of the market under an integrated expected undiscounted cost function solving the optimal control problem. Then, we extend the setup to a game theoretic model where multiple suppliers and consumers interact continuously by setting prices in a dynamic market with friction. We investigate the equilibrium, and analyze the efficiency of the market under an integrated expected social cost function. We provide an intriguing efficiency–volatility no-free-lunch trade-off theorem.

## I Introduction

#### Iv-D2 Numerical Simulation

Here we present a couple of simulations showing the dynamics when and . The high volatility in Fig. 4 compared to the low volatility in Fig. 4 can be observed. One can also notice the effect of volatility on stability.

Also in Fig. 4 the optimal actions of the regulator agent can be observed at 4 points on the trajectory. At P1, the demand goes up due to stochasticity and the regulator acts with full force to increase the price, so that stability can be obtained. At P2, price gets high, and the demand is taken under control; gradually the regulator decreases the price. Between 60 seconds and 80 seconds, we see that supply follows a higher level than the demand. The regulator acts to take the price down to a local minimum at P3. Then, until P4 the regulator gradually increases the price until it comes to a local maximum at P4.

Now we present two more simulations with . The effect of the initial state on the trajectory is observed here. In Fig. 6 initially, demand is higher than the supply, whereas in Fig. 6 demand is lower than the supply. As expected, the price process becomes very volatile in early stages to stabilise the market.

Finally, we present an experimental result showing the relation between and the average absolute difference between supply and demand dynamics. Recall that high costs are paid when this difference is high, and as seen in Fig. 7, as increases the average absolute difference increases. The axis is drawn on a logarithmic scale in order to capture the graph on lower values of .

## V Decentralized Control Formulation

We define a continuous dynamic game for consumers and suppliers. The agents continuously submit their bids as price-quantity graphs, and the system announces the resulting price. Agents buy or sell corresponding shares of supplies according to their bids. One important notion is that future demand and supply processes are dependent on the price process, which is determined instantly by the agents’ price-quantity graphs shaped by their actions.

We have the set of agents . We define the family of three dimensional state processes for the consumers and two dimensional state processes for the suppliers. The initial conditions are mutually independently distributed bounded random variables which are independent of the standard Wiener processes . The process is the demand dynamics for agent , the process is the supply it receives, and the process is the parameter it applies to its pre-announced price-quantity graph function . For the supplier side is the current supply and is the parameter for the price-quantity graph . Here and are the price-quantity graphs that the consumers and the suppliers submit to the market clearing price functional for the instant price determination. The dynamics for the consumers and the suppliers for are given as

 ddit= fdi(dit,pt,ϕdi(pt;pdit))dt+σddwdit,1≤i≤Nd, (22) dpdit= uditdt,1≤i≤Nd, dsit= fsi(sit,pt,ϕsi(pt;psit))dt+σsdwsit,1≤i≤Ns, dpsit= usitdt,1≤i≤Ns, pt= fm({ϕdi(⋅,⋅),1≤i≤Nd;ϕsj(⋅,⋅),1≤j≤Ns}).

The actions of the agents control the size of the increments for . The functional is allowed to be a function of and , values of the demand of the consumer agent , the price and its price-quantity graph; and is allowed to be a functional of , values of the supply of the supplier , the price and its price-quantity graph.

Following [5], the individual loss function of a consumer and a supplier are defined respectively:

 gd(⋅)= pt⋅sdit−v⋅min(dit,sdit)+cbo(sdit−dit), (23) gs(⋅)= c(sit)−pt⋅sit.

Finally, the cost functions associated with each consumer, each supplier and corresponding control actions , and , are specified to be

 Jd(di0,sdi0,p0,udi)= E∫T0[pt⋅sdit−v⋅min(dit,sdit)+cbo(sdit−dit)]dt,1≤i≤Nd, (24) Js(si0,p0,usi)= E∫T0[c(sit)−pt⋅sit]dt,1≤i≤Ns.

We employ AII for initial values and the disturbance processes, and AII on the functions and . Moreover,

A5: , is a strictly decreasing function of , whereas , is strictly increasing. The price-quantity graphs for the consumers are decreasing functions of in the form of , whereas the price-quantity graphs are increasing in the form of , for the suppliers. Functions and are Lipschitz continuous on with Lipschitz constants , and . Consequently, for some , the market clearing price function is a linear function in the form of .

This assumption limits the model to a price process parameterized by and obtained by price-quantity graph functions submitted by the consumer and supplier agents: