We present a mean field game model to study the question of how centralization of reward and computational power occur in Bitcoin-like cryptocurrencies. Miners compete against each other for mining rewards by increasing their computational power. This leads to a novel mean field game of jump intensity control, which we solve explicitly for miners maximizing exponential utility, and handle numerically in the case of miners with power utilities. We show that the heterogeneity of their initial wealth distribution leads to greater imbalance of the reward distribution, or a ``rich get richer'' effect. This concentration phenomenon is aggravated by a higher bitcoin mining reward, and reduced by competition. Additionally, an advantaged miner with cost advantages such as access to cheaper electricity, contributes a significant amount of computational power in equilibrium, unaffected by competition from less efficient miners. Hence, cost efficiency can also result in the type of centralization seen among miners of cryptocurrencies.

## Cryptocurrencies, Mining & Mean Field Games

## Optimal trade execution in an order book model with stochastic liquidity parameters

We analyze an optimal trade execution problem in a financial market with stochastic liquidity. To this end we set up a limit order book model in which both order book depth and resilience evolve randomly in time. Trading is allowed in both directions. In discrete time, we discuss an explicit recursion that, under certain structural assumptions, characterizes minimal execution costs and observe some qualitative differences with related models. In continuous time, due to the stochastic dynamics of the order book depth and resilience, optimal execution strategies are typically of infinite variation, and the first thing to be discussed it how to extend the state dynamics and the cost functional to allow for general semimartingale strategies. We then derive a quadratic BSDE that under appropriate assumptions characterizes minimal execution costs, identify conditions under which an optimal execution strategy exists and, finally, illustrate our findings in several examples. This is a joint work with Julia Ackermann and Thomas Kruse.

## A cross-border market model

On the XBID-market 13 European countries can trade electricity between each other. Like other intraday electricity markets, this is handled using a limit order book. However, cross-border trading is limited via the total amount of available transmission capacities during a trading session. We present a cross-border market model between two countries and want to give insight into the interactions on this market. We introduce a so-called reduced-form representation of the market and a capacity process which may restrict cross-border trades in each direction. Assuming that the capacity process is non-restricted, we are able to derive heavy traffic approximations of the standing volumes and the capacity process. We will further motivate a candidate for the heavy traffic approximation of the restricted market model.

## Finance and Statistics: Trading Analogies for Sequential Learning

The goal of sequential learning is to draw inference from data that is gathered gradually through time. This is a typical situation in many applications, including finance. A sequential inference procedure is `anytime-valid’ if the decision to stop or continue an experiment can depend on anything that has been observed so far, without compromising statistical error guarantees. A recent approach to anytime-valid inference views a test statistic as a bet against the null hypothesis. These bets are constrained to be supermartingales - hence unprofitable - under the null, but designed to be profitable under the relevant alternative hypotheses. This perspective opens the door to tools from financial mathematics. In this talk I will discuss how notions such as supermartingale measures, log-optimality, and the optional decomposition theorem shed new light on anytime-valid sequential learning. (This talk is based on joint work with Wouter Koolen (CWI), Aaditya Ramdas (CMU) and Johannes Ruf (LSE).)

## Pathwise Gaussian Volterra processes in Hilbert space

We discuss a rough volatility model with fractional drift and noise allowing for more flexibility in modelling roughness. Motivated by an extension to infinite stochastic volatility models for commodity futures markets, we are led to a study of Gaussian Volterra processes. We suggest a definition of a pathwise stochastic integral based on combining the regularity of the kernel and the covariance of the noise. Likewise, we define pathwise integration with respect to multi-parameter covariance-like functions, and apply this to derive an explicit representation of the covariance of the Gaussian Volterra process. This is joint work with Fabian Harang (Oslo).

## Weak Approximations and VIX Option Prices Expansions in Rough Forward Variances Models

## Portfolio Liquidation Games with Self-Exciting Order Flow

We analyze novel portfolio liquidation games with self-exciting order flow. Both the $N$-player game and the mean-field game are considered. We assume that players' trading activities have an impact on the dynamics of future market order arrivals thereby generating an additional transient price impact. Given the strategies of her competitors each player solves a mean-field control problem. We characterize open-loop Nash equilibria in both games in terms of a novel mean-field FBSDE system with unknown terminal condition. Under a weak interaction condition we prove that the FBSDE systems have unique solutions. Using a novel sufficient maximum principle that does not require convexity of the cost function we finally prove that the solution of the FBSDE systems do indeed provide existence and uniqueness of open-loop Nash equilibria. The talk is based on joint work with Guanxing Fu and Ulrich Horst.

## Set Values of Mean Field Games

When a mean field game satisfies certain monotonicity conditions, the mean field equilibrium is unique and the corresponding value function satisfies the so called master equation. In general, however, there can be multiple equilibriums, and in the literature one typically studies the asymptotic behaviors of individual equilibriums of the corresponding $N$-player game. We instead study the set of values over all (mean field) equilibriums, which we call the set value of the game. We shall establish two crucial properties of the set value: (i) the dynamic programming principle; (ii) the convergence of the set values from the $N$-player game to the mean field game. We emphasize that the set value is very sensitive to the choice of the admissible controls. For the dynamic programming principle, one needs to use closed loop controls (not open loop controls) and it involves some very subtle path dependence issue. For the convergence, one has to restrict to the same type of equilibriums for the $N$-player game and for the mean field game. The talk is based on a joint work with Zach Feinstein and Birgit Rudloff and another ongoing joint work with Melih Iseri.

## On Set-valued Backward SDEs and Related Issues in Set-valued Stochastic Analysis

In this talk we try to establish an analytic framework for studying Set-Valued Backward Stochastic Differential Equations (SVBSDE for short), motivated largely by the current studies of dynamic set-valued risk measures for multi-asset or network-based financial models. Our framework will be based on the notion of Hukuhara difference between sets, in order to compensate the lack of “inverse” operation of the traditional Minkowski addition, whence the vector space structure, in traditional set-valued analysis. We shall examine and establish a useful foundation of set-valued stochastic analysis under this algebraic framework, and identify the challenges that may arise in the study of SVBSDEs.

## A simple microstructural explanation of the concavity of price impact

I will present a simple model of market microstructure which explains the concavity of price impact. In the proposed model, the local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear, with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact on midprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The main practical conclusion of the model is that, throughout a meta-order, the volumes at the best bid and ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two more concrete predictions of the model, one of which can be tested using publicly available market data and does not require the (difficult to obtain) information about meta-orders. I will present the theoretical results and will support them with the empirical analysis.