We consider a stochastic game-theoretic model of a discrete-time asset market with short-lived assets and endogenous asset prices. We prove that the strategy which invests in the assets proportionally to their expected relative payoffs asymptotically minimizes the expected time needed to reach a large wealth level. The result is obtained under the assumption that the relative asset payoffs and the growth rate of the total payoff during each time period are independent and identically distributed.
We consider a continuous-time game-theoretic model of an investment market with short-lived assets and endogenous asset prices. The first goal of the paper is to formulate a stochastic equation which determines wealth processes of investors and to provide conditions for the existence of its solution. The second goal is to show that there exists a strategy such that the logarithm of the relative wealth of an investor who uses it is a submartingale regardless of the strategies of the other investors, and the relative wealth of any other essentially different strategy vanishes asymptotically. Thi…
We consider a game-theoretic model of a market where investors compete for payoffs yielded by several assets. The main result consists in a proof of the existence and uniqueness of a strategy, called relative growth optimal, such that the logarithm of the share of its wealth in the total wealth of the market is a submartingale for any strategies of the other investors. It is also shown that this strategy is asymptotically optimal in the sense that it achieves the maximal capital growth rate when compared to competing strategies. Based on the results obtained, we study the asymptotic structure …
Optimization methods are used to determine equilibria of investment in cryptocurrencies. The basic assumptions involve existence of a core group (the "wealthy") that fears the loss of substantial assets through government seizure. Speculators constitute another group that tends to introduce volatility and risk for the wealthy. The wealthy must divide their assets between the home currency and the cryptocurrency, while the government decides on the probability of seizing a fraction the assets of this group. Under the assumption that each group exhibits risk aversion through a utility function, …
We construct a diffusion approximation of a repeated game in which agents make bets on outcomes of i.i.d. random vectors and their strategies are close to an asymptotically optimal strategy. This model can be interpreted as trading in an asset market with short-lived assets. We obtain sufficient conditions for a strategy to maintain a strictly positive share of total wealth over the infinite time horizon. For the game with two players, we find necessary and sufficient conditions for the wealth share process to be transient or recurrent in this model, and also in its generalization with Markovi…
This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct express…
The investment risk minimization problem with budget and return constraints has been the subject of research using replica analysis but there are shortcomings in the extant literature. With respect to Tobin's separation theorem and the capital asset pricing model, it is necessary to investigate the implications of a risk-free asset and examine its influence on the optimal portfolio. Accordingly, in this work, we explore the investment risk minimization problem in the presence of a risk-free asset with budget and return constraints. Moreover, we discuss opportunity loss, the Pythagorean theorem…
This thesis mainly focuses on two problems in capital structure and individual's life-cycle portfolio choice. In the first problem, we derive a stochastic control model to optimize banks' dividend and recapitalization policies and calibrate that to a sample of U.S. banks in the situation where we model banks' true accounting asset values as partially observed variables due to the opaqueness in banks' assets. By the calibrated model, the noise in reported accounting asset values hides about one-third of the true asset return volatility and raises the banks' market equity value by 7.8\% because …
This paper investigates a hybrid stochastic differential reinsurance and investment game between one reinsurer and two insurers, including a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium and invest its wealth in a financial market that contains a risk-free asset and a risky asset. The two insurers, as the followers of the Stackelberg game, can purchase proportional reinsurance from the reinsurer and invest in the same financial market. The competitive relationsh…
We study T. Cover's rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to a $\$1$ deposit into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e. $200\%$ stocks and $-100\%$ bonds) and then continuously executing rebalancing trades to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalan…
We consider a stochastic game-theoretic model of an investment market in continuous time with short-lived assets and study strategies, called survival, which guarantee that the relative wealth of an investor who uses such a strategy remains bounded away from zero. The main results consist in obtaining a sufficient condition for a strategy to be survival and showing that all survival strategies are asymptotically close to each other. It is also proved that a survival strategy allows an investor to accumulate wealth in a certain sense faster than competitors.
We call an investment strategy survival, if an agent who uses it maintains a non-vanishing share of market wealth over the infinite time horizon. In a discrete-time multi-agent model with endogenous asset prices determined through a short-run equilibrium of supply and demand, we show that a survival strategy can be constructed as follows: an agent should assume that only their actions determine the prices and use a growth optimal (log-optimal) strategy with respect to these prices, disregarding the actual prices. Then any survival strategy turns out to be close to this strategy asymptotically.…
We study the ex-ante minimization of market inefficiency, defined in terms of minimum deviation of market prices from fundamental values, from a centralized planner's perspective. Prices are pressured from exogenous trading actions of leverage targeting banks, which rebalance their portfolios in response to asset shocks. We characterize market inefficiency in terms of two key drivers, the banks' systemic significance and the statistical moments of asset shocks, and develop an explicit expression for the matrix of asset holdings which minimizes such inefficiency. Our analysis shows that to redu…
In this paper, we consider a frequency-based portfolio optimization problem with $m \geq 2$ assets when the expected logarithmic growth (ELG) rate of wealth is used as the performance metric. With the aid of the notion called dominant asset, it is known that the optimal ELG level is achieved by investing all available funds on that asset. However, such an "all-in" strategy is arguably too risky to implement in practice. Motivated by this issue, we study the case where the portfolio weights are chosen in a rather ad-hoc manner and a buy-and-hold strategy is subsequently used. Then we show that,…
A new approach in stochastic optimization via the use of stochastic gradient Langevin dynamics (SGLD) algorithms, which is a variant of stochastic gradient decent (SGD) methods, allows us to efficiently approximate global minimizers of possibly complicated, high-dimensional landscapes. With this in mind, we extend here the non-asymptotic analysis of SGLD to the case of discontinuous stochastic gradients. We are thus able to provide theoretical guarantees for the algorithm's convergence in (standard) Wasserstein distances for both convex and non-convex objective functions. We also provide expli…
