An Optimal Trading Rule Under a Switchable Mean-Reversion Model

This work provides an optimal trading rule that allows buying and selling an asset sequentially over time. The asset price follows a switchable mean-reversion model with a Markovian jump. Such a model can be applied to assets with a “staircase” price behavior and yet is simple enough to allow an analytic solution. The objective is to determine a sequence of trading times to maximize an overall return. The corresponding value functions are characterized by a set of quasi-variational inequalities. A closed-form solution is obtained under suitable conditions. The sequence of trading times can be given in terms of a set of threshold levels. Finally, numerical examples are given to demonstrate the results.     

具有比例摩擦金融市场的一般经济均衡与资产定价(英文)

本文建立具有比例摩擦金融市场的简单两时期模型．经济人具有均值－方差偏好,并且在交易金融资产的过程中支付交易费用．本文证明了两种金融资产的一般经济均衡与资产定价的基本估值公式．     

Smooth solutions to portfolio liquidation problems under price-sensitive market impact

We consider the stochastic control problem of a financial trader that needs to unwind a large asset portfolio within a short period of time. The trader can simultaneously submit active orders to a primary market and passive orders to a dark pool. Our framework is flexible enough to allow for price-dependent impact functions describing the trading costs in the primary market and price-dependent adverse selection costs associated with dark pool trading. We prove that the value function can be characterized in terms of the unique smooth solution to a PDE with singular terminal value, establish its explicit asymptotic behavior at the terminal time, and give the optimal trading strategy in feedback form.     

Robin型二阶m 点边值问题正解的存在性

设 a∈C［0,1］, b∈C(［0,1］,(-∞, 0)). 设\-1(t)为线性边值问题  u″+a(t)u′+b(t)u=0， u′(0)=0,\ u(1)=1  的唯一正解. 该文研究非线性二阶常微分方程m 点边值问题  u″+a(t)u′+b(t)u+h(t) f(u)=0，\= u′(0)=0, u(1)-∑[DD(]m-2[]i=1[DD)]α\-i u(ξ\-i)=d  正解的存在性. 其中 d 为参数, ξ\-i∈(0,1), α\-i∈(0,∞) 为满足 ∑[DD(]m-2[]i=1[DD)]α\-i\-1(ξ\-i)<1的常数, i∈{1,\:，m-2}. 在适当的条件下证得: 存在正常数 d\+*, 使 当0d\+*时无正解.     

Martingale optimal transport and robust hedging in continuous time

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge–Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.     

Generalized Hopf bifurcation in Hilbert space

We consider a family of semilinear evolution equations in Hilbert space of the form with, in general, unbounded operators *A(λ), F(λ·) depending analytically on a real parameter λ. We assume that the origin is a stationary solution, i.e. F(λ,0) = 0, for all λ ε R and that the linearization (with respect to u) at the origin is given by du/dt + A(λ)u = 0. Our essential assumption is the following: A(λ) possesses one pair of simple complex conjugate eigenvalues μ(λ) = Re μ(λ) ± i Im μ(λ) such that Im μ(0) > 0 and for some m ε N or If m = 1 the curves of eigenvalues μ(λ) cross the imaginary axis transversally at ±i Im μ(0). In this case a unique branch of periodic solutions emanates from the origin at λ = 0 which is commonly called Hopf bifurcation. If μ(λ) and the imaginary axis are no longer transversal, i.e. m > 1, we call a bifurcation of periodic solutions, if it occurs, a generalized Hopf bifurcation. It is remarkable that up to m such branches may exist. Our approach gives the number of bifurcating solutions, their direction of bifurcation, and its asymptotic expansion. We regain the results of D. Flockerzi who established them in a completely different way for ordinary differential equations.     

Dual representation of superhedging costs in illiquid markets

This paper studies superhedging of contingent claims in illiquid markets where trading costs may depend nonlinearly on the traded amounts and portfolios may be subject to constraints. We give dual expressions for superhedging costs of financial contracts where claims and premiums are paid possibly at multiple points in time. Besides classical pricing problems, this setup covers various swap and insurance contracts where premiums are paid in sequences. Validity of the dual expressions is proved under new relaxed conditions related to the classical no-arbitrage condition. A new version of the fundamental theorem of asset pricing is given for unconstrained models with nonlinear trading costs.     

Relative entropy maximization and directed diffusion equations

Motivated by the concept of maximum entropy methods in signal and image processing, we introduce and discuss a class of ‘directed diffusion equations’ with suitable boundary conditions. The paradigmatic ‘directed diffusion equation’ is The relative entropy $ Sb[f](t): = - \int_\Omega {f(t,x)} \;\ln \;(f(t,x)/b(x))dx $ is rapidly increasing along solution trajectories of (i). This suggests that solving (i) will yield efficient procedures for entropy maximization. We also discuss the asymptotic behavior of solutions of (i)—this is readily done because (i) has a large family of Ljapunov functionals.     

机构投资者的最优变现策略

在投资、变现等大宗交易过程中,资产交易价格与交易策略密切相关,因此,交易的完成过程需要很高的技巧.文章讨论了机构投资者的最优变现策略问题,假设证券价格服从几何布朗运动,以均值方差效用为目标函数,得到了最优变现策略所满足的二阶微分方程,并由差分法得到其数值解.最后,由参数的敏感性分析知:最优变现策略与瞬时冲击、市场波动率及风险厌恶系数等参数有关,但与永久冲击无关,且最优变现策略对市场波动率和瞬时冲击的变化较敏感.     

The Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation: Long‐time dynamics of nonzero boundary conditions

We consider the Gerdjikov‐Ivanov–type derivative nonlinear Schrödinger equation on the line. The initial value q(x,0) is given and satisfies the symmetric, nonzero boundary conditions at infinity, that is, q(x,0)→q_± as x→±∞, and |q_±|=q₀>0. The goal of this paper is to study the asymptotic behavior of the solution of this initial value problem as t→∞. The main tool is the asymptotic analysis of an associated matrix Riemann‐Hilbert problem by using the steepest descent method and the so‐called g‐function mechanism. We show that the solution q(x,t) of this initial value problem has a different asymptotic behavior in different regions of the xt‐plane. In the regions and , the solution takes the form of a plane wave. In the region , the solution takes the form of a modulated elliptic wave.     

On a free boundary problem in electrostatics

Let Ω_i ? ?^N, i = 0, 1, be two bounded separately star-shaped domains such that $ \Omega _0 \supset \bar \Omega _1 $. We consider the electrostatic potential u defined in $ \Omega : = \Omega _0 \backslash \bar \Omega _1 $: The geometry of the two boundary components Γ₀ and Γ₁ is not given, but instead the electrostatic potential u is supposed to satisfy the further boundary conditions Using a best possible maximum principle, we show that this free boundary problem has a unique solution which is radially symmetric.     

Risk-Sensitive Dynamic Asset Management

This paper develops a continuous time portfolio optimization model where the mean returns of individual securities or asset categories are explicitly affected by underlying economic factors such as dividend yields, a firm's return on equity, interest rates, and unemployment rates. In particular, the factors are Gaussian processes, and the drift coefficients for the securities are affine functions of these factors. We employ methods of risk-sensitive control theory, thereby using an infinite horizon objective that is natural and features the long run expected growth rate, the asymptotic variance, and a single risk-aversion parameter. Even with constraints on the admissible trading strategies, it is shown that the optimal trading strategy has a simple characterization in terms of the factor levels. For particular factor levels, the optimal trading positions can be obtained as the solution of a quadratic program. The optimal objective value, as a function of the risk-aversion parameter, is shown to be the solution of a partial differential equation. A simple asset allocation example, featuring a Vasicek-type interest rate which affects a stock index and also serves as a second investment opportunity, provides some additional insight about the risk-sensitive criterion in the context of dynamic asset management. Accepted 10 December 1997     

Mean–variance asset–liability management with asset correlation risk and insurance liabilities

Consider an insurer who invests in the financial market where correlations among risky asset returns are randomly changing over time. The insurer who faces the risk of paying stochastic insurance claims needs to manage her asset and liability by taking into account of the correlation risk. This paper investigates the impact of correlation risk to the optimal asset–liability management (ALM) of an insurer. We employ the Wishart process to model the stochastic covariance matrix of risky asset returns. The insurer aims to minimize the variance of the terminal wealth given an expected terminal wealth subject to the risk of paying out random liabilities of compound Poisson process. This ALM problem then becomes a linear–quadratic stochastic optimal control problem with stochastic volatilities, stochastic correlations and jumps. The recognition of an affine form in the solution process enables us to derive the explicit closed-form solution to the optimal ALM portfolio policy, obtain the efficient frontier, and identify the condition that the solution is well behaved.     

Some Limit Theorems for Extremal and Union Shot-Noise Processes

We study the limiting behaviour of suitably normalized union shot-noise processes , where F is a set-valued function on R^d × ?? is a sequence of i.i.d. random elements on some measurable space [?? ??] and Ψ = {x_i, i≥ 1} stands for a stationary d-dimensional point process whose intensity λ tends to infinity. General results concerning weak convergence of parametrized union shot-noise processes Ξ?(t) as ? ↓ 0 are obtained (Theorem 5.1 and its corollaries), if the point process λ^{1 d}Ψ has a weak limit and F satisfies some technical conditions. An essential tool for proving these results is the notion of regular variation of multivalued functions. Some examples illustrate the applicability of the results.     

A method to construct 1‐rotational factorizations of complete graphs and solutions to the oberwolfach problem

The concept of a 1‐rotational factorization of a complete graph under a finite group was studied in detail by Buratti and Rinaldi. They found that if admits a 1‐rotational 2‐factorization, then the involutions of are pairwise conjugate. We extend their result by showing that if a finite group admits a 1‐rotational ‐factorization with even and odd, then has at most conjugacy classes containing involutions. Also, we show that if has exactly conjugacy classes containing involutions, then the product of a central involution with an involution in one conjugacy class yields an involution in a different conjugacy class. We then demonstrate a method of constructing a 1‐rotational ‐factorization under given a 1‐rotational 2‐factorization under a finite group . This construction, given a 1‐rotational solution to the Oberwolfach problem , allows us to find a solution to when the ’s are even (), and when is an odd prime, with no restrictions on the ’s.     

Half‐inverse problems for the quadratic pencil of the Sturm–Liouville equations with impulse

In this paper, we consider the inverse spectral problem for the impulsive Sturm–Liouville differential pencils on [0, π] with the Robin boundary conditions and the jump conditions at the point . We prove that two potentials functions on the whole interval and the parameters in the boundary and jump conditions can be determined from a set of eigenvalues for two cases: (i) the potentials given on and (ii) the potentials given on , where 0 < α < 1 , respectively. Inverse spectral problems, Sturm–Liouville operator, spectrum, uniqueness.     

大额资产清仓的自动化交易策略

市场的机构投资者经常需要清仓手中持有的大额资产, 因此清仓的交易策略成为了关心的问题. 以工商银行的股票为例,给出适用于计算机执行的自动化清仓策略. 首先将高频的工商银行股票历史数据在每个交易日分别划分出48个交易期, 将问题简化为处理每个交易日交易期的数据. 在此基础上, 综合考虑用神经网络模拟预测清仓时股票价格随时间下降的风险和用信息流理论模型衡量的价格冲击和交易时刻, 并通过优化模型得到清仓持续的交易日天数. 此后, 再制定出每个交易日的具体自动化交易策略.在制定日内交易策略 时, 首先用神经网络对交易时刻做出预测, 然后综合考虑使用 VWAP 预测出的交易量和通过 Kalman 滤波方法修正过的期权定价公式预测出的各时刻股票的初始价格, 最终给出详细的交易策略及交易的成本.     

具有增益的概率网络金融计划模型

本文绘出一类具有增益的概率网络金融计划模型．许多多阶段金融计划问题可纳入这类模型．在这类模型中,随机变量的分布函数与Alexander过滤交易规则密切联系在一起,金融市场交易信号由神经网络产生,目标函数的最优值按其期望值计算．文中提出临界流和临界路的概念,给出目标函数下界等于其期望值的充分必要条件和期望最优解的求解方法．     

On the honeymoon Oberwolfach problem

The honeymoon Oberwolfach problem HOP asks the following question. Given newlywed couples at a conference and round tables of sizes , is it possible to arrange the participants at these tables for meals so that each participant sits next to their spouse at every meal and sits next to every other participant exactly once? A solution to HOP is a decomposition of , the complete graph with additional copies of a fixed 1‐factor , into 2‐factors, each consisting of disjoint ‐alternating cycles of lengths . It is also equivalent to a semi‐uniform 1‐factorization of of type ; that is, a 1‐factorization such that for all , the 2‐factor consists of disjoint cycles of lengths . In this paper, we first introduce the honeymoon Oberwolfach problem and then present several results. Most notably, we completely solve the case with uniform cycle lengths, that is, HOP. In addition, we show that HOP has a solution in each of the following cases: ; is odd and ; as well as for all . We also show that HOP has a solution whenever is odd and the Oberwolfach problem with tables of sizes has a solution.     

Upper Tail Large Deviations of Regular Subgraph Counts in Erdős-Rényi Graphs in the Full Localized Regime

For a -regular connected graph H the problem of determining the upper tail large deviation for the number of copies of H in , an Erdős-Rényi graph on n vertices with edge probability p, has generated significant interest. For and , where is the number of vertices in H, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of H in exceeds its expectation by a constant factor is predicted to hold at a speed , and the rate function is conjectured to be given by the solution of a mean-field variational problem. After a series of developments in recent years, covering progressively broader ranges of p, the upper tail large deviations for cliques of fixed size were proved by Harel, Mousset, and Samotij in the entire localized regime. This paper establishes the conjecture for all connected regular graphs in the whole localized regime. © 2021 Wiley Periodicals LLC.