### 機率 Probability 地 點 ： B 1 0 3 理 學 院

TMS Annual Meeting

### 數 學 年 會

## 2018 ^{數 學 年 會}

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### 演講摘要

Speech Abstracts

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1 4 : 3 0 - 1 4 : 5 5

### Some Stochastic Control Problems in the Study of Finance

### Shuenn-Jyi Sheu

### Department of Mathematics and Department of Applied Mathematics National Central University and National Chengchi University

### E-mail: sheusj@math.ncu.edu.tw

Merton(1969) studied a continuous time portfolio optimization problem us-ing dynamic programmus-ing approach to solve the problem. In Merton (1971) a general Markovian model was discussed and the HJB equation was derived which is a nonlinear PDE with complicated nonlinearity. Solving the HJB equa-tion left open for many years. The development of martingale method in Pliska (1986) and Karatzas-Lehoczky-Sethi (1986) provides a powerful alternative ap-proach to find a solution when the market is complete. For the case of incom-plete market, market completion, as a consequence of martingale method, has been studied in Pages(1987), He-Pearson(1991) and Karatzas-Lehoczky-Shreve-Xu(1991), and also some recent works of Haugh-Kogan-Wang(2006), Rogers (2003), Klein-Rogers and Rogers-Zacvkowski(2013).

In this talk, a review is given to the recent developments of the studies of Merton portfolio optimization problems following the dynamic programming approach. They include risk-sensitive portfolio optimization problem, upside chance and downside risk probabilities optimization and optimal consumption problem. The line of developments follow the ideas of Fleming(1995), which sug-gests to reformulate the risk-sensitive optimization problem as a risk-sensitive stochastic control problem. We will also discuss recent ideas to find a solution for the finite time consumption problem by rewriting the HJB equation as an inf-sup type Isaacs equation, suggested by the idea of market completion. The Merton problem for a model with risk income will be also discussed. When the market is complete, a solution of the HJB equation can be found. When the market is incomplete, the solution of the HJB equation remains a challenge.

We also mention some investment problems with risky income from insur-ance, and for model with delay. The talk is based on joint works with H. Nagai, H. Hata, L.H Sun and Z. Zhang.

**Keywords: Merton problem; dynamic programming; Hamilton-Jacobi-Bellman**
equationsstochastic control.

### 70

### On the Stochastic Heat Equations

### Shang-Yuan Shiu Department of Mathmatics National Central University E-mail: shiu@math.ncu.edu.tw

We consider the following stochastic heat equation:

*∂*_{t}*u*_{t}*(x) = ∂*_{xx}*u*_{t}*(x) + σ(u*_{t}*(x)) ˙W (t, x),*

*x∈ (−∞, ∞) or x ∈ [−1, 1] with certain boundary conditions subject to initial*
*data u*0*(x).* We will discuss how initial data and the noise term effect the
behaviors of the solution. This is based on several different papers.

**Keywords: Stochastic heat equations, intermittency, dissipation.**

### 71

### Some Limit Distributions of Discounted Branching Random Walks

### Jyy-I Hong

### Department of Applied Mathematics National Chengchi University

### E-mail: hongjyyi@gmail.com

We consider a Galton-Watson discounted branching random walk*{Z**n**, ζ**n**}**n**≥0*,
*where z**n* *is the population of the nth generation and ζ**n* is a collection of the
positions on *R of the the Z**n* *individuals in the nth generation, and let Y**n* be
*the position of a randomly chosen individual from the nth generation and Z**n**(x)*
*be the number of points in ζ**n* *that are less than or equal to x, for x∈ R. In*
*this talk, we present the limit theorems for the distributions of Y**n* and ^{Z}^{n}_{Z}^{(x)}

*n* in
both supercritical and explosive cases.

**Keywords: branching random walks, branching processes, coalescence, **
su-percritical, explosive

**References**

*[1] K. B. Athreya. Discounted branching random walks. Advanced Applied*
**Probability, 17, 1985, 53-66.**

[2] K. B. Athreya and J.-I. Hong. An application of the coalescence theory to
**branching random walks. Journal of Applied Probability, 50, 2013, 893-899.**

### 72

*The max-ℓ* ^{2} mixing of reversible Markov chains

### Guan-Yu Chen

### Department of Applied Mathematics National Chiao Tung University E-mail: gychen@math.nctu.edu.tw

*The ℓ*^{2}-distance is one frequently used measurement to analyze the
conver-gence of Markov chains to their stationarity. For reversible Markov chains, their
*ℓ*^{2}-distances can be formulated by the spectral information of their transition
matrices. The corresponding mixing time and cutoff phenomenon for reversible
Markov chains were first systemically studied by C. and Saloff-Coste in [3]. Later
*in [1], C., Hsu and Sheu revealed more intrinsic mechanisms of ℓ*^{2}-cutoffs and
polished the cutoff criterion of C. and Saloff-Coste. Such a refinement makes
*some further theoretical analyses feasible including the comparison of ℓ*^{2}-cutoffs
between discrete time chains and continuous time chains.

*The max-ℓ*^{2} distance was first considered by C. and Saloff-Coste in [2]. An
*equivalent condition for the max-ℓ*^{2}cutoff was then built on the product of the
*max-ℓ*^{2} mixing time and the spectral gap of the transition matrix. Based on
*the theoretical work in [1], we derive another proof for the max-ℓ*^{2} cutoff
crite-rion in [2] and provide a formula of its cutoff time using the spectral information.

**Keywords: Markov chains, reversibility, ℓ**^{2}-distance

**References**

*[1] Guan-Yu Chen, Jui-Ming Hsu, and Yuan-Chung Sheu. The L*^{2}-cutoffs for
*reversible Markov chains. Ann. Appl. Probab., 27(4):2305-2341, 2017.*

[2] Guan-Yu Chen and Laurent Saloff-Coste. The cutoff phenomenon for
*er-godic Markov processes. Electron. J. Probab., 13:no. 3, 26-78, 2008.*

*[3] Guan-Yu Chen and Laurent Saloff-Coste. The L*^{2}-cutoff for reversible
*Markov processes. J. Funct. Anal., 258(7):2246-2315, 2010.*

### 73

### State-Dependent M/G/1 Queue with Multiple Vacations

### Gi-Ren Liu

### Department of Mathematics National Cheng Kung University E-mail: girenliu@mail.ncku.edu.tw

In this talk, we consider an M/G/1 queue, whose state depends on the workload. When the workload is equal to zero, the state will switch to the sleep state from the awake state. In order to avoid the ping-pong effect, the duration of the sleep state is determined by a sleep timer and the accumulated workload during the sleeping period. Hence, the waiting time for each service request depends on the system state. The random Sleep-Awake schedule has been used extensively in the design of modern communication systems for reducing the energy cost. For example, during the power saving mode, the message delivery will be delayed at some level. In view of that there exists a trade-off between the energy saving ratio and the service-communication delay, the waiting time analysis can provide a guideline for the operators to set up the length of sleep time for the Sleep-Awake scheme to maintain reasonable service delay and reduce the impact of buffer overflow. A recent work on the waiting time analysis for a restricted-length M/G/1 queue with multiple vacations will be presented in this talk. The details of its application on the design of green routers can be referred to [1] and [2].

**Keywords: Waiting Time Analysis, Sleep-Awake scheme, M/G/1 queue,**
multiple vacations.

**References**

[1] G.-R. Liu, P. Lin and M. K. Awad. Modeling Energy Saving Mechanism for Green Routers. IEEE Transactions on Green Communications and Net-working (2018). Vol. 2, 817-829.

[2] M. K. Awad, P. Lin and G.-R. Liu. Distributed and Load Adaptive Energy Management Algorithm for Ethernet Green Routers. Journal of Internet Technology (2018). Vol. 19, 781-794.

### 74

### The localized phase transition of a polymer

### Chien-Hao Huang Department of Mathematics

### National Taiwan University E-mail: chienhaohuang@ntu.edu.tw

A polymer is penalized by long excursions. We discuss the free energy and the order of phase transitions. The behavior of infinite-volume system is also discussed.

**Keywords: Polymers, phase transitions, localization**

### 75

### 最佳化 Optimization 地 點 ： M 3 1 0 數 學 館

TMS Annual Meeting

### 數 學 年 會

## 2018 ^{數 學 年 會}

### D e c . 8 / 0 9 : 3 0 - 2 1 : 0 0

### D e c . 9 / 0 9 : 3 0 - 1 5 : 5 0

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