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数理解析研究所講究録
第2078巻 2018年 22-24

Risk measure とその周辺
名古屋市立大学大学院芸術工学研究科

中国清華大学

Masayuki Kageyama 影山正幸

*Graduate School of Design and Architecture,
Nagoya City University
Tsinghua University
名古屋市立大学

Qi Wang 王キ
Nagoya City University概要

長らく,Markowitz [9] の研究以来,分散がリスクを図る指標であった.しかし,1990年代に人り,裾の思い分布に対しては,分散だけでは不十分である事が指摘されるようになり,リスク測度に関する研究がArtzner [1] らにより開始され,多くの研究がこれまでなされてきた[2, 3, 4, 5, 6, 7, 8, 11, 12, 13]. 本稿では [10] に従って,Distortion Risk Functional を紹介する.

1

Risk measure の定義

定義1.1 (Artzner et al.[1]) 確率変数 Y_{1}, Y_{2} \in L^{1} に対して, $\rho$ : L^{1} \rightarrow 亜が次の4つの性質を満たすとき, $\rho$ はcoherent 性を満たすという.

1.

(Monotonicity) $\rho$(Y_{1}) \leq $\rho$(Y_{2}) whenever Y_{1} \geq Y_{2} a.s.,2.

(Translation Equivariance) $\rho$(Y+c)= $\rho$(Y)-c, if c\in \mathbb{R}3.

(Positive homogeneity) $\rho$( $\lambda$ Y)= $\lambda \rho$(Y) if $\lambda$>0,

4.

(Convexity) $\rho$((1- $\lambda$)Y_{0}+ $\lambda$ Y_{1}) \leq (1- $\lambda$) $\rho$(Y_{0})+ $\lambda \rho$(Y_{1}) for 0\leq $\lambda$\leq 1.

これらの公理の必然性については,[1] を参照されたい.

*kageyama@sda.nagoya‐cu.ac.jp

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定義1.2 (Artzner et \mathrm{a}1.[1] )V@ R $\gamma$ (I)=G_{-I}^{-1}( $\gamma$) :=\displaystyle \inf\{x\in \mathbb{R}|G_{-I}(x) \geq $\gamma$\} (0\leq $\gamma$\leq 1) ,CV

@ R$\gamma$ ( )I:=\displaystle\frac{1} -$\gam a$}\int_{$\gam a$}^{1 @ ( ) (0\leq $\gamma$\leq 1) .
Rp I dpV(1)

ただし, Gi(x) :=P(I\leq x)(x\in \mathbb{R}) .
定理1.1 (Artzner et \mathrm{a}|.[1] )CV@R_{ $\gamma$}(I) はcoherent 性を満たす.

定理1.2 (Rocafellar et \mathrm{a}|.[11] )CV @ R $\gamma$ ( I )=\displaystyle \inf_{b\in \mathb {R} \{b+\frac{1}{1- $\gamma$}E[ -I-b]^{+}]\} .

(2)

ただし, [x]^{+} :=\displaystyle \max\{x, 0\}.

定理1.3 (Phlug and Pichler[10])すべての Y\in L^{1} に対して,写像$\gamma$\mapsto (l— $\gamma$ )CV@R $\gamma$ (Y)はconcave.

2

Distortion Risk Functional

定義2.1 (Distortion Risk Functional [10])\mathcal{R}_{ $\sigma$}(Y)$\sigma$:=\displaystyle \int_{0}^{1} $\sigma$(u)G_{-Y}^{-1}(u)du=\int_{0}^{1} (u)V@Ru(Y)du.
$\sigma$: [0, 1)\rightarrow [0, \infty) は非負,非減少で \displaystyle \int_{0}^{1} $\sigma$(u)du=1 を満たす関数とする. $\sigma$ はdistortion densityと呼ばれる.

CV@R_{ $\gamma$}(Y) はDistortion Risk Functional になっている.その際の distortion density は$\sigma$_{ $\gamma$}(u)=定理2.1 ([10])\left{begin{ary}l 0&\mathr{i}\mathr{f}u<$\gam $\ frac{1}-$\gam $}&\mathr{i}\mathr{f}u\geq$ am $. \end{ary}\ight.

任意の Distortion Risk Functional は,次のようにCV@R $\gamma$ で表現できる.

\displaystyle\mathcal{R}_{$\sigma$}(Y)=\int_{0}^{1}\mathrm{V} @ \mathrm{R} $\gam a$(Y)$\sigma$($\gam a$)d$\gam a$=\displaystyle\int_{0}^{1}

CV @ R $\gamma$ (Y)$\mu$_{ $\sigma$}(d $\gamma$) .

ただし, $\mu$_{ $\sigma$} は [0 , 1 ] 上の確率測度とする.

定理2.2 ([10])\mathcal{R}_{ $\sigma$} をある一つの distortion risk functional とするとき,次式が成り立つ.

\displaystyle \mathcal{R}_{ $\sigma$}=\sup\{E(YZ)|E(Z)=1 , CV@R (Z) \leq \displaystyle \frac{1}{1- $\gam a$}\int_{ $\gam a$}^{1} $\sigma$(u)du for all $\gamma$\in [0 , 1$\gamma$

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定理2.3 ([10])CV@R $\gamma$ (Y) は次式のように表現できる.

CV@R (Y)$\gamma$=\displaystyle \sup{ E(YZ)|E(Z)=1 , CV@Rp(Z) \leq \displayst le\frac{1} -$\gam a$} for all p\in [$\gamma$, 1] }.

参考文献[1] P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, “ Coherent measures of risk
Math. Finance, 9, 1999, 203‐228.

[2] P. Artzner, F. Dclbacn, J. M. Eber, D. Heath and H. Ku, ‘’ Coherent multiperiod risk
adjusted values and Bellman’s principle

Annals of Operations Research, 152, 2007, 5‐22.

[3] N. Bäuerle, A. Popp, “Risk‐sensitive stopping problems for continuous‐time markov
chainsto appear in Stochastics.

[4] N. Bäuerle, U. Rieder, “Partially observable risk‐sensitive Markov Decision Processes
Mathematics of Operations Research, 42 (4), 2017, 1180‐1196.

[5] H. Föllmer and I. Penner, “ Convex Measures of Risk and Trading Constraints”, Financeand Stochastics, Vol. 6, No. 4, 2002, 429‐447.

[6] H. Föllmer and I. Penner, “ Convex Risk Measure and the Dynamics of Their Penalty
Functions”, Statistics éỉ Decision, Vol. 24, 2006, 61‐96.

[7] A. Inoue, “ On the Worst Conditional Expectation

Journal on Applied Mathematics, Vol.

286, No. 1, 2003, 237‐247.

[8] S. Kusuoka, ” On law invariant coherent risk measures

Advances in Mathematical Eco‐nomics, Vol.3, Springer, Tokyo, (2001), 83‐95.

[9] H. M. Markowitz, Portfolio Selection:Efficient Diversification of Investment, Wiley, New
York, 1958.

[10] G. Ch. Phlug, A. Pichler, Multistage Stochastic optimization, Springer, 2014.
[11] R. T. Rockafellar and S. Uryasev, ‘’ optimization of Conditional Value‐at‐Risk”, Journalof Risk, Vol. 2, 3, 2000, 21‐42.

[12] R. T. Rockafellar and S. Uryasev, “ Conditional Value‐at‐Risk for General Loss Distribu‐tions”, Journal of Banking \not\in y Finance, Vol. 26, No. 7, 2002, 1443‐1471.

[13] C. Wu and Y. Lin, ‘)Minimizing Risk Models in Markov Decision Processes with Policies
Depending on Target Values
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