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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Stochastic portfolio theory ： ウィキペディア英語版 Stochastic portfolio theory Stochastic portfolio theory (SPT) is a mathematical theory for analyzing stock market structure and portfolio behavior introduced by E. Robert Fernholz in 2002. It is descriptive as opposed to normative, and is consistent with the observed behavior of actual markets. Normative assumptions, which serve as a basis for earlier theories like modern portfolio theory (MPT) and the capital asset pricing model (CAPM), are absent from SPT. SPT uses continuous-time random processes (in particular, continuous semi-martingales) to represent the prices of individual securities. Processes with discontinuities, such as jumps, have also been incorporated into the theory. == Stocks, portfolios and markets == SPT considers stocks and stock markets, but its methods can be applied to other classes of assets as well. A stock is represented by its price process, usually in the logarithmic representation. In the case the market is a collection of stock-price processes $X\_i,$ for $i=1,\; \backslash dots,\; n,$ each defined by a continuous semimartingale :$d\; \backslash log\; X\_i(t)\; =\; \backslash gamma\_i(t)\; \backslash ,\; dt\; +\; \backslash sum\_^d\; \backslash xi\_(t)\; \backslash ,\; dW\_(t)$ where $W\; :=\; (W\_1,\; \backslash dots,\; W\_d)$ is an $n$-dimensional Brownian motion (Wiener) process with $d\; \backslash geq\; n$, and the processes $\backslash gamma\_i$ and $\backslash xi\_$ are progressively measurable with respect to the Brownian filtration $\backslash \; =\; \backslash$. In this representation $\backslash gamma\_i(t)$ is called the (compound) growth rate of $X\_i,$ and the covariance between $\backslash log\; X\_i$ and $\backslash log\; X\_j$ is $\backslash sigma\_(t)=\backslash sum\_^d\; \backslash xi\_(t)\; \backslash xi\_(t).$ It is frequently assumed that, for all $i,$ the process $\backslash xi\_^2(t)\; +\; \backslash cdots\; +\; \backslash xi\_^2(t)$ is positive, locally square-integrable, and does not grow too rapidly as $t\; \backslash rightarrow\backslash infty.$ The logarithmic representation is equivalent to the classical arithmetic representation which uses the rate of return $\backslash alpha\_i(t),$ however the growth rate can be a meaningful indicator of long-term performance of a financial asset, whereas the rate of return has an upward bias. The relation between the rate of return and the growth rate is :$\backslash alpha\_(t)\; =\; \backslash gamma\_i(t)\; +\; \backslash frac$ The usual convention in SPT is to assume that each stock has a single share outstanding, so $X\_i(t)$ represents the total capitalization of the $i$-th stock at time $t,$ and $X(t)\; =\; X\_1(t)\; +\; \backslash cdots\; +\; X\_n(t)$ is the total capitalization of the market. Dividends can be included in this representation, but are omitted here for simplicity. An investment strategy $\backslash pi\; =\; (\backslash pi\_1\; ,\; \backslash cdots,\; \backslash pi\_n)$ is a vector of bounded, progressively measurable processes; the quantity $\backslash pi\_i(t)$ represents the proportion of total wealth invested in the $i$-th stock at time $t$, and $\backslash pi\_0(t)\; :=\; 1\; -\; \backslash sum\_^n\; \backslash pi\_i(t)$ is the proportion hoarded (invested in a money market with zero interest rate). Negative weights correspond to short positions. The cash strategy $\backslash kappa\; \backslash equiv\; 0\; (\backslash kappa\_0\; \backslash equiv\; 1)$ keeps all wealth in the money market. A strategy $\backslash pi$ is called portfolio, if it is fully invested in the stock market, that is $\backslash pi\_1(t)\; +\; \backslash cdots\; +\; \backslash pi\_n\; (t)\; =\; 1$ holds, at all times. The value process $Z\_$ of a strategy $\backslash pi$ is always positive and satisfies :$$ d \log Z_(t) = \sum_^n \pi_i(t) \, d\log X_i(t) + \gamma_\pi^ *(t) \, dt where the process $\backslash gamma\_^$ * is called the excess growth rate process and is given by :$$ \gamma_^ *(t) := \frac \sum_^n \pi_i(t) \sigma_(t) -\frac \sum_^n \pi_i(t) \pi_j(t) \sigma_(t) This expression is non-negative for a portfolio with non-negative weights $\backslash pi\_i(t)$ and has been used in quadratic optimization of stock portfolios, a special case of which is optimization with respect to the logarithmic utility function. The market weight processes, :$$ \mu_i(t) := \frac where $i=1,\; \backslash dots,\; n$ define the market portfolio $\backslash mu$. With the initial condition $Z\_\backslash mu(0)\; =\; X(0),$ the associated value process will satisfy $Z\_(t)\; =\; X(t)$ for all $t.$ A number of conditions can be imposed on a market, sometimes to model actual markets and sometimes to emphasize certain types of hypothetical market behavior. Some commonly invoked conditions are: # A market is nondegenerate if the eigenvalues of the covariance matrix $(\backslash sigma\_\; (t))\_$ are bounded away from zero. It has bounded variance if the eigenvalues are bounded. # A market is coherent if $\backslash operatorname\_\; t^\; \backslash log(\backslash mu\_i(t))\; =\; 0$ for all $i\; =\; 1,\; \backslash dots,\; n.$ # A market is diverse on $(T)$ if there exists $\backslash varepsilon\; >\; 0$ such that $\backslash mu\_(t)\; \backslash leq\; 1\; -\backslash varepsilon$ for $t\; \backslash in\; (T).$ # A market is weakly diverse on $(T)$ if there exists $\backslash varepsilon\; >\; 0$ such that $\backslash frac\backslash int\_0^T\; \backslash mu\_(t)\backslash ,\; dt\; \backslash leq\; 1\; -\; \backslash varepsilon$ Diversity and weak diversity are rather weak conditions, and markets are generally far more diverse than would be tested by these extremes. A measure of market diversity is market entropy, defined by :$S(\backslash mu(t))\; =\; -\backslash sum\_^\; \backslash mu\_i(t)\; \backslash log(\backslash mu\_i(t)).$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Stochastic portfolio theory」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.