On a class of diverse market models

Abstract

A market model in Stochastic Portfolio Theory is a finite system of strictly positive stochastic processes. Each process represents the capitalization of a certain stock. If at any time no stock dominates almost the entire market, which means that its share of total market capitalization is not very close to one, then the market is called diverse. There are several ways to outperform diverse markets and get an arbitrage opportunity, and this makes these markets interesting. A feature of real-world markets is that stocks with smaller capitalizations have larger drift coefficients. Some models, like the volatility-stabilized model, try to capture this property, but they are not diverse. In an attempt to combine this feature with diversity, we construct a new class of market models. We find simple, easy-to-test sufficient conditions for them to be diverse and other sufficient conditions for them not to be diverse.

Appendix

Let us state Feller’s test for explosions. It is taken from Durrett (1996, Section 6.2). Fix an interval $(\alpha , \beta ) \subseteq \mathbb {R}$ and a point $x_0 \in (\alpha , \beta )$. Let $X(\cdot ) = (X(t), t \ge 0)$ be a real-valued stochastic process satisfying the following stochastic differential equation:

$$\begin{aligned} \mathrm {d}X(t) = b(X(t))\mathrm {d}t + \sigma (X(t))\mathrm {d}B(t),\ \ t \ge 0,\ \ X(0) = x_0. \end{aligned}$$

Here, $B = (B(t), t \ge 0)$ is a one-dimensional standard Brownian motion, and $b, \sigma : (\alpha , \beta ) \rightarrow \mathbb {R}$ are continuous functions such that $\sigma ^2(x) > 0$ for all $x \in (\alpha , \beta )$. Define the natural scale as

$$\begin{aligned} \varphi (x) := \int \limits _{x_0}^x\exp \left[ \int \limits _{x_0}^y-\frac{2b(z)}{\sigma ^2(z)}dz\right] dy,\ \ x \in (\alpha , \beta ). \end{aligned}$$

Also, let

$$\begin{aligned} m(x) = \frac{1}{\varphi '(x)\sigma ^2(x)},\ \ x \in (\alpha , \beta ). \end{aligned}$$

Finally, define the following integrals:

$$\begin{aligned} I_{\alpha } := \int \limits _{\alpha }^{x_0}(\varphi (x) - \varphi (\alpha ))m(x)\, dx,\ \ I_{\beta } = \int \limits _{x_0}^{\beta }(\varphi (\beta ) - \varphi (x))m(x)\, dx. \end{aligned}$$

Proposition 1

• (i) If $\varphi (\alpha ) = -\infty$, or $\varphi (\alpha ) > -\infty$ and $I_{\alpha } = \infty$, then the process $X(\cdot )$ a.s. does not hit $\alpha$. Otherwise, it does hit $\alpha$ with positive probability.

• (ii) If $\varphi (\beta ) = \infty$, or $\varphi (\beta ) < \infty$ but $I_{\beta } = \infty$, then the process $X(\cdot )$ a.s. does not hit $\beta$. Otherwise, it does hit $\beta$ with positive probability.

Proof of Lemma 1

From (2), we have:

$$\begin{aligned} \mathrm {d}X_i(t) = X_i(t)\left[ \left( -g(\mu _i(t)) + \frac{1}{2}\right) \mathrm {d}t + \mathrm {d}W_i(t)\right] ,\ \ i = 1, \ldots , n \end{aligned}$$

and

$$\begin{aligned} \mathrm {d}S(t) = \sum \limits _{i=1}^nX_i(t)\left[ \left( -g(\mu _i(t)) + \frac{1}{2}\right) \mathrm {d}t + \mathrm {d}W_i(t)\right] \!. \end{aligned}$$

Therefore,

$$\begin{aligned} d<X_i, S>_t = X_i^2(t)\mathrm {d}t,\ \ d<S>_t = \sum \limits _{i=1}^nX_i^2(t)\, \mathrm {d}t. \end{aligned}$$

Apply Ito’s formula to $X_i(t)/S(t)$. Let $f(x, y) = x/y$, then

$$\begin{aligned} f_x = \frac{1}{y},\ f_y = -\frac{x}{y^2},\ f_{xx} = 0,\ f_{xy} = -\frac{1}{y^2},\ f_{yy} = \frac{2x}{y^3}. \end{aligned}$$

Therefore, we have:

$$\begin{aligned} \mathrm {d}\mu _i(t)&= \mathrm {d}f(X_i(t), S(t)) = f_x(X_i(t), S(t))\mathrm {d}X_i(t) + f_y(X_i(t), S(t))\mathrm {d}S(t) \\&\quad +\, \frac{1}{2}f_{xx}(X_i(t), S(t))\mathrm {d}<X_i>_t + f_{xy}(X_i(t), S(t))\mathrm {d}<X_i, S>_t \\&\quad +\, \frac{1}{2}f_{yy}(X_i(t), S(t))\mathrm {d}<S>_t \\&= \frac{1}{S(t)}X_i(t)\left[ \left( -g(\mu _i(t)) + \frac{1}{2}\right) \mathrm {d}t + \mathrm {d}W_i(t)\right] \\&\quad -\, \frac{X_i(t)}{S^2(t)}\sum \limits _{j=1}^nX_j(t)\left[ \left( -g(\mu _j(t)) + \frac{1}{2}\right) \mathrm {d}t + \mathrm {d}W_j(t)\right] \\&\quad -\, \frac{1}{S^2(t)}X_i^2(t)\mathrm {d}t + \frac{X_i(t)}{S^3(t)}\sum \limits _{j=1}^nX_j^2(t)\mathrm {d}t. \end{aligned}$$

We can express this in terms of market weights as

$$\begin{aligned}&\mu _i(t)\left( -g(\mu _i(t)) + \frac{1}{2}\right) \mathrm {d}t + \mu _i(t)\mathrm {d}W_i(t) - \mu _i(t)\sum \limits _{j=1}^n\mu _j(t)\left( -g(\mu _j(t)) + \frac{1}{2}\right) \mathrm {d}t\\&\qquad - \mu _i(t)\sum \limits _{j=1}^n\mu _j(t)\mathrm {d}W_j(t) - \mu _i^2(t)\mathrm {d}t + \mu _i(t)\sum \limits _{j=1}^n\mu _j^2(t)\mathrm {d}t \\&\quad = \biggl [\psi (\mu _i(t)) - \mu _i(t)\sum \limits _{j=1}^n\psi (\mu _j(t))\biggr ]\mathrm {d}t + \sum \limits _{j=1}^n(\delta _{ij}\mu _i - \mu _i\mu _j)\mathrm {d}W_j(t). \end{aligned}$$

Lemma 2

Assume $X = (X_t, t \ge 0)$ is a progressively measurable continuous stochastic process such that

$$\begin{aligned} X_t = x + \int _0^t\gamma _u\mathrm {d}u + \int _0^t\rho _u\mathrm {d}W_u, \end{aligned}$$

where $B$ is an $(\mathcal F_t)_{t \ge 0}$-Brownian motion, and the processes $\gamma = (\gamma _t)_{t \ge 0}$ and $\rho = (\rho _t)_{t \ge 0}$ are progressively measurable. Assume that, for some constants $0 < \varkappa _1 \le \varkappa _2$, we have the following estimate:

$$\begin{aligned} 0 < \varkappa _1 \le \frac{|\rho _t|}{\sigma (X_t)} \le \varkappa _2 < \infty ,\ \ t \ge 0, \end{aligned}$$

where $\sigma : \mathbb {R}\rightarrow (0, \infty )$ is a continuous real-valued function. Then the following time-change process

$$\begin{aligned} \Delta (t) := \int _0^t\frac{\rho ^2_s}{\sigma ^2(X_s)}ds. \end{aligned}$$

is a strictly increasing function, $\Delta (0) = 0,\ \Delta (\infty ) = \infty$. Define its inverse:

$$\begin{aligned} \tau (s) := \inf \{t \ge 0 \mid \Delta (t) \ge s\}. \end{aligned}$$

The family $(\mathcal F_{\tau (s)})_{s \ge 0}$ of $\sigma$-algebras is a filtration satisfying the usual conditions. Moreover, the process $Z = (Z_s = X_{\tau (s)})_{s \ge 0}$ satisfies the equation

$$\begin{aligned} \mathrm {d}Z_s = \gamma _{\tau (s)}\frac{\sigma ^2(Z_s)}{\rho ^2_{\tau (s)}}\mathrm {d}s + \sigma (Z_s)\mathrm {d}B_s, \end{aligned}$$

where $B = (B_s)_{s \ge 0}$ is another $(\mathcal F_{\tau (s)})_{s \ge 0}$-Brownian motion.

Proof

Analogous to Hajek (1985, Section 3). For $s \ge 0$, we have:

$$\begin{aligned} Z_s = X_{\tau (s)} = x + \int _0^{\tau (s)}\gamma _u\mathrm {d}u + \int _0^{\tau (s)}\rho _u\mathrm {d}W_u. \end{aligned}$$

(22)

Let us change variables in the first integral from the right-hand side of (22): $u = \tau (v),\ v = \Delta (u),\ 0 \le v \le s$. Then

$$\begin{aligned} \mathrm {d}u = \tau '(v)\mathrm {d}v = \frac{1}{\Delta '(\tau (v))}\mathrm {d}v = \frac{\sigma ^2(X_{\tau (v)})}{\rho ^2_{\tau (v)}}\mathrm {d}v = \frac{\sigma ^2(Z_v)}{\rho ^2_{\tau (v)}}\mathrm {d}v. \end{aligned}$$

Therefore,

$$\begin{aligned} \int _0^{\tau (s)}\gamma _u\mathrm {d}u = \int _0^s\gamma _{\tau (v)}\frac{\sigma ^2(Z_v)}{\rho ^2_{\tau (v)}}\mathrm {d}v. \end{aligned}$$

Now, consider the Itô integral from the right-hand side of the (22): let

$$\begin{aligned} M_t = \int _0^t\frac{\rho _u}{\sigma (X_u)}\mathrm {d}W_u,\ \ t \ge 0. \end{aligned}$$

Then $M = (M_t)_{t \ge 0}$ is a continuous $(\mathcal F_t)_{t \ge 0}$-martingale with $<M>_t = \Delta (t)$, and

$$\begin{aligned} \int _0^t\rho _u\mathrm {d}W_u = \int _0^t\sigma (X_u)\mathrm {d}M_u. \end{aligned}$$

By Karatzas and Shreve (1991, Theorem 3.4.6), $M_t = B_{\Delta (t)}$ for $t \ge 0$, where $B = (B_s)_{s \ge 0}$ defined by $B_s = M_{\tau (s)},\ s \ge 0$ is another $(\mathcal F_{\tau (s)})_{s \ge 0}$-Brownian motion, and this filtration satisfies the usual conditions. By Karatzas and Shreve (1991, Proposition 3.4.8),

$$\begin{aligned} \int _0^t\sigma (X_u)\mathrm {d}M_u = \int _0^{\Delta (t)}\sigma (X_{\tau (v)})\mathrm {d}B_v = \int _0^{\Delta (t)}\sigma (Z_v)\mathrm {d}B_v. \end{aligned}$$

Since $\Delta (\tau (s)) = s$, we have:

$$\begin{aligned} \int _0^{\tau (s)}\rho _u\mathrm {d}W_u = \int _0^{\tau (s)}\sigma (X_u)\mathrm {d}M_u = \int _0^{s}\sigma (Z_v)\mathrm {d}B_v. \end{aligned}$$

Thus, for $s \ge 0$ we have:

$$\begin{aligned} Z_s = x + \int _0^s\gamma _{\tau (v)}\frac{\sigma ^2(Z_v)}{\rho ^2_{\tau (v)}}\mathrm {d}v + \int _0^{s}\sigma (Z_v)\mathrm {d}B_v, \end{aligned}$$

which completes the proof.

Lemma 3

Assume $X = (X_t)_{t \ge 0}$ and $Y = (Y_t)_{t \ge 0}$ are two progressively measurable continuous stochastic processes which satisfy

$$\begin{aligned} \mathrm {d}X_t = \beta _t\mathrm {d}t + \sigma (X_t)\mathrm {d}W_t,\ X_0 = x;\ \ \mathrm {d}Y_t = b(Y_t)\mathrm {d}t + \sigma (Y_t)\mathrm {d}W_t,\ Y_0 = x. \end{aligned}$$

Here, $\beta = (\beta _t)_{t \ge 0}$ is a progressively measurable process, and $b,\ \sigma : \mathbb {R}\rightarrow \mathbb {R}$ are real-valued continuous functions. If $\beta _t \le b(X_t)$ a.s. for $t \ge 0$, then $X_t \le Y_t$ a.s. for $t \ge 0$. If $\beta _t \ge b(X_t)$ a.s. for $t \ge 0$, then $X_t \ge Y_t$ a.s. for $t \ge 0$.

Proof

Follows from Ikeda and Watanabe (1989, Lemma 6.1).