ABSTRACT

Using the portfolio model, we introduce a general stochastic process that is not necessarily a diffusion/jump process and the random variable is not necessarily normally distributed.

1. Introduction

The literature on stochastic processes (especially in finance) relied mainly on Levy processes such as Wiener process, Poisson process, and the Variance-Gamma process. Examples include Madan and Seneta [1], Focardo and Fabozzi [2], among many others. Much of the literature assumes a Wiener process (Brownian motion), which implies normally distributed and independent stationary increments. The Brownian motion is extensively used in stochastic finance especially in investment models (see, for example, Alghalith [3]).

However, these assumptions of diffusion/jump process and Gaussian/Poisson distribution (or any specific probability distribution) can be relaxed. That is, we can introduce a general stochastic process that is more general than the Levy process without losing significant analytical convenience. Consequently, this paper offers three major contributions. First, it relaxes the assumption of a diffusion/jump process. Secondly, it relaxes the Gaussian/Poisson distribution or any specific probability distribution. Thirdly, it provides solutions without reliance on the existing duality or variational methods. Moreover, we introduce a general model that can be applied to any specific topic.

2. The Model

In general, a continuous stochastic process

can be written as a function of a control variable, state variables and a random variable as the following (the first two integrals can be zero)

(1)

where is the control variable, is a vector of state variables or coefficients, is a stochastic factor, and is a random variable (not necessarily a Brownian motion) and thus the assumption of normal distribution (or any specific probability distribution) is not required. Moreover, in contrast to Levy processes, is not necessarily a linear (diffusion) function. In addition, we assume is admissible and progressively measurablewhere is the filtration.

The objective is to maximize the expected utility of with respect to

where is a differentiable, bounded and concave utility function. Using the method of Alghalith [4], the solution yields

(2)

where the subscript denote the derivatives.

Consider this exact Taylor polynomial (and suppressing the notations) (Equation (3))

Taking expectations of both sides yields (Equation (4))

Thus,

(5)

3. Example—The Investment/Consumption Model

It is well-known that the stock price is a function of the expected return, the volatility and a random variable

(6)

where is stochastic economic factor. However,

is not necessarily normally distributed and is not necessarily a linear function. Consequently, the wealth function is given by

(7)

where is the portfolio process, is the consumption process, is the initial wealth, is the risk-free rate of return. Thus,

(8)

The objective is to maximize the expected utility of wealth and consumption with respect to the portfolio and consumption

The solutions are

(9)

(10)

Using an exact Taylor expansion (and suppressing the notations), we obtain

(11)

Thus,

(12)

Therefore we can obtain expressions for the optimal portfolio and consumption

(13)

(14)

We can obtain explicit solutions under specific forms of the utility function. For example, under mean-variance (quadratic) preference, we can obtain explicit solutions since is constant and is linear. It is worth noting that even with Levy process general explicit solutions were not provided by the literature; thus, the assumption of a Levy process does not offer a significant analytical convenience.

REFERENCES