# A Stochastic Differential Equation Inventory Model

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## Abstract

Inventory for an item is being replenished at a constant rate whilst simultaneously being depleted by demand growing randomly and in relation to the inventory level. A stochastic differential equation is put forward to model this situation with solutions to it derived when analytically possible. Probabilities of reaching designated a priori inventory levels from some initial level are considered. Finally, the existence of stable inventory states is investigated by solving the Fokker–Planck equation for the diffusion process at the steady state. Investigation of the stability properties of the Fokker–Planck equation reveals that a judicious choice of control strategy allows the inventory level to remain in a stable regime.

## Keywords

Diffusion process Itô solution Fokker–Planck equation Time-dependent Ornstein–Uhlenbeck process## Introduction

It has been recognized for some time that the demand for some items may be proportional to the inventory on display. Baker and Urban [1] argued that the demand rate of an item is of a polynomial functional form, dependent on the inventory level. A thorough review of such demand models has been carried out by Urban [2]. In his 2005 article Urban stated that two distinct functional models for the demand rate were dominant, Type I Models, where the demand rate advanced on the initial inventory alone, and Type II Models, where the demand rate was a continuous (predominantly a power) function of the inventory level. Tsoularis [3] used the type of demand function in this article to solve an optimal inventory control problem but no deep analysis of the stochastic differential equation was undertaken.

*d*, is a continuous function of the inventory level,

*x*, assuming the quadratic form:

*d*

_{1}and

*d*

_{2}are both positive constants that establish the concavity \( (d^{{\prime \prime }} (x) < 0) \) of the function (1). The growth in demand is at its most rapid for small inventory levels, governed primarily by the coefficient

*d*

_{1}, rising gradually at a decreasing rate \( (d^{{\prime }} (x) > 0,\;d^{{\prime \prime }} (x) < 0) \), due to the influence of the coefficient

*d*

_{2}, until the inventory reaches the level, \( x = \frac{{d_{1} }}{{2d_{2} }}(d^{{\prime }} (x) = 0) \), where the demand rate is at its peak value, \( \frac{{d_{1}^{2} }}{{4d_{2} }} \). This behaviour of the demand rate is a feature of the Type II Models reviewed by Urban [2]. However, when the inventory exceeds \( \frac{{d_{1} }}{{2d_{2} }} \), the growth in demand declines rapidly \( (d^{{\prime }} (x) < 0,\;d^{{\prime \prime }} (x) < 0) \) and ceases altogether \( (d(x) = 0) \) when \( x = \frac{{d_{1} }}{{d_{2} }} \). This constitutes a departure from Type II Models and allows the realistic possibility of saturation in demand when the product inventory reaches a sufficiently high level.

*d*

_{1}, to be a random variable that evolves according to

*σ*is the diffusion coefficient measuring the intensity of the disturbance. In discrete time, white noise is a sequence of independent uncorrelated random variables. In continuous time, however, the autocorrelation is the Dirac delta function,

*E*[

*η*(

*t*)

*η*(

*t*+

*s*)] =

*δ*(

*s*). The white noise, although not an actual physical process, is a useful approximation to physical situations where noise is inherently present in the dynamics of a process [4].

*d*(

*x*) in an infinitesimal interval,

*dt*, is

*dt*, demand grows by

*d*(

*x*)

*dt*and the inventory is replenished at a rate

*udt*, so the infinitesimal inventory change is (

*u*-

*d*(

*x*))

*dt*. Using (4) we can formulate the following stochastic differential equation (SDE):

*σx*.

## The Stochastic Differential Inventory Equation

*u*, assumes a constant value throughout some interval [0,

*t*]. Moreover (6) will be valid for \( x \in [0,D) \) only, that is, the inventory will obey (6) so long as

*x*is bounded form above by

*D*, and (6) will be no longer valid for \( x \ge D \). The mean value of the random parameter

*d*

_{1},

*α*, is the demand growth rate per unit of inventory (the random parameter whilst the inventory is low, and

*D*is the level of inventory that places an upper limit on demand growth, so no further growth in demand is possible beyond this value. If demand saturation is unlikely to occur at small values then \( D \to \infty \) and demand grows linearly with inventory, which in this case evolves thus:

- (i)The Lipschitz condition:for some constant independent of time$$ \left| {\left( {u - \alpha x_{1} \left( {1 - \frac{{x_{1} }}{D}} \right)} \right) - \left( {u - \alpha x_{2} \left( {1 - \frac{{x_{2} }}{D}} \right)} \right)} \right| + \left| {\sigma x_{1} - \sigma x_{2} } \right| \le L\left| {x_{1} - x_{2} } \right| $$
*t*,*L*, in some time interval, [0,*T*] and \( x_{1} ,x_{2} \in \left[ {x_{0} ,D} \right] \). This is essentially a smoothness condition which is fulfilled in the sufficient as the functions, \( u - \alpha x\left( {1 - \frac{x}{D}} \right) \) and*σx*, are continuously differentiable. - (ii)The growth condition:in some finite time interval, [0,$$ \left| {u - \alpha x\left( {1 - \frac{x}{D}} \right)} \right|^{2} + \left| {\sigma x} \right|^{2} \le L^{2} (1 + \left| x \right|^{2} ) $$
*T*]. This condition is imposed to prevent the solution to (6) becoming infinite in [0,*T*], which is always the case when*u*and*σ*are bounded from above by*L*.

*L*

^{2}), \( \mathop {lim}\limits_{n \to \infty } E\left[ {\sum\nolimits_{i = 1}^{n} {\sigma x(t_{i - 1} )(w(t_{i} ) - w(t_{i - 1} ))} - \int_{0}^{t} {\sigma xdw} } \right]^{2} = 0. \)

## Solution to SDE (6)

In this section a solution to the temporally homogeneous process (6) is presented for two distinct cases: (i) when the order rate, *u*, is a nonzero constant, and (ii) when *u* = 0.

### Solution to SDE (6) with *u *≠ 0

*Fu*. We shall not make any further attempt to solve Eq. (9) in this article.

*E*[

*x*(

*t*)]:

### Solution to SDE (6) when the Noise *σ* is Small

*σ*, is small. In such cases, it is reasonable to assume that the solution to the SDE will be a stochastic perturbation of the deterministic solution as \( \sigma \to 0 \). We assume a solution to (6) of the form

*y*(

*t*) is the solution to the deterministic differential equation

*y*(

*t*):

*σ*to obtain an infinite set of stochastic differential equations. We write below only the first two which are often adequate in practice:

### Solution to SDE (6) with *u *= 0

*u*= 0, (6) is a Bernoulli equation which is transformed via the substitution \( z = \frac{1}{y} \) to the standard form

*w*(

*t*), appears in the exponent.

## Solution to SDE (7)

In this section a solution to the temporally homogeneous SDE (7) is presented for two distinct cases: (i) when the order rate, *u*, is a nonzero constant, and (ii) when *u* = 0.

### Solution to SDE (7) with *u *≠ 0

*u*= 0” section, first introduce the integrating factor, \( F(t) = \exp \left( {\frac{{\sigma^{2} t}}{2} - \sigma w(t)} \right) \), then proceed along the same lines as before we arrive at the final solution:

*N*~ (0,

*t*−

*τ*), and the standard formula, \( E[e^{Z} ] = \exp \left( {E[Z] + \frac{1}{2}Var(Z)} \right) \) [6], for a normally distributed random variable

*Z*, is applicable here. The expected value of

*x*(

*t*) is then given by the following formula, which is independent of

*σ*:

*x*

_{0}if \( x_{0} = \frac{u}{\alpha } \).

### Solution to SDE (7) with *u *= 0

*u*= 0 (7) reduces to:

## The Boundary at *x *= 0

*x*=0, is an intrinsic boundary of the diffusion process (6) because the diffusion coefficient,

*σx*, vanishes there. Its classification is dependent on the integrability of the scale function, \( S(\xi ) = \int_{0}^{{x_{0} }} {s(\xi )d\xi } \), where

*x*= 0,

*x*= 0 is a natural boundary according to the Russian literature classification scheme [7]. According to another classification scheme proposed by Feller [5, 8], one classifies the point,

*x*= 0, based on the convergence of the integral

*x*= 0 is an entrance boundary. An entrance boundary cannot be reached from the interior of the state space, that is the inventory cannot vanish in finite time from some initial value, \( x_{0} \ne 0 \). The inventory however, can start from \( x_{0} = 0 \) and quickly build up to nonzero values.

*u*= 0, and

*x*= 0. The boundary,

*x*= 0, is attracting and the inventory never attains the boundary zero in finite time.

When \( D \to \infty \), then \( s(\xi ) = \xi^{{\frac{2\alpha }{{\sigma^{2} }}}} \exp \left( {\frac{2u}{{\sigma^{2} \xi }}} \right) \), and the scale function, \( S(0) = \int_{0}^{{x_{0} }} {\xi^{{\frac{2\alpha }{{\sigma^{2} }}}} \exp \left( {\frac{2u}{{\sigma^{2} \xi }}} \right)d\xi = \infty } \). The integral, \( \int_{0}^{x} {\left( {\int_{\xi }^{{x_{0} }} {s(z)dz} } \right)} \frac{1}{{\sigma^{2} \xi^{2} s(\xi )}}d\xi < \infty \), and the boundary, *x* = 0, is a natural boundary under the Russian classification scheme and an entrance boundary in the sense of Feller. For *u* = 0, \( S(0) = \int\limits_{0}^{{x_{0} }} {\xi^{{\frac{2\alpha }{{\sigma^{2} }}}} } d\xi < \infty \), and *x* = 0 is an attracting boundary.

## First Passage Times and Probabilities of Exit Through Absorbing Barriers

*x*

_{0}at time

*t*= 0, remains in the interval (

*x*

_{l}

*, x*

_{r}), which is assumed to contain

*x*

_{0},

*x*

_{l}<

*x*

_{0}<

*x*

_{r}. By erecting artificial absorbing barriers at

*x*

_{l}and

*x*

_{r}we investigate the probability that the inventory crosses over either barrier,

*x*

_{l}or

*x*

_{r}, and the mean first passage time to

*x*

_{l}or

*x*

_{r}. Define

## Passage Probabilities for SDE (6) with *D* < ∞

*x*

_{l}or

*x*

_{r}, starting from

*x*

_{0}, with the boundary conditions:

### Mean Passage Times Through Boundaries with *D* < ∞

*x*

_{l}or

*x*

_{r}. The solution to (28) with integration constants

*k*

_{1},

*k*

_{2}is

### Passage Probabilities for SDE (7) with *D* = ∞

*u*, being several orders of magnitude larger than \( \sigma^{2} \), the integrals in (31) behave like Laplace integrals [9] of the form, \( \int_{a}^{b} {x^{{\frac{2\alpha }{{\sigma^{2} }}}} e^{{\frac{2u}{{\sigma^{2} x}}}} dx} \). As the functions, \( x^{{\frac{2\alpha }{{\sigma^{2} }}}} \) and \( \frac{d}{dx}\left( {\frac{1}{x}} \right) = - \frac{1}{{x^{2} }}, \) vanish nowhere in the interval (

*a*,

*b*), an asymptotic expression for the Laplace integral is possible:

### Mean Passage Times Through Boundaries with *D* = ∞

## Stationary Solution of the Fokker–Planck Equation and Existence of Stable States

The Kolmogorov forward equation of Fokker–Planck equation governs the evolution of the transition probability density, \( f(x\left( t \right)|x_{0} = x\left( 0 \right)) \), henceforth denoted by *f*.

### Stationary Solution to the Fokker–Planck Equation for SDE (6)

*N*is the integration (normalization) constant.

*u*= 0, it can be seen from (38) that

*N*does not exist, and in this case,\( f^{*} (x) = \delta (x) \), in \( [0,D) \). This is because the boundary,

*x*= 0, is an attracting boundary and the stationary probability mass will be concentrated entirely on zero.

### Stationary Solution to the Fokker–Planck Equation for SDE (7)

*N*, is now furnished by the much simpler expression:

### Extrema of Stationary Densities

*D*when either \( \sigma^{2} \le \alpha , \) or \( \sigma^{2} > \hbox{max} \left( {\alpha ,\frac{u}{D}} \right). \)

Differentiation of (43) with respect to *x* reveals that \( x_{1}^{*} \) is a relative maximum and \( x_{2}^{*} > x_{1}^{*} \) is a relative minimum. The inventory tends to move away from the relative minimum, \( x_{2}^{*} \), towards the relative maximum, \( x_{1}^{*} \), which represents the stable inventory state of the diffusion process.

### Probabilistic Potentials

*D*) given by

*u*, always exceeds demand growth (constantly positive drift).

The probabilistic potential, \( \phi (x) \), is analogous to the potential (Lyapunov) function in Classical Mechanics. The root \( \xi_{1} \) is a stable minimum and the second root, \( \xi_{2} > \xi_{1} \), is an unstable maximum. The inventory will tend to drift towards levels that minimize \( \phi (x) \) and maximize \( f^{*} (x) \). But the maximum, \( x_{1}^{*} \), of \( f^{*} (x) \) does not in general coincide with the minimum, \( \xi_{1} \), of \( \phi (x) \), unless the diffusion term is just an additive constant, independent of *x*. In practice, stable inventory values will be those that fall within the valley of \( \phi (x) \) and the peak of \( f^{*} (x) \). If \( \phi (x) \) does not have a minimum but \( f^{*} (x) \) still has a maximum, then there is a non-negligible probability that the inventory will fall anywhere in the range, \( [0,D) \), which is clearly an undesirable consequence when *D* is large in relation to the existing stock.

### Approximation of the Stationary Density by a Normal Density in the Vicinity of Its Extremum

*A*, under the Gaussian function,

*g*(

*x*), is given by

*x*) is the well known error function, \( {\text{erf}}\left( x \right) = \frac{2}{\sqrt \pi }\int_{0}^{x} {e^{{ - t^{2} }} dt} \) [11]. The effective width,

*ε*, of the peak of the Gaussian function,

*g*(

*x*), is the width of the rectangle that has the same height as its peak, \( f^{*} (x_{1}^{*} ) \), and the same area,

*A*. So

The Gaussian density, *g*(*x*), is a credible approximation to \( f^{*} (x) \) when both \( \phi (x) \) and \( f^{*} (x) \) possess extreme values in [0,D), so that the effective width for *g*(*x*) represents the basin of stability.

## A Numerical Example

We close the paper by a simple numerical demonstration of the key findings. Let \( x_{0} = 5, \, u = 5, \, D = 400, \, \alpha = 0. 2 , { }\sigma = 0.3. \)

*g*(

*x*). The maximum density inventory is \( x_{1}^{*} \approx 18 \) from (44) and the effective width is \( \varepsilon \approx 18 \) from (50), hence the range of stable inventory values is approximately \( [9,27] \). Finally, Fig. 3 illustrates the probabilistic potential, \( \phi (x) \), maximized at \( \xi_{1} \approx 27 \) from (46).

Finally, suppose the inventory planner wants to estimate for instance, the probability that the inventory, starting from \( x_{0} = 5 \) will either double to \( x_{r} = 10 \) or drop to \( x_{l} = 4 \), when the replenishment rate is *u* = 5 and \( \alpha = 0.2, \, \sigma = 0.3, \, D = 400 \). From (25), (26) and (27) we obtain the probability estimates, \( \pi (x_{l} = 4) = 0.0177 \) and \( \pi (x_{r} = 10) = 0.9823 \). If the replenishment rate drops to *u* = 2 for instance, the probabilities become roughly equal, \( \pi (x_{l} = 4) = 0.5089 \) and \( \pi (x_{r} = 10) = 0.4911 \).

## Discussion

We have presented in this work a continuous time mathematical model for a randomly evolving inventory of an item for which demand is growing at a gradually slowing rate in relation to the inventory’s availability, whilst being simultaneously satisfied at a constant order rate. The model put forward is a temporally homogeneous stochastic differential equation described by (6) and in a simpler reduced version by (7), with Itô solutions provided when analytically possible. To assess the direction the inventory is likely to take, theoretical estimates of the probabilities of attaining arbitrary inventory levels from some current inventory state under a fixed replenishment scheme are explicitly determined in “First Passage Times and Probabilities of Exit Through Absorbing Barriers” section. Finally, in “Stationary Solution of the Fokker–Planck Equation and Existence of Stable States” section the issue of long term stable stock levels is thoroughly addressed and the constraint on the replenishment rate, *u*, for a stable inventory regime is explicitly obtained. The size of the stable regime derived from the approximation of the probability density to a Gaussian density, will depend on the magnitude of the diffusion parameter, *σ*. The probabilities of reaching prescribed inventory levels and the determination of the stability regime are useful practical parameters for the inventory planner.

## Notes

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