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Journal of Applied Mathematics
Volume 2012, Article ID 838397, 13 pages
http://dx.doi.org/10.1155/2012/838397
Research Article

The Sum and Difference of Two Lognormal Random Variables

Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Hong Kong

Received 15 May 2012; Accepted 19 July 2012

Academic Editor: Mehmet Sezer

Copyright © 2012 C. F. Lo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of N lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

1. Introduction

Given two correlated lognormal stochastic variables, what is the stochastic dynamics of the sum or difference of the two variables?”; or equivalently “What is the probability distribution of the sum or difference of two correlated lognormal stochastic variables?” The solution to this long-standing problem has wide applications in many fields such as telecommunication studies [16], financial modelling [79], actuarial science [1012], biosciences [13], physics [14], and so forth. Although the lognormal distribution is well known in the literature [15, 16], yet almost nothing is known of the probability distribution of the sum or difference of two correlated lognormal variables. However, it is commonly agreed that the distribution of either the sum or difference is neither normal nor lognormal.

The aforesaid problem can be formulated as follows. Given two lognormal stochastic variables 𝑆1 and 𝑆2 obeying the following stochastic differential equations: 𝑑𝑆𝑖𝑆𝑖=𝜎𝑖𝑑𝑍𝑖,𝑖=1,2,(1.1) where 𝜎2𝑖=Var(ln𝑆𝑖), 𝑑𝑍𝑖 denotes a standard Weiner process associated with 𝑆𝑖, and the two Weiner processes are correlated as 𝑑𝑍1𝑑𝑍2=𝜌𝑑𝑡, the time evolution of the joint probability distribution function 𝑃(𝑆1,𝑆2,𝑡;𝑆10,𝑆20,𝑡0) of the two correlated lognormal variables is governed by the backward Kolmogorov equation 𝜕𝜕𝑡0+𝐿𝑃𝑆1,𝑆2,𝑡;𝑆10,𝑆20,𝑡0=0for𝑡>𝑡0,(1.2) where 1𝐿=2𝜎21𝑆210𝜕2𝜕𝑆210+𝜌𝜎1𝜎2𝑆10𝑆20𝜕2𝜕𝑆10𝜕𝑆20+12𝜎22𝑆220𝜕2𝜕𝑆220(1.3) subject to the boundary condition 𝑃𝑆1,𝑆2,𝑡;𝑆10,𝑆20,𝑡0𝑆𝑡=𝛿1𝑆10𝛿𝑆2𝑆20.(1.4) This joint probability distribution function tells us how probable the two lognormal variables assume the values 𝑆1 and 𝑆2 at time 𝑡>𝑡0, provided that their values at 𝑡0 are given by 𝑆10 and 𝑆20. Since 𝑃(𝑆1,𝑆2,𝑡;𝑆10,𝑆20,𝑡0) is known in closed form as follows: 𝑃𝑆1,𝑆2,𝑡;𝑆10,𝑆20,𝑡0=12𝜋𝜎1𝜎21𝜌2𝑆1𝑆2expln𝑆1ln𝑆10+(1/2)𝜎21𝑡𝑡022𝜎211𝜌2𝑡𝑡0𝜌×expln𝑆1ln𝑆10+(1/2)𝜎21𝑡𝑡0ln𝑆2ln𝑆20+(1/2)𝜎22𝑡𝑡0𝜎1𝜎21𝜌2𝑡𝑡0×expln𝑆2ln𝑆20+(1/2)𝜎22𝑡𝑡022𝜎221𝜌2𝑡𝑡0,(1.5) the probability distribution of the sum or difference, namely 𝑆±𝑆1±𝑆2, of the two correlated lognormal variables can be obtained by evaluating the integral 𝑃±𝑆±,𝑡;𝑆10,𝑆20,𝑡0=0𝑑𝑆1𝑑𝑆2𝑃𝑆1,𝑆2,𝑡;𝑆10,𝑆20,𝑡0𝛿𝑆1±𝑆2𝑆±.(1.6) Despite that many methods have been developed to address the problem, a closed-form representation for the probability distribution of the sum or difference is still missing. Hence, we must resort to numerical methods to perform the integration. Nevertheless, the numerically exact solution does not provide any information about the stochastic dynamics of the sum or difference explicitly.

In the lack of knowledge about the probability distribution of the sum or difference of two correlated lognormal variables, several analytical approximation methods which focus on finding a good approximation for the desired probability distribution have been proposed in the literature [16, 8, 1727]. Essentially, these analytical approximations assume a specific distribution that the sum or difference of the two correlated lognormal variables follow, and then use a variety of methods to identify the parameters for that specific distribution. However, no mathematical justification for the specific distribution was apparently given. In spite of this shortcoming, these approximations attract considerable attention and have been extended to tackle the algebraic sums of 𝑁 correlated lognormal variables, too.

In this communication, we apply the Lie-Trotter operator splitting method [28] to derive an approximation for the dynamics of the sum or difference of two correlated lognormal variables. It is shown that both the sum and difference can be described by a shifted lognormal stochastic process. Approximate probability distributions of both the sum and difference of the lognormal variables are determined in closed form, and illustrative numerical examples are presented to demonstrate the accuracy of these approximate distributions. Unlike previous studies which treat the sum and difference in a separate manner, our proposed method thus provides a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. In addition, in terms of the approximate solutions, we are able to obtain an analytical series expansion of the exact solutions, which can allow us to improve the approximation systematically. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of 𝑁 lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

2. Lie-Trotter Operator Splitting Method

It is observed that the probability distribution of the sum or difference of the two correlated lognormal variables, that is, 𝑃±(𝑆±,𝑡;𝑆10,𝑆20,𝑡0), also satisfies the same backward Kolmogorov equation given in (1.2), but with a different boundary condition 𝑃±𝑆±,𝑡;𝑆10,𝑆20,𝑡0𝑆𝑡=𝛿10±𝑆20𝑆±.(2.1) To solve for 𝑃±(𝑆±,𝑡;𝑆10,𝑆20,𝑡0), we first rewrite the backward Kolmogorov equation in terms of the new variables 𝑆±0𝑆10±𝑆20 as 𝜕𝜕𝑡0+𝐿++𝐿0+𝐿𝑃±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0=0,(2.2) where 𝐿+=18𝜎2+𝑆+02𝜎+221𝜎22𝑆+0𝑆0+𝜎2𝑆02𝜕2𝜕𝑆0+2𝐿0=14𝜎21𝜎22𝑆+02+𝑆02+𝜎21+𝜎22𝑆+0𝑆0𝜕2𝜕𝑆+0𝜕𝑆0𝐿=18𝜎2+𝑆02𝜎+221𝜎22𝑆+0𝑆0+𝜎2𝑆+02𝜕2𝜕𝑆02𝜎±=𝜎21+𝜎22±2𝜌𝜎1𝜎2.(2.3) The corresponding boundary condition now becomes 𝑃±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0𝑆𝑡=𝛿±0𝑆±.(2.4) Accordingly, the formal solution of (2.2) is given by 𝑃±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0=exp𝑡𝑡0𝐿++𝐿0+𝐿𝛿𝑆±0𝑆±.(2.5)

Since the exponential operator exp{(𝑡𝑡0𝐿)(++𝐿0+𝐿)} is difficult to evaluate, we apply the Lie-Trotter operator splitting method [28] to approximate the operator by (see the appendix) 𝑂LT±=exp𝑡𝑡0𝐿±exp𝑡𝑡0𝐿0+𝐿,(2.6) and obtain an approximation to the formal solution 𝑃±(𝑆±,𝑡;𝑆+0,𝑆0,𝑡0), namely 𝑃LT±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0=𝑂LT±𝛿𝑆±0𝑆±=exp𝑡𝑡0𝐿±𝛿𝑆±0𝑆±,(2.7) where the relation exp{(𝑡𝑡0𝐿)(0+𝐿)}𝛿(𝑆±0𝑆±)=𝛿(𝑆±0𝑆±) is utilized. For (𝑆0/𝑆+0)21, which is normally valid unless 𝑆10 and 𝑆20 are both close to zero, the operators 𝐿+ and 𝐿 can be approximately expressed as 𝐿±12𝜎2±𝑆0±2𝜕2𝜕𝑆0±2(2.8) in terms of the two new variables: 𝑆+0=𝑆+0+𝜎21𝜎22𝜎2+𝑆0,𝑆0=𝑆0+𝜎2𝜎21𝜎22𝑆+0,(2.9) where 𝜎+=𝜎+/2and𝜎=(𝜎21𝜎22)/(2𝜎). Without loss of generality, we assume that 𝜎1>𝜎2. Obviously, both 𝑆+ and 𝑆 are lognormal (LN) random variables defined by the stochastic differential equations 𝑑𝑆±=𝜎±𝑆±𝑑𝑍±,(2.10) and their closed-form probability distribution functions are given by 𝑓LN𝑆±𝑆,𝑡;±0,𝑡0=1𝑆±2𝜋𝜎2±𝑡𝑡0𝑆expln±/𝑆±0+(1/2)𝜎2±𝑡𝑡022𝜎2±𝑡𝑡0(2.11) for 𝑡>𝑡0. As a result, it can be inferred that within the Lie-Trotter splitting approximation both 𝑆+ and 𝑆 are governed by a shifted lognormal process. It should be noted that for the Lie-Trotter splitting approximation to be valid, 𝜎2±(𝑡𝑡0) needs to be small.

Alternatively, we can also approximate the operator 𝐿 by 𝐿12𝜎2𝑅0𝜕2𝜕𝑅02,(2.12)where𝜎=(𝜎21𝜎22)𝑆+0/2 and 𝑅0=𝑆0+12𝜎2𝜎21𝜎22𝑆+0.(2.13) It is not difficult to recognize that 𝑅 follows the square-root (SR) stochastic process defined by the stochastic differential equation 𝑑𝑅=𝜎𝑅𝑑𝑍,(2.14) and has the closed-form probability distribution function 𝑓SR𝑅,𝑡;𝑅0,𝑡0=2𝜎2𝑡𝑡0𝑅0𝑅2𝑅exp+𝑅0𝜎2𝑡𝑡0𝐼14𝑅𝑅0𝜎2𝑡𝑡0(2.15) for 𝑡>𝑡0, where 𝐼1() is the modified Bessel function of the first kind of order one. Accordingly, we have shown that within the Lie-Trotter splitting approximation, which requires 𝜎2(𝑡𝑡0) to be small, 𝑆 can be described by a shifted square-root process, too.

Moreover, in terms of the approximate solutions 𝑃LT±(𝑆±,𝑡;𝑆+0,𝑆0,𝑡0), we can express the exact solutions 𝑃±(𝑆±,𝑡;𝑆+0,𝑆0,𝑡0) in the following form: 𝑃±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0=𝑃LT±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0+𝑡𝑡0𝑑𝜉exp𝜉𝑡0𝐿±𝐿0+𝐿𝑃±𝑆±,𝑡;𝑆+0,𝑆0=,𝜉1+𝑡𝑡0𝑑𝜉1±𝑡0𝜉1+𝑡𝑡0𝑑𝜉1𝑡𝜉1𝑑𝜉2±𝑡0𝜉1±𝑡0𝜉2+𝑡𝑡0𝑑𝜉1𝑡𝜉1𝑑𝜉2𝑡𝜉2𝑑𝜉3±𝑡0𝜉1±𝑡0𝜉2±𝑡0𝜉3+𝑡𝑡0𝑑𝜉1𝑡𝜉1𝑑𝜉2𝑡𝜉2𝑑𝜉3𝑡𝜉3𝑑𝜉4𝑃LT±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0,(2.16) where ±𝐿(𝜏)=exp𝜏±𝐿0+𝐿𝜏𝐿exp±=𝐿0+𝐿+𝜏𝐿1!0+𝐿,𝐿±+𝜏2𝐿2!0+𝐿,𝐿±,𝐿±+𝜏3𝐿3!0+𝐿,𝐿±,𝐿±,𝐿±+.(2.17) The integrals over the temporal variables {𝜉𝑖;𝑖=1,2,3,} can be evaluated analytically. If we keep terms up to the order of (𝑡𝑡0)2, then 𝑃±(𝑆±,𝑡;𝑆+0,𝑆0,𝑡0) can be approximated by 𝑃±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0=𝑃LT±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0+𝑡𝑡0𝐿0+𝐿𝑃LT±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0+12𝑡𝑡02𝐿0+𝐿2𝐿0+𝐿,𝐿±𝑃LT±𝑆±,𝑡;𝑆+0,𝑆0,𝑡0.(2.18) This analytical series expansion can allow us to improve the approximate solutions systematically.

3. Illustrative Numerical Examples

In Figure 1 we plot the approximate closed-form probability distribution function of the sum 𝑆+ given in (2.11) for different values of the input parameters. We start with 𝑆10=110, 𝑆20=100, 𝜎1=0.25, and 𝜎2=0.15 in Figure 1(a). Then, in order to examine the effect of 𝑆20, we decrease its value to 70 in Figure 1(b) and to 40 in Figure 1(c). In Figures 1(d), 1(e), and 1(f) we repeat the same investigation with a new set of values for 𝜎1 and 𝜎2, namely 𝜎1=0.3 and 𝜎2=0.2. Without loss of generality, we set 𝑡𝑡0=1 for simplicity. The (numerically) exact results which are obtained by numerical integrations are also included for comparison. It is clear that the proposed approximation can provide accurate estimates for the exact values. Moreover, to have a clearer picture of the accuracy, we plot the corresponding errors of the estimation in Figure 2. We can easily see that major discrepancies appear around the peak of the probability distribution function, and that the estimation deteriorates as the correlation parameter 𝜌 decreases from 0.5 to 0.5. It is also observed that the errors increase with the ratio 𝑆0/𝑆+0 as expected but they seem to be not very sensitive to the changes in 𝜎1 and 𝜎2.

fig1
Figure 1: Probability density versus 𝑆1+𝑆2: The solid lines denote the distributions of the approximate shifted lognormal process, and the dash lines show the exact results. (a) 𝑆10=110, 𝑆20=100, 𝜎1=0.25, and 𝜎2=0.15; (b) 𝑆10=110, 𝑆20=70, 𝜎1=0.25, and 𝜎2=0.15; (c) 𝑆10=110, 𝑆20=40, 𝜎1=0.25, and 𝜎2=0.15; (d) 𝑆10=110, 𝑆20=100, 𝜎1=0.3, and 𝜎2=0.2; (e) 𝑆10=110, 𝑆20=70, 𝜎1=0.3, and 𝜎2=0.2; (f) 𝑆10=110, 𝑆20=40, 𝜎1=0.3, and 𝜎2=0.2.
fig2
Figure 2: Error versus 𝑆1+𝑆2: The error is calculated by subtracting the approximate estimate from the exact result. (a) 𝑆10=110, 𝑆20=100, 𝜎1=0.25, and 𝜎2=0.15; (b) 𝑆10=110, 𝑆20=70, 𝜎1=0.25, and 𝜎2=0.15; (c) 𝑆10=110, 𝑆20=40, 𝜎1=0.25, and 𝜎2=0.15; (d) 𝑆10=110, 𝑆20=100, 𝜎1=0.3, and 𝜎2=0.2; (e) 𝑆10=110, 𝑆20=70, 𝜎1=0.3, and 𝜎2=0.2; (f) 𝑆10=110, 𝑆20=40, 𝜎1=0.3, and 𝜎2=0.2.

Next, we apply the same sequence of analysis to the two approximate closed-form probability distribution functions of the difference 𝑆 given in (2.11) and (2.15). Similar observations about the accuracy of the proposed approximation can be made for the difference 𝑆, too (see Figures 3 and 4). However, contrary to the case of 𝑆+, the estimation performs better for positive correlation. Of the two different approximation schemes for the 𝑆, the shifted LN process seems to have a comparatively better performance than the shifted SR process, as evidenced by the numerical results.

fig3
Figure 3: Probability density versus 𝑆1𝑆2: the dash lines denote the distributions of the approximate shifted lognormal process, the dotted lines indicate the distributions of the approximate shifted square-root process, and the solid lines show the exact results. (a) 𝑆10=110, 𝑆20=100, 𝜎1=0.25, and 𝜎2=0.15; (b) 𝑆10=110, 𝑆20=70, 𝜎1=0.25, and 𝜎2=0.15; (c) 𝑆10=110, 𝑆20=40, 𝜎1=0.25, and 𝜎2=0.15; (d) 𝑆10=110, 𝑆20=100, 𝜎1=0.3, and 𝜎2=0.2; (e) 𝑆10=110, 𝑆20=70, 𝜎1=0.3, and 𝜎2=0.2; (f) 𝑆10=110, 𝑆20=40, 𝜎1=0.3, and 𝜎2=0.2.
fig4
Figure 4: Error versus 𝑆1𝑆2: the error is calculated by subtracting the approximate estimate from the exact result. The dash lines denote the errors of the approximate shifted square-root process, and the solid lines show the errors of the approximate shifted lognormal process. (a) 𝑆10=110, 𝑆20=100, 𝜎1=0.25, and 𝜎2=0.15; (b) 𝑆10=110, 𝑆20=70, 𝜎1=0.25, and 𝜎2=0.15; (c) 𝑆10=110, 𝑆20=40, 𝜎1=0.25, and 𝜎2=0.15; (d) 𝑆10=110, 𝑆20=100, 𝜎1=0.3, and 𝜎2=0.2; (e) 𝑆10=110, 𝑆20=70, 𝜎1=0.3, and 𝜎2=0.2; (f) 𝑆10=110, 𝑆20=40, 𝜎1=0.3, and 𝜎2=0.2.

4. Conclusion

In this paper we have presented a new unified approach to model the dynamics of both the sum and difference of two correlated lognormal stochastic variables. By the Lie-Trotter operator splitting method, both the sum and difference are shown to follow a shifted lognormal stochastic process, and approximate probability distributions are determined in closed form. Illustrative numerical examples are presented to demonstrate the validity and accuracy of these approximate distributions. In terms of the approximate probability distributions, we have also obtained an analytical series expansion of the exact solutions, which can allow us to improve the approximation in a systematic manner. Moreover, we believe that this new approach can be extended to study both (1) the algebraic sum of 𝑁 lognormals, and (2) the sum and difference of other correlated stochastic processes, for example, two correlated CEV processes, two correlated CIR processes, and two correlated lognormal processes with mean-reversion.

Appendix

Lie-Trotter Splitting Approximation

Suppose that one needs to exponentiate an operator 𝐶 which can be split into two different parts, namely 𝐴 and 𝐵. For simplicity, let us assume that 𝐵𝐶=𝐴+, where the exponential operator exp(𝐶) is difficult to evaluate but exp(𝐴) and exp(𝐵) are either solvable or easy to deal with. Under such circumstances, the exponential operator exp(𝜀𝐶), with 𝜀 being a small parameter, can be approximated by the Lie-Trotter splitting formula [28]: 𝜀𝐶𝜀𝐴𝜀𝐵𝜀exp=expexp+𝑂2.(A.1) This can be seen as the approximation to the solution at 𝑡=𝜀 of the equation 𝑑𝑌𝑌/𝑑𝑡=(𝐴+𝐵) by a composition of the exact solutions of the equations 𝑑𝐴𝑌𝑌/𝑑𝑡= and 𝑑𝐵𝑌𝑌/𝑑𝑡= at time 𝑡=𝜀.

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