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Profit Optimization of Cattle Growth with Variable Prices


We apply a class of stochastic differential equations to model the growth of individual animals in randomly fluctuating environments using real weight data of males of the Mertolengo cattle breed. The use of these more realistic models can help farmers to optimize their profit. To this end we obtain the probability distribution, the first two moments and others quantities of interest of the profit obtained by raising and selling an animal under the more general, and more realistic, situation where the raising costs and the price per kg paid to the farmers depends on the animal’s age and weight category. We also present sensitivity results on how the expected profit and the optimal selling age vary with small changes on the estimates of the model parameters. We conclude that farmers are selling the animals a little earlier than the optimal selling age, which results in a lower profit per animal. The sensitivity analysis of the parameters shows that small changes on the parameters result in very small effects on the optimal profit and negligible effects on the optimal selling age.

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  1. Braumann CA (2019) Introduction to stochastic differential equations with applications to modelling in Biology and Finance. John Wiley & Sons

  2. Darmani Kuhi H, Porter T, Lopez S, Kebreab E, Strathe AB, Dumas A, Dijkstra J, France J (2010) A review of mathematical functions for the analysis of growth in poultry. Worlds Poult Sci J 66(2):227–240

    Article  Google Scholar 

  3. Filipe PA, Braumann CA (2008) Modelling individual animal growth in random environments. In: Proceedings of the 23rd International Workshop on Statistical Modelling, Eilers, P.H.C. (Ed.), Utrecht, pp 232–237

  4. Filipe PA, Braumann CA, Brites NM, Roquete CJ (2010) Modelling animal growth in random environments: an application using nonparametric estimation. Biom J 52(5):653–666

    MathSciNet  Article  Google Scholar 

  5. Filipe PA, Braumann CA, Carlos C (2015) Profit optimization for cattle growing in a randomly fluctuating environment. Optimization: A Journal of Mathematical Programming and Operations Research 64(6):1393–1407

    MathSciNet  Article  Google Scholar 

  6. Filipe PA, Braumann CA, Roquete CJ (2007) Modelos de crescimento de animais em ambiente aleatório. In: Ferrão ME, Nunes C, Braumann CA (eds) Estatística Ciência Interdisciplinar, Actas do XIV Congresso Anual da Sociedade Portuguesa de Estatística. Edições SPE, pp 401–410

  7. Filipe PA, Braumann CA, Roquete CJ (2012) Multiphasic individual growth models in random environments. Methodol Comput Appl Probab 14(1):49–56

    MathSciNet  Article  Google Scholar 

  8. Garcia O (1983) A stochastic differential equation model for the height of forest stands. Biometrics 39:1059–1072

    MathSciNet  Article  Google Scholar 

  9. Ghosh H, Prajneshu (2019) Optimum fitting of Richards growth model in random environment. J Stat Theory Pract 13(1):6

    MathSciNet  Article  Google Scholar 

  10. Goldsworthy P, Colegrove MP (1974) Growth and yield of highland maize in Mexico. J Agric Sci 83:213–221

    Article  Google Scholar 

  11. Gompertz B (1825) On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos Trans R Soc Lond 115:515–585

    Google Scholar 

  12. Kebreab E, Schuli-Zeuthen M, López S, Dias RS, De Lange CFM, France J (2007) Comparative evaluation of mathematical functions to describe growth and efficiency of phosphorus utilization in growing pigs. J Anim Sci 85:2498–2507

    Article  Google Scholar 

  13. Kozusko F, Bajzer Z (2003) Combining Gompertzian growth and cell population dynamics. Math Biosci 185:153–167

    MathSciNet  Article  Google Scholar 

  14. Ohnishi S, Akamine T (2006) Extension of von Bertalanffy growth model incorporating growth patterns of soft and hard tissues in bivalve molluscs. Fish Sci 72(4):787–795

    Article  Google Scholar 

  15. Prajneshu, Ghosh H (2019) Stochastic differential equation models and their applications to agriculture: an overview. Statistics and Applications 17(1):73–83

    Google Scholar 

  16. Qiming Lv, Pitchford J (2007) Stochastic von Bertalanffy models, with applications to fish recruitment. J Theor Biol 244:640–655

    MathSciNet  Article  Google Scholar 

  17. Rennolls K (1995) Forest height growth modelling. For Ecol Manag 71:217–225

    Article  Google Scholar 

  18. Richards F (1959) A flexible growth function for empirical use. J Exp Bot 10:290–300

    Article  Google Scholar 

  19. Russo T, Baldi P, Parisi A, Magnifico G, Mariani S, Cataudella S (2009) Lévy processes and stochastic von Bertalanffy models of growth, with application to fish population analysis. J Theor Biol 258(4):521–529

    Article  Google Scholar 

  20. Sengül T, Kiraz S (2005) Non-Linear models for growth curves in large white turkeys. Turk J Vet Anim Sci 29:331–337

    Google Scholar 

  21. Valentine H (1985) Tree-growth models: derivations employing the pipe-model theory. J Theor Biol 117:579–585

    Article  Google Scholar 

  22. Vasicek O (1977) An equilibrium characterization of the term structure. J Financ Econ 5:177–188

    Article  Google Scholar 

  23. Verhulst PF (1838) Notice sur la loi que la population poursuit dans son accroissement. Correspondance Mathématique et Physique 10:113–121

    Google Scholar 

  24. von Bertalanffy L (1957) Quantitative laws in metabolism and growth. Q Rev Biol 34:786–795

    Google Scholar 

  25. Yoshioka H, Yaegashi Y, Yoshioka Y, Tsugihashi K (2019) A short note on analysis and application of a stochastic open-ended logistic growth model. Lett Biomath 6(1):67–77

    MathSciNet  Article  Google Scholar 

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We are grateful to ACBM and José Pais (ACBM head engineer) for providing the data and for continuous support. We are also grateful to RuralBit for the help in extracting the data from Genpro database. We would also like to thank the Associate Editor and two anonymous reviewers for the careful review and suggestions that allowed us to broaden the scope and improve the final version of the paper.


The authors belong to the research center CIMA (Centro de Investigação em Matemática e Aplicações, Instituto de Investigação e Formação Avançada, Universidade de Évora), supported by FCT (Fundação para a Ciência e a Tecnologia, Portugal), project UID/04674/2020. This work was developed within the Operational Group PDR2020-1.0.1-FEADER-031130 - Go BovMais - Productivity improvement in the system of bovine raising for meat, funded by PDR 2020.

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Correspondence to Gonçalo Jacinto.

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Appendix 1

For the SGM, we have \(X_t=\exp {(Y_t)}\), and then \(\mathbb {E}[X_t]=\mathbb {E}[e^{Y_t}]=\exp \left( \mu _t+\frac{\sigma ^2_t}{2}\right)\). For \(u<v\),

$$\begin{aligned} Cov\left[ X_u,X_v \right]= & \ \mathbb {E}\left[ \exp {(Y_u)}\exp {(Y_v)}\right] -\mathbb {E}\left[ \exp {(Y_u)}\right] \mathbb {E}\left[ \exp {(Y_v)}\right] \\= & \ \mathbb {E}\left[ \exp {(Y_u+Y_v)}\right] -\exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \\= & \ \exp \left( \mathbb {E}\left[ Y_u+Y_v\right] +\frac{1}{2}Var\left[ Y_u+Y_v\right] \right) -\exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \\= & \ \exp \left( \mu _u+\mu _v+\frac{\sigma ^2_u}{2}+\frac{\sigma ^2_v}{2}+Cov[Y_u,Y_v]\right) -\exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \\= & \ \exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \left( \exp \left( Cov[Y_u,Y_v]\right) -1\right) . \end{aligned}$$

Denote by \(I_t=\int _{t_0}^te^{\beta s} dW_s\), which has a Gaussian distribution with mean 0 and variance (in result of the integrand function being deterministic) given by

$$K_t=Var[I_t]=\int _{t_0}^t e^{2 \beta } ds=\frac{1}{2\beta }\left( e^{2\beta t}-e^{2\beta t_0}\right) .$$

Let \(E_t=e^{-\beta (t-t_0)}\) and \(y_0=Y_{t_0}\). Then

$$Y_t=\mu _t+\sigma e^{-\beta t}I_t$$

has a Gaussian distribution with mean

$$\mu _t=\alpha +(y_0-\alpha )e^{-\beta (t-t_0)}=\alpha +(y_0-\alpha )E_t$$

and variance

$$\sigma^2_t=\sigma^2e^{-2\beta t}K_t=\frac{\sigma^2}{2\beta}(1-E^2_t).$$

Since the stochastic integrals, as functions of the upper limit of integration, have non-correlated increments, we get, for \(u<v\),

$$\begin{aligned} Cov[I_u,I_v]&=E[I_uI_v]=\mathbb {E}\left[ I_u\left( I_u+\int _u^ve^{\beta s}dW_s\right) \right] \\&=\mathbb {E}[I^2_u]+E\left[ I_u\int _u^v e^{\beta s}dW_s\right] \\&=\mathbb {E}[I^2_u]+E[I_u]E\left[ \int _u^v e^{\beta s}dW_s\right] \\&=K_u. \end{aligned}$$

Then \(Cov[Y_u,Y_v]\) is given by

$$\begin{aligned} Cov[Y_u,Y_v]&=E[(Y_u-\mu _u)(Y_v-\mu _v)]\\&=\sigma e^{-\beta u}\sigma e^{-\beta v}E[I_uI_v]\\&=\frac{\sigma ^2}{2\beta }\frac{E_v}{E_u}(1-E^2_u)\\&=\frac{E_v}{E_u}\sigma ^2_u,\, \text {for } u<v. \end{aligned}$$

Appendix 2

We are going to determine an expression for the covariance \(Cov\left[ X_u,X_v \right]\) with \(u<v\) for the SBRM, where \(X_t=(Y_t)^3\). So, since \(Y_t=\mu _t+\sigma e^{-\beta t}I_t\), with \(I_t=\int _{t_0}^te^{\beta s} dW_s\), we have

$$\begin{aligned} Cov\left[ X_u,X_v \right] &=\mathbb {E}\left[ (Y_u)^3(Y_v)^3\right] -\mathbb {E}\left[ (Y_u)^3\right] \mathbb {E}\left[ (Y_v)^3\right] \\&=\mathbb {E}\left[ (\mu _u+\sigma e^{-\beta u}I_u)^3(\mu _v+\sigma e^{-\beta v}I_v)^3\right] -\left( \mu _u (\mu _u^2+3\sigma _u^2)\right) \left( \mu _v (\mu _v^2+3\sigma _v^2)\right) \\&=\mathbb {E}\left[ \left( \mu _u^3+3\mu _u^2 \sigma e^{-\beta u} I_u + 3\mu _u \sigma ^2 e^{-2\beta u} I_u^2 + \sigma ^3 e^{-3\beta u} I_u^3\right) \left( \text {similar expression with }v\right) \right] \\&\quad -\left( \mu _u (\mu _u^2+3\sigma _u^2)\right) \left( \mu _v (\mu _v^2+3\sigma _v^2)\right) . \end{aligned}$$

We just need to expand the product inside the last expectation and compute the expectations of the form \(\mathbb {E}\left[ I_u^i I_v^j\right]\) (\(i,j=0,1,2,3\)) that appear in such expansion.

The stochastic integral \(I_t\), as a function of the upper limit of integration, has uncorrelated increments and, since in this case is Gaussian, it has independent increments. Therefore, since \(u<v\), in \(I_v=I_u+(I_v-I_u)\) we have \(I_u\) and \(I_v-I_u\) independent. Since \(I_t\) is Gaussian with mean 0 and variance \(K_t=\frac{1}{2\beta }\left( e^{2\beta t}-e^{2\beta t_0}\right)\), we have

$$\mathbb {E}\left[ I_u\right] =\mathbb {E}\left[ I_u^3\right] =\mathbb {E}\left[ I_v\right] =\mathbb {E}\left[ I_v^3\right] =0,\quad \mathbb {E}\left[ I_u^2\right] =K_u,\quad \mathbb {E}\left[ I_v^2\right] =K_v.$$

Note that \(I_v-I_u=\int _{u}^{v} e^{\beta s}ds\) is Gaussian with mean zero and variance \(K_v-K_u\) and is independent of \(I_u\). Using this and the fact that \(\mathbb {E}\left[ I_t^4\right] =3K_t^2\), doing the expansions in \(\mathbb {E}\left[ I_u^i I_v^j\right] =\mathbb {E}\left[ I_u^i (I_u+ (I_v-I_u))^j\right]\) (\(i,j=1,2,3\)) and simplifying, one obtains expressions for the remaining expectations. It can be seen that

$$\mathbb {E}\left[ I_u^i I_v^j\right] =0 \text { whenever } i+j \text { is odd }.$$

For example,

$$\begin{aligned}\mathbb {E}[I_uI^2_v]&=\mathbb {E}\left[ I_u\left( I_u+(I_v-I_u)\right) ^2\right] =\mathbb {E}[I^3_u]+2\mathbb {E}\left[ I^2_u(I_v-I_u)\right] +\mathbb {E}\left[ I_u(I_v-I_u)^2\right] \\&=0+2\times K_u \times 0+0\times (K_v-K_u)=0. \end{aligned}$$

Let us now compute \(\mathbb {E}\left[ I_u^i I_v^j\right]\) (\(i,j=1,2,3\)) for the cases where \(i+j\) is even. We have

$$\begin{aligned}\mathbb {E}[I_uI_v]&=\mathbb {E}\left[ I_u(I_u+(I_v-I_u)\right] =\mathbb {E}[I^2_u]+\mathbb {E}\left[ I_u(I_v-I_u)\right] \\&=\mathbb {E}[I^2_u]+\mathbb {E}\left[ I_u\right] \mathbb {E}\left[ I_v-I_u\right] =K_u+0\times 0=K_u \end{aligned}$$
$$\begin{aligned}\mathbb {E}[I_uI^3_v]&=\mathbb {E}\left[ I_u\left( I_u+(I_v-I_u)\right) ^3\right] =\mathbb {E}[I^4_u]+3\mathbb {E}\left[ I^3_u(I_v-I_u)\right] +3\mathbb {E}\left[ I^2_u\left( I_v-I_u\right) ^2\right] +\mathbb {E}\left[ I_u\left( I_v-I_u\right) ^3\right] \\&=3K^2_u+0+3K_u(K_v-K_u)+0=3K_uK_v \end{aligned}$$

and, using similar arguments, we can easily obtain

$$\mathbb {E}[I^2_uI^2_v]=2K^2_u+K_uK_v, \quad \mathbb {E}[I^3_uI_v]=3K^2_u, \qquad \mathbb {E}[I^3_uI^3_v]=3K^2_u(2K_u+3K_v).$$

Putting all together and simplifying, we have for \(u<v\):

$$\begin{aligned} Cov\left[ X_u,X_v \right] =9(\mu ^2_v+\sigma ^2_v)(\mu ^2_u+\sigma ^2_u)\sigma ^2_u e^{-\beta (v-u)}+18\mu _u\mu _v\sigma ^4_ue^{-2\beta (v-u)}+6\sigma ^6_u e^{-3\beta (v-u)}. \end{aligned}$$

Appendix 3

To derive the covariance between the selling price and the raising costs we have from (36)

$$\begin{aligned} Cov[V(t),C(t)]&=R C_3 \sum _{j=1}^k P_{t,j}\left( \int _{t_0}^t \mathbb {E}\left[ X_t I_{\big[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}\big[}(X_t)X_s\right] ds\right) \nonumber \\&\quad -R C_3 \sum _{j=1}^k P_{t,j}\left( \int _{a_{t,j-1}/R}^{a_{t,j}/R} x f_{X_t}(x) dx\right) \left( \int _{t_0}^t\mathbb {E}[X_s]ds\right) \end{aligned}$$

and what remains to be obtained in this expression is its first integral. We will do it now for the SGM.

Using a change of variable given by \(y=\ln {x}\) and \(z=\ln {u}\) we have

$$\begin{aligned} \mathbb {E}\left[ X_t I_{[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}[}(X_t)X_s\right] ds&=\int _{a_{t,j-1/R}}^{a_{t,j/R}}x\left( \int _{-\infty }^{\infty }u f_{X_t,X_s}(x,u)du\right) dx \nonumber \\&=\int _{\ln {\frac{a_{t,j-1}}{R}}}^{\ln {\frac{a_{t,j}}{R}}}e^y\left( \int _{-\infty }^{\infty }e^z f_{Y_t,Y_s}(y,z)dz\right) dy \end{aligned}$$

Note that, for \(s<t\), the joint distribution of \((Y_t,Y_s)\) is a Gaussian distribution with mean \((\mu _t,\mu _s)\), variances \((\sigma ^2_t,\sigma ^2_s)\) and correlation coefficient

$$\begin{aligned} \rho _{s,t}=\sqrt{\frac{K_s}{K_t}}=\sqrt{\frac{e^{2\beta (s-t_0)}-1}{e^{2\beta (t-t_0)}-1}} \end{aligned}$$

which we are going to abbreviate to \(\rho\). The joint probability density function is given by

$$f_{Y_t,Y_s}(y,z)=\frac{(2\pi \sigma _t \sigma _s)^{-1}}{\sqrt{1-\rho ^2}}\exp \left( \frac{-1}{2(1-\rho ^2)}\left( \left( \frac{y-\mu _t}{\sigma _t}\right) ^2-2\rho \left( \frac{y-\mu _t}{\sigma _t}\right) \left( \frac{z-\mu _s}{\sigma _s}\right) +\left( \frac{z-\mu _s}{\sigma _s}\right) ^2\right) \right)$$

To obtain the inner integral of (54), using the variable change \(\theta =\frac{z-\mu _s}{\sigma _s\sqrt{1-\rho ^2}}\), we obtain, after some calculations

$$\begin{aligned}\int _{-\infty }^{+\infty }e^z f_{Y_t,Y_s}(y,z)dz&=\int _{-\infty }^{+\infty }e^{\sigma _s\theta \sqrt{1-\rho ^2}+\mu _s}\frac{1}{2\pi \sigma _t}\exp \left( \frac{1}{2}\left( \left( \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\right) ^2-2\theta \left( \rho \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\right) +\theta ^2\right) \right) d\theta \\&=\frac{1}{\sqrt{2\pi }\sigma _t}\exp \left( \mu _s-\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\right) ^2 +\frac{1}{2}\left( \rho \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}+\sigma _s\sqrt{1-\rho ^2}\right) ^2\right) \\&\quad \int _{-\infty }^{+\infty }\frac{1}{\sqrt{2\pi }}\exp \left( -\frac{1}{2}\left( \theta -\rho \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}-\sigma _s\sqrt{1-\rho ^2}\right) ^2 \right) d\theta \\&=\frac{1}{\sqrt{2\pi }\sigma _t}\exp \left( \mu _s-\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\right) ^2 +\frac{1}{2}\left( \rho \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}+\sigma _t\sqrt{1-\rho ^2}\right) ^2\right) \\&=\exp \left( \mu _s+\frac{\sigma ^2_s}{2}\right) \frac{1}{\sqrt{2\pi }\sigma _t}\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}-\rho \sigma _s\right) ^2\right) . \end{aligned}$$

Having a closed expression for the inner integral, replacing in (54) and using another variable change \(p=\frac{y-(\mu _t+\rho \sigma _t\sigma _s+\sigma ^2_t)}{\sigma _t}\) we obtain

$$\begin{aligned}\int _{\ln \left( \frac{a_{t,j-1}}{R}\right) }^{\ln \left( \frac{a_{t,j}}{R}\right) }e^y\left( \int _{-\infty }^{\infty }e^z f_{Y_t,Y_s}(y,z)dz\right) dy \nonumber &=\exp \left( \mu _s+\frac{\sigma ^2_s}{2}\right) \int _{\ln \left( \frac{a_{t,j-1}}{R}\right) }^{\ln \left( \frac{a_{t,j}}{R}\right) }e^y\frac{1}{\sqrt{2\pi }\sigma _t}\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}-\rho \sigma _s\right) ^2\right) dy \nonumber \\&=\exp \left( \mu _s+\frac{\sigma ^2_s}{2}\right) \exp \left( \mu _t+\frac{\sigma ^2_t}{2}+\rho \sigma _t\sigma _s\right) \int _{A_{j-1}(t)}^{A_{j}(t)}\frac{1}{\sqrt{2\pi }}\exp \left( -\frac{1}{2}p^2\right) dp \end{aligned}$$

with \(A_j(t)=\frac{\ln {\frac{a_{t,j}}{R}}-(\mu _t+\rho \sigma _t\sigma _s+\sigma ^2_t)}{\sigma _t}\). Since the covariance between the two variables is given by \(\rho _{s,t}\sigma _t\sigma _s=\frac{E_t}{E_s}\sigma ^2_s\), we have

$$\begin{aligned}\int _{t_0}^t \mathbb {E}\left[ X_t I_{[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}[}(X_t)X_s\right] ds&=\int _{t_0}^t\left( \int _{a_{t,j-1/R}}^{a_{t,j/R}}x\left( \int _{-\infty }^{\infty }u f_{X_t,X_s}(x,u)du\right) dx \right) ds\\&=\int _{t_0}^t\left( \exp \left( \mu _s+\frac{\sigma ^2_s}{2}\right) \exp \left( \mu _t+\frac{\sigma ^2_t}{2}+\rho \sigma _t\sigma _s\right) \int _{A_{j-1}(t)}^{A_{j}(t)}\frac{1}{\sqrt{2\pi }}\exp \left( -\frac{1}{2}p^2\right) dp\right) ds\\&=\exp \left( \mu _t+\frac{\sigma ^2_t}{2}\right) \int _{t_0}^t\left( \exp \left( \mu _s+\frac{\sigma ^2_s}{2}+\frac{E_t}{E_s}\sigma ^2_s\right) \right. \\& \quad\left. \left( \Phi \left( \frac{\ln (\frac{a_{t,j}}{R})-(\mu _t+\sigma ^2_t+\frac{E_t}{E_s}\sigma ^2_s)}{\sigma _t}\right) -\Phi \left( \frac{\ln (\frac{a_{t,j-1}}{R})-(\mu _t+\sigma ^2_t+\frac{E_t}{E_s}\sigma ^2_s)}{\sigma _t}\right) \right) \right)ds \end{aligned}$$

for the SGM, which is to be solved numerically.

Appendix 4

To derive the covariance between the selling price and the raising costs given by (36), we have seen in Subsection 3.3 that the only remaining thing to be obtained for the specific models is the integral \(\int _{t_0}^t \mathbb {E}\left[ X_t I_{\big[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}\big[}(X_t)X_s\right] ds\). We will do it now for the SBRM. Put \(A_{t,j}=\left( \frac{a_{t,j}}{R}\right) ^{1/3}\) and \(B_{j}(t)=\frac{A_{t,j}-\mu _t}{\sigma _t}=\frac{\left( \frac{a_{t,j}}{R}\right) ^{1/3}-\mu _t}{\sigma _t}\).

Using a change of variable given by \(y=x^{1/3}\) and \(z=u^{1/3}\), we have

$$\begin{aligned} \mathbb {E}\left[ X_t I_{\big[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}\big[}(X_t)X_s\right] ds&=\int _{a_{t,j-1/R}}^{a_{t,j/R}}x\left( \int _{-\infty }^{\infty }u f_{X_t,X_s}(x,u)du\right) dx \nonumber \\&=\int _{A_{t,j-1}}^{A_{t,j}}y^3\left( \int _{-\infty }^{\infty }z^3 f_{Y_t,Y_s}(y,z)dz\right) dy. \end{aligned}$$

To obtain the inner integral of (57), using two variable changes \(\theta =\frac{z-\mu _s}{\sigma _s\sqrt{1-\rho ^2}}\) and later \(\nu =\theta -\rho \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\), and the expression (55) for the correlation coefficient, we obtain for \(s<t\), after some computations

$$\begin{aligned}\int _{-\infty }^{+\infty }z^3 f_{Y_t,Y_s}(y,z)dz&=\int _{-\infty }^{+\infty }\frac{\left(\sigma _s\theta \sqrt{1-\rho ^2}+\mu _s\right)^3}{2\pi \sigma _t}\exp \left( -\frac{1}{2}\left( \left( \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\right) ^2-2\theta \left( \rho \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\right) +\theta ^2\right) \right) d\theta \\&=\frac{\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}\right) ^2\right) }{2\pi \sigma _t} \int _{-\infty }^{+\infty }\left(\sigma _s\theta \sqrt{1-\rho ^2}+\mu _s\right)^3\exp \left( -\frac{1}{2}\left( \theta -\rho \frac{y-\mu _t}{\sigma _t\sqrt{1-\rho ^2}}\right) ^2 \right) d\theta \\&=\frac{\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}\right) ^2\right) }{\sqrt{2\pi }\sigma _t} \int _{-\infty }^{+\infty }\left(\sigma _s\nu \sqrt{1-\rho ^2}+\mu _s+\rho \frac{\sigma _s}{\sigma _t}\left(y-\mu _t\right)\right)^3\frac{\exp \left( -\frac{1}{2}\nu ^2\right) }{\sqrt{2\pi }}d\nu \\&=\frac{\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}\right) ^2\right) }{\sqrt{2\pi }\sigma _t}\left( 3\sigma ^2_s(1-\rho ^2)\left( \mu _s+\rho \frac{\sigma _s}{\sigma _t}(y-\mu _t)\right) +\left( \mu _s+\rho \frac{\sigma _s}{\sigma _t}(y-\mu _t)\right) ^3\right). \end{aligned}$$

Having a closed expression for the inner integral, replacing in (57) and using another variable change \(p=\frac{y-\mu _t}{\sigma _t}\), we obtain

$$\begin{aligned}\int _{a_{t,j-1/R}}^{a_{t,j/R}}x\left( \int _{-\infty }^{\infty }u f_{X_t,X_s}(x,u)du\right) dx=&\int _{A_{t,j-1}}^{A_{t,j}}y^3\left( \int _{-\infty }^{\infty }z^3 f_{Y_t,Y_s}(y,z)dz\right) dy \nonumber \\=&\int _{A_{t,j-1}}^{A_{t,j}}y^3\frac{\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}\right) ^2\right) }{\sqrt{2\pi }\sigma _t}\left( \left( 3\sigma ^2_s(1-\rho ^2)\left( \mu _s+\rho \frac{\sigma _s}{\sigma _t}(y-\mu _t)\right) \right. \right. \nonumber \\&+\left.\left( \mu _s+\rho \frac{\sigma _s}{\sigma _t}(y-\mu _t)\right) ^3\right) dy \\=&\int _{B_{j-1}(t)}^{B_j(t)} (\sigma _t p+\mu _t)^3 \left( 3\sigma ^2_s(1-\rho ^2)\left( \mu _s+\rho \sigma _s p\right) +\left( \mu _s+\rho \sigma _s p\right) ^3\right) \frac{e^{-\frac{p^2}{2}}}{\sqrt{2\pi }}dp \\=&\int _{B_{j-1}(t)}^{B_j(t)} \left(\sigma ^3_tp^3+3\sigma ^2_t\mu _tp^2+3\sigma _t\mu ^2_tp+\mu ^3_t\right)(b_3(s)p^3+b_2(s)p^2+b_1(s)p+b_0)\frac{e^{-\frac{p^2}{2}}}{\sqrt{2\pi }}dp \nonumber \\=&\int _{B_{j-1}(t)}^{B_j(t)} \left( \sum _{m=0}^{6} c_m(s,t)p^m\right) \frac{e^{-\frac{p^2}{2}}}{\sqrt{2\pi }}dp=\sum _{m=0}^6c_m(s,t)Q_m(t), \end{aligned}$$

with (we remind that \(\rho\) is an abbreviation of the correlation coefficient \(\rho _{s,t}\) given by (55))

$$Q_m(t)=\int _{B_{j-1}(t)}^{B_j(t)}p^m\phi (p)dp$$
$$b_3(s,t)=\rho _{s,t}^3 \sigma _s^3$$
$$b_2(s,t)=3\rho _{s,t}^2 \sigma _s^2 \mu _s$$
$$b_1(s,t)=3\rho _{s,t} \sigma _s \mu _s^2 +3\rho _{s,t} (1-\rho _{s,t}^2)\sigma _s^3$$
$$b_0(s,t)=\mu _s^3 +3(1-\rho _{s,t}^2)\sigma _s^2 \mu _s$$

and therefore with

$$c_0(s,t)=\mu ^3_tb_0(s,t)$$
$$c_1(s,t)=3\sigma _t\mu _t^2b_0(s,t)+\mu _t^3b_1(s,t)$$
$$c_2(s,t)=3\sigma _t^2 \mu _t b_0(s,t)+3\sigma _t\mu _t^2b_1(s,t)+\mu _t^3b_2(s,t)$$
$$c_3(s,t)=\sigma _t^3 b_0(s,t)+3\sigma _t^2 \mu _t b_1(s,t)+3\sigma _t\mu _t^2b_2(s,t)+\mu _t^3b_3(s,t)$$
$$c_4(s,t)=\sigma _t^3 b_1(s,t)+3\sigma _t^2 \mu _t b_2(s,t)+3\sigma _t\mu _t^2b_3(s,t)$$
$$c_5(s,t)=\sigma _t^3 b_2(s,t)+3\sigma _t^2 \mu _t b_3(s,t)$$
$$c_6(s,t)=\sigma _t^3 b_3(s,t).$$

To obtain the \(Q_m(t)\), we can see that

$$Q_0(t)=\Phi (B_j(t))-\Phi (B_{j-1}(t))$$
$$Q_1(t)=\phi (B_{j-1}(t))-\phi (B_{j}(t)).$$

Integrating by parts , we get

$$Q_{m+2}(t)=(m+1)Q_m(t)+B^{m+1}_{j-1}(t)\phi (B_{j-1}(t))-B^{m+1}_{j}(t)\phi (B_{j}(t)),$$

which can be used recursively to easily give the remaining \(Q_m(t)\) (\(m=2,3,4,5,6\)).

Putting it all together we obtain

$$\begin{aligned}&\int _{t_0}^t \mathbb {E}\left[ X_t I_{[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}[}(X_t)X_s\right] ds=\sum _{m=0}^6 Q_m(t)\int _{t_0}^t c_m(s,t)ds=\sum _{m=0}^6 Q_m(t)d_m(t), \end{aligned}$$


$$d_m(t)=\int _{t_0}^tc_m(s,t)ds$$

can be obtained in closed form long expressions.

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Jacinto, G., Filipe, P.A. & Braumann, C.A. Profit Optimization of Cattle Growth with Variable Prices. Methodol Comput Appl Probab (2021).

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  • Stochastic differential equations
  • Cattle growth
  • Profit optimization
  • Sensitivity analysis
  • Maximum likelihood estimators.

Mathematics Subject Classification

  • 60H10
  • 60E05
  • 62G07
  • 91B70
  • 92D99