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Profit Optimization of Cattle Growth with Variable Prices

Abstract

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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Acknowledgements

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.

Funding

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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Appendices

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 \(u<v\),

$$\begin{aligned} Cov\left[ X_u,X_v \right]= & \ \mathbb {E}\left[ \exp {(Y_u)}\exp {(Y_v)}\right] -\mathbb {E}\left[ \exp {(Y_u)}\right] \mathbb {E}\left[ \exp {(Y_v)}\right] \\= & \ \mathbb {E}\left[ \exp {(Y_u+Y_v)}\right] -\exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \\= & \ \exp \left( \mathbb {E}\left[ Y_u+Y_v\right] +\frac{1}{2}Var\left[ Y_u+Y_v\right] \right) -\exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \\= & \ \exp \left( \mu _u+\mu _v+\frac{\sigma ^2_u}{2}+\frac{\sigma ^2_v}{2}+Cov[Y_u,Y_v]\right) -\exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \\= & \ \exp \left( \mu _u+\frac{\sigma ^2_u}{2}\right) \exp \left( \mu _v+\frac{\sigma ^2_v}{2}\right) \left( \exp \left( Cov[Y_u,Y_v]\right) -1\right) . \end{aligned}$$
(51)

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

$$\begin{aligned} Cov\left[ X_u,X_v \right] &=\mathbb {E}\left[ (Y_u)^3(Y_v)^3\right] -\mathbb {E}\left[ (Y_u)^3\right] \mathbb {E}\left[ (Y_v)^3\right] \\&=\mathbb {E}\left[ (\mu _u+\sigma e^{-\beta u}I_u)^3(\mu _v+\sigma e^{-\beta v}I_v)^3\right] -\left( \mu _u (\mu _u^2+3\sigma _u^2)\right) \left( \mu _v (\mu _v^2+3\sigma _v^2)\right) \\&=\mathbb {E}\left[ \left( \mu _u^3+3\mu _u^2 \sigma e^{-\beta u} I_u + 3\mu _u \sigma ^2 e^{-2\beta u} I_u^2 + \sigma ^3 e^{-3\beta u} I_u^3\right) \left( \text {similar expression with }v\right) \right] \\&\quad -\left( \mu _u (\mu _u^2+3\sigma _u^2)\right) \left( \mu _v (\mu _v^2+3\sigma _v^2)\right) . \end{aligned}$$
(52)

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 \(u<v\):

$$\begin{aligned} Cov\left[ X_u,X_v \right] =9(\mu ^2_v+\sigma ^2_v)(\mu ^2_u+\sigma ^2_u)\sigma ^2_u e^{-\beta (v-u)}+18\mu _u\mu _v\sigma ^4_ue^{-2\beta (v-u)}+6\sigma ^6_u e^{-3\beta (v-u)}. \end{aligned}$$
(53)

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

$$\begin{aligned} \mathbb {E}\left[ X_t I_{[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}[}(X_t)X_s\right] ds&=\int _{a_{t,j-1/R}}^{a_{t,j/R}}x\left( \int _{-\infty }^{\infty }u f_{X_t,X_s}(x,u)du\right) dx \nonumber \\&=\int _{\ln {\frac{a_{t,j-1}}{R}}}^{\ln {\frac{a_{t,j}}{R}}}e^y\left( \int _{-\infty }^{\infty }e^z f_{Y_t,Y_s}(y,z)dz\right) dy \end{aligned}$$
(54)

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

$$\begin{aligned} \rho _{s,t}=\sqrt{\frac{K_s}{K_t}}=\sqrt{\frac{e^{2\beta (s-t_0)}-1}{e^{2\beta (t-t_0)}-1}} \end{aligned}$$
(55)

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

$$\begin{aligned}\int _{\ln \left( \frac{a_{t,j-1}}{R}\right) }^{\ln \left( \frac{a_{t,j}}{R}\right) }e^y\left( \int _{-\infty }^{\infty }e^z f_{Y_t,Y_s}(y,z)dz\right) dy \nonumber &=\exp \left( \mu _s+\frac{\sigma ^2_s}{2}\right) \int _{\ln \left( \frac{a_{t,j-1}}{R}\right) }^{\ln \left( \frac{a_{t,j}}{R}\right) }e^y\frac{1}{\sqrt{2\pi }\sigma _t}\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}-\rho \sigma _s\right) ^2\right) dy \nonumber \\&=\exp \left( \mu _s+\frac{\sigma ^2_s}{2}\right) \exp \left( \mu _t+\frac{\sigma ^2_t}{2}+\rho \sigma _t\sigma _s\right) \int _{A_{j-1}(t)}^{A_{j}(t)}\frac{1}{\sqrt{2\pi }}\exp \left( -\frac{1}{2}p^2\right) dp \end{aligned}$$
(56)

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

$$\begin{aligned} \mathbb {E}\left[ X_t I_{\big[\frac{a_{t,j-1}}{R},\frac{a_{t,j}}{R}\big[}(X_t)X_s\right] ds&=\int _{a_{t,j-1/R}}^{a_{t,j/R}}x\left( \int _{-\infty }^{\infty }u f_{X_t,X_s}(x,u)du\right) dx \nonumber \\&=\int _{A_{t,j-1}}^{A_{t,j}}y^3\left( \int _{-\infty }^{\infty }z^3 f_{Y_t,Y_s}(y,z)dz\right) dy. \end{aligned}$$
(57)

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

$$\begin{aligned}\int _{a_{t,j-1/R}}^{a_{t,j/R}}x\left( \int _{-\infty }^{\infty }u f_{X_t,X_s}(x,u)du\right) dx=&\int _{A_{t,j-1}}^{A_{t,j}}y^3\left( \int _{-\infty }^{\infty }z^3 f_{Y_t,Y_s}(y,z)dz\right) dy \nonumber \\=&\int _{A_{t,j-1}}^{A_{t,j}}y^3\frac{\exp \left( -\frac{1}{2}\left( \frac{y-\mu _t}{\sigma _t}\right) ^2\right) }{\sqrt{2\pi }\sigma _t}\left( \left( 3\sigma ^2_s(1-\rho ^2)\left( \mu _s+\rho \frac{\sigma _s}{\sigma _t}(y-\mu _t)\right) \right. \right. \nonumber \\&+\left.\left( \mu _s+\rho \frac{\sigma _s}{\sigma _t}(y-\mu _t)\right) ^3\right) dy \\=&\int _{B_{j-1}(t)}^{B_j(t)} (\sigma _t p+\mu _t)^3 \left( 3\sigma ^2_s(1-\rho ^2)\left( \mu _s+\rho \sigma _s p\right) +\left( \mu _s+\rho \sigma _s p\right) ^3\right) \frac{e^{-\frac{p^2}{2}}}{\sqrt{2\pi }}dp \\=&\int _{B_{j-1}(t)}^{B_j(t)} \left(\sigma ^3_tp^3+3\sigma ^2_t\mu _tp^2+3\sigma _t\mu ^2_tp+\mu ^3_t\right)(b_3(s)p^3+b_2(s)p^2+b_1(s)p+b_0)\frac{e^{-\frac{p^2}{2}}}{\sqrt{2\pi }}dp \nonumber \\=&\int _{B_{j-1}(t)}^{B_j(t)} \left( \sum _{m=0}^{6} c_m(s,t)p^m\right) \frac{e^{-\frac{p^2}{2}}}{\sqrt{2\pi }}dp=\sum _{m=0}^6c_m(s,t)Q_m(t), \end{aligned}$$
(58)

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

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

where

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

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). https://doi.org/10.1007/s11009-021-09889-z

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Keywords

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

Mathematics Subject Classification

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