ABSTRACT
In this section, we provide a few examples of physical, economical, or engineering situations where considering dynamical systems described by differential equation in the presence of “random noise” is necessary and justifies at the intuitive level the formal development of stochastic differential equation, which will be presented in the following sections. Example 7.1 (Population growth model).
Consider a population growing in a random environment affected by the weather, climate, and other environmental conditions. The rate of growth, r(t), is, in general, time-dependent, but also dependent on the environmental, also time-dependent “noise”. So the size of the population, N(t), can be described by the differential equation https://www.w3.org/1998/Math/MathML"> d N ( t ) d T = ( r ( t ) + “ n o i s e ” ( t ) ) N ( t ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0077-01.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
with the initial condition N(0) = N 0 describing the original size of the population.
Example 7.2 (Electrical circuit with thermal noise).The current intensity itor(t) in an electrical circuit (shown in Figure 7.1) involving a capacitor with capacitance C, a resistor with resistivity R, and inductor with inductance L, is described by the integro-differential equation https://www.w3.org/1998/Math/MathML"> L d i ( t ) d t + R i ( t ) + 1 C ∫ 0 t i ( s ) d s = V ( t ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0077-02.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> An example of electronic circuit involving a capacitor, resistor and inductor, in the presence of external voltage. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/Fig07_01.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
where U(t) is the voltage applied to the circuit. In many situations, the voltage U(t) has a random “noise” component, which can be the results of transmission through a random medium or the thermal noise of the electrons moving around. In this case, after differentiation, the above integro-differential equation can be rewritten in the 78 form of a second-order differential equation https://www.w3.org/1998/Math/MathML"> L i ″ ( t ) + R i ′ ( t ) + 1 C i ( t ) = V ′ ( t ) + “ n o i s e ″ ( t ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0078-01.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
Example 7.3 (Filtering and prediction).Assume we observe a stochastic Q(t) generated by a random mechanism at times t 1 < t 2 < … t n . The observations are transmitted through channel in the presence of “noise” , so the result of observations is a sequence of random variables https://www.w3.org/1998/Math/MathML"> Z ( t k ) = Q ( t k ) + “ n o i s e ” ( t k ) , k = 1 , 2 , … , n , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0078-02.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
There are two questions related to this situation:
How do we recover Q(t k ), k = 1, 2,…, n, from observations of Z(t k ), k = 1, 2,…, n (the filtering problem)?
How do we estimate Q(t), t > t n , based on our observations (the prediction problem)?
These problems and be stated in the language of stochastic differential equations and the answer is provided by the so-called Kalman-Bucy filter construction to be discussed later.
Example 7.4 (Stochastic control).The macroeconomic question “How much should a nation save?” can be formulated as a problem of optimization for a system of differential equation with “noise”. Denote by L(t) the size of the labor force, K(t)–the capital, P(t) – production rate, and C(t) — consumption rate at time t. With the labor force dynamics described by the population growth equation from Example 7.1, the set of generally accepted equations tying together the above quantities are as follows: https://www.w3.org/1998/Math/MathML"> d L ( t ) d t = ( r ( t ) + “ n o i s e ” ( t ) ) L ( t ) , d K ( t ) d t = P ( t ) - C ( t ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0078-03a.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
79 and https://www.w3.org/1998/Math/MathML"> P ( t ) = A K α ( t ) L β ( t ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0079-01.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
for some values of the constants A, α, and β. Given the “utility function” U(c), with the “utiliy” U(c)Δt interpreted as consumption of goods at rate c in time interval Δt, and the “bequest” function ψ, the optimization problem can be phrased as follows: Determine the consumption rate C(t) which maximizes the “total utility” https://www.w3.org/1998/Math/MathML"> E [ ∫ 0 T U ( C ( t ) ) e - ρ t d t ] + ψ ( K ( T ) ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003216759/e7d667ff-34e1-4696-9d72-d545d9499230/content/unequ0079-04.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>
