Question

Newton’s Law of Cooling and the Ornstein-Uhlenbeck Process

The Law of Cooling says the temperature difference between an object (say a hot cup of coffee) and the ambient temperature (the temperature in the room) declines exponentially:

If T(t) is the temperature of the object at time t, we have the ODE:

d/dt(T(t) – T_room) = b (T(t) – T_room)    b < 0, T_room is a constant.

Equivalently,   dT/dt = b (T – T_room)

T(0) is the starting temperature of the object (T(0) > T_room); b is the rate.  

[1] Solve this ODE to obtain T as a function of t.

If we make this process stochastic in the simplest way possible – by adding a Brownian motion to the increment – we obtain the Ornstein-Uhlenbeck process:

dS   =   b (S - S) dt + s dW

S(0) = S₀     b, s, and S are constants

b < 0

The condition b < 0 means that when S(t) is larger than S the drift will pull S(t) down (closer to S). When S(t) is smaller than S the drift will pull S(t) up (closer to S).

[2] Solve the SDE.

Hint: Consider the function   A(S,t) = exp(-bt) S. Compute the increment dA using Ito’s Formula. You should end up with:   

dA =   f(t) dt + g(t) dW for functions f(t) and g(t).

Is the increment dA normally distributed?

It should be easy to integrate this type of SDE.

[3] Explain why the random variable S(T) is normally distributed.

[4] How does the standard deviation of S(T) grow with T? Does it grow without bound?

Answer 1