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Exercise 7.13 (M)

Do the Taylor expansion and averaging of Eq. (7.86) to show that m(x, t, T) satisfies the differential equation (with variables suppressed)

Now add the interpretation that X(t) indicates the position of an individual following the stochastic differential equation given above and that

Prfbeing killed in the next dt|X(t) = x}= y(x)dt + o(dt) (7.88)

so that u(x, t, T) represents the probability of surviving from t to T, given that X(t) = x. (If you don't want to think of this as position, think of X(t) as energy reserves, with death by starvation in the next dt determined by y(x), or any other analogy that works for you.) Use the method of thinking along sample paths to get directly to Eq. (7.87) (hint: to survive from t to T, the individual must first survive from t to t + dt and then from t + dt to T).

Equations (7.83) and (7.87) are called the Feynman-Kac formula. In 1948, when Richard Feynman presented his path integral formulation of quantum mechanics (which involves the Schroedinger equation, also a diffusion-like equation), Mark Kac recognized that path integrals could thus be used to solve the usual diffusion equation (see Connections).

There is also associated forward equation for the probability density of the process:

f (y, s, x, t)dy = Pr{y < X(s) < y + dy and the individual is still alive X(t) = xg

Following the same procedures as in the previous section leads us to fs = 2 (a(y; s)f )yy -(b(y; s)f)y " V(x)f

with the appropriate delta function as a condition as s approaches t.

Finally, let us consider one more equation, in this case assuming that X(t) represents the population size of a harvested stock and that at time s when stock size is X(s) the economic return is r(X(s), s). If the discount rate is 6, the long-term discounted rate of return given that X(t) = x is u(x, t) = E^

where, as before, the expectation refers to an average over the sample paths that begin at X(t) = x. We now break the integral into two pieces

and recognize that the first integral on the right hand side is r(x, t)e_4idt + o(dt)

and that the second integral can be conditioned into an average over dX of the average over the new starting point X(t + dt) = x + dX:

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