9 EVENT UNCERTAINTY 1699 Event UncertaintyA class of problem involving uncertainty can be modeled using either deter-ministic optimal control or stochastic control of a jump process – defined as astochastic process whose realization can be a discontinuous function of time.We illustrate both of these approaches using a problem in which the prob-ability of a random event depends on the value of a state variable that thedecision maker controls. For example, the probability of a climate-relatedcatastrophe might depend on the stock of GHSs. At a point in time thisstock is predetermined, so the decision maker is not able to affect the prob-ability of the catastrophe over the next short interval of time. However, bychoosing an emissions path, the decision maker chooses the trajectory of thestock of GHGs, and thereby influences the probability of catastrophe in thefuture.9.1 PreliminariesThe building block of this kind of model is the hazard function, which givesthe probability of an event (such as a catastrophe) over the next small intervalof time, conditional on the event not having yet occurred. Denote τ asa random variable, the time at which this event occurs, and denote thecumulative distribution function as F (t)=Pr{τ ≤ t}. We assume that thisfunction is differentiable, so f(t) ≡dF (t)dtis the probability density function.For small dt, f (t)dt is approximately the probability that the event occursduring the interval of time (t, t + dt).If A and B are two random events, then using the rule for conditional prob-abilities we havePr (A | B)=Pr (A ∩ B)Pr (B).This formula states that the probability of event A, given that event Bhas occurred, equals the probability that both A and B occur divided bythe unconditional probability that B occurs. Think of event A as being“the disaster occurs over the interval (t, d + dt)” and event B being “theprobability does not occur by time t”. Applying the formula for conditional9 EVENT UNCERTAINTY 170probabilities, we havePr {event occurs during (t + dt) | event has not occurred by t} (150)= h(t)dt ≡f(t)dt1 − F (t).The function h(t) is the hazard rate. If we interpret dt = 1 as an “instant”,then the hazard rate equals the probability that the event occurs during thenext instant, given that it has not yet occurred.One of the earliest applications of event uncertainty involves the problemof life-time consumption when death occurs at a random time (cite Yari).For this problem, the probability of the “event”, death, is exogenous: thedecision maker cannot alter the future risk, but he can adjust his consumptiondecisions to take the risk into account. We use this problem to illustrate themethod of analysis, before turning to the problem of interest, in which thedecision maker affects future risk.If death occurs at time t, the present discounted value of the agent’s welfareat the initial time 0 ist0e−rsU (cs) ds + d (t) B (k(t)) . (151)In this problem, the instantaneous flow of utility is the increasing and concavefunction U (c) and the pure rate of time preference (the discount rate) is r.The amount of wealth that the decision maker bequeaths to his heirs is k(t),and B(k) is the utility that the decision maker obtains from this bequest;d(t) discounts utility of the bequest at time t back to the first period. Thefunction d(t) could have a variety a shapes. For example, it might besmall for t close to 0 and close to T but large for intermediate values of t;this shape would arise if the decision maker does not worry about leaving abequest before he has a family (small t) or after the family is grown (large t),but is concerned about the bequest during his middle age, when his familyis young.The decision maker is able to invest wealth at a constant rate of return i andreceives an exogenous income stream Y (t), so his wealth obeys the differentialequation˙k (t)=ik(t)+Y (t) − c(t), k(0) = k0, given. (152)9 EVENT UNCERTAINTY 171The probability distribution function for the random time of death, t,isF (t),with F (0) = 0 and F (T ) = 1. The first equality states that the decisionmaker is certainly alive at time 0, and the second states that he is certainlydead by time T . The decision maker is risk neutral and maximizes theexpectation of lifetime utilitymax{c}Ett0e−rsU (cs) ds + d (t) B (k(t)) (153)=max{c}T0 t0e−rsU (cs) ds + d (t) B (k(t))f(t)dt.The term in square brackets is lifetime utility conditional on death occurringat time t; we multiply this term times the probability of death at time t,f(t), and integrate over t to obtain the expectation of lifetime utility.Denotez(t)=t0e−rsU (cs) ds =⇒dzdt= e−rtU (ct) (154)and integrate the first term on the right side of equation 153 by parts, usingf(t)=F(t)toobtainT0z(t)f(t)dt =T0z(t)F(t)dt = z(t)F (t) |T0−T0F (t)dzdtdt=T0e−rtU (ct) dt −T0F (t)e−rtU (ct) dt=T0(1 − F (t)) e−rtU (ct) dt.The third equality uses the fact that z(0) = 0 and F (T )=1. Usingtheexpression after the last equality, we write the expected payoff in equation153 asmax{c}T0(1 − F (t)) e−rtU (ct)+d (t) B (k(t)) f (t)dt. (155)At each point in time, t, if the agent is alive he obtains the present valueutility of consumption e−rtU (ct),andifhediesatthatpointintimeheobtains the present value utility of the bequest d (t) B (k(t)). The probabilityof the first event is (1 − F (t)) and the probability of the second event is f(t).The expected payoff is the integral over time of the two payoffs, weighted9 EVENT UNCERTAINTY 172by their probabilities. Equation 152 is the constraint to the optimizationproblem.These manipulations transform the original problem involving risk into analmost standard deterministic problem. Recall that in a deterministic prob-lem the optimal control rule can be written either as a function of time, i.e.as an open loop control rule, or as function of the state variable (and possi-bly also of time), i.e. as a feedback rule. The two solutions are equivalentbecause in a deterministic setting the decision maker does not acquire newinformation as time goes on. The control problem consisting of expression155 and the constraint 152 appears to be a deterministic problem, becausethe random time of death has been “concentrated out”, i.e. removed bytaking expectations.Thus, it appears that the decision maker can choose at the initial time t =0the entire trajectory of consumption. This appearance is correct,