We will develop this point of view which

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We will develop this point of view (which unfortunately can’t be made rigorous, but provides the properintuition). Hence, reviewing from above, with Brownian motion we would have:dz(t) =z(t+dt)z(t)N(0,dt).(1.17)Note that since Brownian motion has continuous sample paths,z(t) =z(t). However, for a Poisson process,we should think of differentials as(t;λ) =π(t+dt;λ)π(t;λ)Poisson(λdt)(1.18)in order to make sure that we capture jumps.Don’t forget that we also have the binary model approximations of Figures 1.3 and 1.4. Those binarymodels provide the proper intuition, and in both cases, sums of them will limit as Brownian motion or aPoisson process. For the differential, we simply replace ∆tbydtin (1.8) and (1.10), givingdzbraceleftbiggdt w.p.1/2dt w.p.1/2(1.19)and(λ)braceleftbigg1w.p.λdt0w.p.1λdt.(1.20)1.2.3Compound Poisson ProcessWhen Poisson processes jump, they jump up by 1. We can generalize this and allow them to jump randomly.Letπ(t;λ) be a Poisson process with jump timest1,t2,.... Construct a new processπY(t;λ), by assigningjumpY1at timet1,Y2at timet2, etc.whereY1, Y2, ...are iid random variables.This process can bewritten asπY(t;λ) =π(t;λ)summationdisplayi=0Yi.(1.21)That is, at timetit is the sum ofπ(t;λ) iid copies ofY, whereπ(t;λ) is a standard Poisson process. Processesof this form can also conveniently be written as integrals,πY(t,λ) =π(t;λ)summationdisplayi=0Yi=integraldisplayt0Ys(s;λ).(1.22)For this reason, we represent the differential form of a compound Poisson process byYdπ(t,λ). That is, wemay writeY=Ydπ.(1.23)Following along the lines of the binary approximation to a Poisson process as in Figure 1.4, an infinitesimalmodel of a compound Poisson process can be thought of asYdπ(λ)braceleftbiggYiw.p.λdt0w.p.1λdt(1.24)and a heuristic infinitesimal picture of this is given in Figure 1.6.1.2.4Ito Stochastic differential equationsStochastic integrals can be defined in different ways. The most useful for us is the Ito stochastic integral. Atthis point, I will not delve into the depths of the stochastic integral (because often people are never able toreturn!), but merely provide the intuition that you should take away when considering stochastic differentialequations.
8CHAPTER 1. BASIC BUILDING BLOCKS AND STOCHASTIC DIFFERENTIAL EQUATION MODELSFigure 1.6: Infinitesimal model of a compound Poisson process.A stochastic differential equation will be written as:dx(t) =a(x(t),t)dt+b(x(t),t)dz(t)(1.25)where in this case, it is being driven by Brownian motionz(t). (At this stage, I will ignore the technicalconditions that must be placed onaandbin order to make such an equation well defined.) We will interpretthis equation as follows:x(t+dt)x(t) =a(x(t),t)dt+b(x(t),t)(z(t+dt)z(t)).(1.26)Sincez(t) has independent increments, anda(x(t),t) andb(x(t),t) are evaluated at timet, they are inde-pendent ofdz(t) =z(t+dt)z(t). This is important! It allows us to do the following simple calculations

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Term
Winter
Professor
JIMPRIMBS
Tags
Derivative, Brownian Motion, dt, Ito, poisson processes

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