Sloppy derivations of Ito s formula and the Fokker-Planck equations

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1 Sloppy derivations of Ito s formula and the Fokker-Planck equations P. G. Harrison Department of Computing, Imperial College London South Kensington Campus, London SW7 AZ, UK pgh@doc.ic.ac.uk April 005 Abstract These steps get you Ito s formula and the Fokker-Planck forward and backward partial differential equations from the stochastic differential equation dx t µx t,tdt + σx t,tdw t, where W t is a Wiener process, using simple, intuitive steps. The derivations are not to be considered as rigorous but hopefully they give the gist. 1 Ito s chain rule When considering the stochastic differential equation sde dx t µx t,tdt + σx t,tdw t where W t is a Wiener process, i.e. has independent Normal increments W t W s with zero mean and variance t s, the magnitude of dw t is considered to be dt. This arises from consideration of the corresponding integral equation Wt X t X 0 + µx u,u + σx u,udw u 0 W 0 where the integrals are Lebesgue. 1 1 It can be shown that, over an interval [a, b], W t has infinite total variation, Wt has finite total variation W t has finite quadratic variation and that W p t has zero total variation for p>. The total variation of W t over [a, b], TVW t is the cumulative sum of its changes in value the number of feet of ascent plus the number of feet of descent on a mountain hike. TVW t lim sup n a t 1... t n b i1 n 1 W ti+1 W ti where t 1,...,t n is a partition of [a, b],t 1 a, t n b of size n 1. In fact this is one of the crucial properties that W t has to make the theory work. It can be generalised to continuous semi-martingales in M. In particular TVWt b a, which suggests that dw t has expectation dt. Moreover, the total quadratic variation of Wt is zero, which implies that dw t is dt w.p. one. 1

2 Now consider the function fx t,t. Expanding hopefully with no justification at all since we don t know properties about the infinitesimal random terms dx t nor about differentiation wrt a random variable, using Taylor s series, up to terms linear in dt gives df fx t+dt,t+dt fx t,t f f dt + dx t + 1 f t X t Xt dx t f t + µx t,t f + σx t,t f dt + σx t,t f dw t X t X t X t This is Ito s formula in differential form. Maybe it s most commonly known in this form because of the simple, if sloppy, derivation. A proper derivation obtains the integral form using precise definitions of Lebesgue integrals and probability measure. A good background reference on measure and probability is Kingman and Taylor s book CUP. The Fokker-Planck equation for the probability density The probability density function for the random variable X t can be obtained under appropriate conditions not investigated here by deriving a partial differential equation pde from the given sde. The time-dependent pdf, when it exists, is the function p defined by IPa<X t b X x b a px, t; x, dx for t, given that the state X x at time. Equivalently, px, t; x, dx IPx<X t x +dx X x We also define the random variable px, t X, which takes the value px, t; x, when X x. Similarly, looking backwards in time, we define the random variable px t x,, which also takes the value px, t; x, when X t x..1 The forward equation In integral form, Ito s equation is f fx t,t fx, + + µx u,u f + σx u,u f X u + σx u,u f dw u X u X u Let fx, u δx x for some given x, so that E[fX u,u X ] pz,u X δz xdz px, u X

3 Taking the expectation of Ito s integral equation, conditional on X,we now obtain px, t X px, X dz pz,u X δz x µz,u + σz,u z δz x z since f 0 and every increment dw u in any limiting approximation to the integral has zero expectation and is independent of X. Changing the order of integration, integrating by parts three times and abbreviating pz,u X by p, µz,u byµ and σz,u byσ, we obtain: giving px, t X px, X px, t X px, X [ ] pµδz x dz δz x µp z ] 1 + [p σ δz x z [ t 1 σ p ] δz x z + dz δz x 1 σ p z + t t µx, upx, u X σx, u px, u X 1 where differentiation with respect to x, a fixed value, is interpreted as differentiation w.r.t. z, evaluated at the point z x. Considering,x,x as fixed, px, t X px, px, u X t X and so px, t X t or px, t; x, t + µx, upx, u X + µx, upx, u; x, This is the Fokker-Planck forward equation. 1 σx, u px, u X 0 1 σx, u px, u; x, 0 3

4 . The backward equation The Fokker-Planck backward equation is obtained by taking fx u,u px, t X u for given x, t. Multiplying Ito s differential equation at time ubyδx u x and taking expectations gives: rx,udfx,u fx rx,u,u + µx,u fx,u + σx,u fx,u where the differentiation with respect to x is interpreted as differentiation of fz,u w.r.t. z, evaluated at the point z x similarly to the case with x above. Thus, since fx,u 0: dfx,u Integrating, we find µx,u fx,u + σx,u px, t; x,t px, t; x, px, t; x,u µx,u px, t; x,u + σx,u and so px, t; x, + µx, px, t; x, + σx, This is the Fokker-Planck backward equation..3 A direct derivation fx,u px, t; x,u px, t; x, 0 As an alternative approach, consider px, t + τ; x, dx px, t; x, + p t τ dx for small τ in the standard cf. CTMCs Chapman-Kolmogorov equation: px, t + τ; x, dx dx The expresion in square brackets is py, t; x, px, t + τ; y, tdy px,t; x, [ px, t + τ; x,tdx ] d IPx<X t+τ x +dx X t x IPx<X t +dx t x +dx X t x IP0 < +µx,tτ + σx,tdw t dx X t x µx,tτ µx,tτ dx IP < dw t + σx,t σx,t σx,t dx µx,tτ σx,t w σx,t 4

5 where w is the pdf of the given random variable dw t, which has mean 0 and variance τ. Omitting the arguments of µ and σ, let so that Then we get: z g µτ σ g 1 z µτ + σz px, t + τ; x, dx dx px g 1 z,t; x, σg g 1 wzdz z Now we expand the functions in the integrand in Taylor series up to order τ. Note that the integrals zwzdz 0 z wzdz τ these being the mean and variance of dw t for time increment dt τ. Thus we need not consider powers of z higher than z. Recalling that the functions µ and σ are abbreviations for µx,t and σx,t respectively, where g 1 z, we obtain: σg 1+τµ + µτσ /σ 1+τµ + σ z For brevity, we omit the last three arguments of p, these always being t; x,. We find, to first order in τ where the subscripts x denote partial differentiation w.r.t. x at argument x, t, e.g. µ x µx,t : px g 1 z p p x g 1 z+p xx g 1 z / p p x µτ +σ σ x z+p xx σ z / and since Hence, σg 1 1 σ x z µ x τ +σσ xx + σ xz σg 1 + σ x z + µ x τ σ xx z px g 1 z σg p p x µτ + σz σσ x z +p xx σ z / 1 σ x z µ x τ +σ xx z / p p x σ + pσ x z p x µ + pµ x τ +4p x σσ x + p xx σ + pσ xx z / to first order in τ p σp x z µp x τ +σ p xx z / 5

6 Performing the integral then yields: px, t + τ; x, dx dx p + p t τ dx p µp x τ +σ p xx τ/ which gives the forward equation. To obtain the backward equation, we use and px, t; x, τdx px, t; x, τdx The expresion in square brackets is px, t; x, p τ dx [ px +, ; x, τdx ] px, t; x +, d IPx + <X x + +dx X τ x IPx + <X τ+τ x + +dx X τ x IP <µx, ττ + σx, τdw τ +dx µτ µτ IP < dw t σ τ + dx σ σ dx µτ σ w σ where µ and σ are evaluated at µx, τ and σx, τ respectively crucially, not functions of, which greatly simplifies things. Now let z µτ/σ so that dz d /σ. The integral then becomes, to first order in τ, px, t; x + µτ + σz, wzdz px, t; x, + p µτ + σz+1 p σ z wzdz px, t; x, + p µτ + 1 p σ τ and so the px, t; x, cancels and, comparing coefficients of τ, we are left with p p µ + 1 p σ which is the Fokker-Planck backwards equation. 3 Conclusion These notes are intended to take the mystery out of stochastic calculus by showing, by simple methods, how some of the main starting points can be reached. As a result, there is no rigour in the derivations, for example limiting operations are interchanged freely and the singularities of the Dirac 6

7 delta-function and its derivatives are ignored obliviously. Rigorous analyses tend to be elegant and apply very generally we only used the fact that the increments dw t are independent, have zero mean and variance equal to the time-increment, unbounded linear variation, finite quadratic variation and zero higher-order variation not that they are also Normally distributed. However, they are hard to grasp by the non-specialist. At the other extreme, practical applications tend to be presented with little or no justification and leave the reader in the dark. These notes are an attempt to bridge the gap. Comments will be very welcome. 7