Continuous Time Stochastic Processes, Schemes and Mind Maps of Stochastic Processes

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Jes˙s Fern·ndez-Villaverde

University of Pennsylvania

November 9, 2013

Filtration

Fix a probability space (Ω, F , P).

DeÖne t 2 [ 0 , ∞) = R +.

Filtration: a family F = fFt : t  0 g of increasing σ algebras contained in F :

Fs  Ft for 8 s  t and Ft  F

Clearly, F∞ = F is the smallest σ algebras containing 8Ft.

(Ω, F , P): Öltered probability space.

Brownian Motions: DeÖnition

A Wiener processes (or Brownian motion) is a stochastic process W having: (^1) continuous sample paths. (^2) independent increments. (^3) W (t)  N ( 0 , t) , 8 t.

Basic Result If a stochastic process fX (t) , t  0 g has continuous sample paths with stationary, independent, and i.i.d. increments, then it is a Wiener process.

Di§erential: dW = (^) dtlim # 0 (W (t + dt) W (t))

Properties of Di§erentials

Moments: (^1) E [dW ] = 0.

(^2) E

h (dW )^2

i = dt.

Also, as dt! 0 (we skip the proof):

(^1) dW  o

p dt

 .

(^2) (dW )^2! E

h (dW )^2

i = dt.

Note that, while W (t) has a continuous path, it is not di§erentiable:

dW dt

o

p dt

dt

! ∞ as dt! 0

Di§usions II

Di§usion are important in arbitrage-free asset pricing. AÔt-Sahalia (2006).

Particularly useful cases are:

(^1) Geometric Brownian motion

dX = μ Xdt + σ XdW (^2) Ornstein-Uhlenbeck process

dX = θ (X μ ) dt + σ (t, x ) XdW

Functions of Stochastic Processes I

Let F (t, x ) be a function that is at least once di§erentiable in t and twice in x. We approximate the total di§erential of F (^) (t, X (^) (t, ω )) by a Taylor expansion:

dF = Ft dt + Fx dX +

2 Ftt^ (dt)

2 Fxx^ (dX^ )

(^2) + Fxt dt (dX ) + ...

We substitute in: dF = Ft dt + Fx [ μ dt + σ dW (^) ]

• 1 2

Ftt (dt)^2

• 1 2

Fxx

h μ^2 (dt)^2 + 2 μσ dtdW + σ^2 (dW )^2

i

+Fxt dt ( μ dt + σ dW ) +H.O.T ...

Functions of Stochastic Processes III

Particular case F (t, x ) = ert^ f (x ):

E (^) [dF (^) ] =

rf + μ f 0 + 1 2

σ^2 f 00

ert^ dt

and when r = 0 (that is, F (t, x ) = f (x ), an often relevant case in economics) E (^) [dF (^) ] =

μ f 0 + 1 2

σ^2 f 00

dt