•Probability space(Ω,F,P).•Random variable,random vector,σ-field.•Probability space(Ω,F,P).•Random variable,random vector,σ-field.◦σ-filed F is a class of subsets ofΩsatisfying(1)includesΩ;(2)closure for complement;(3)close for countable union.•Probability space(Ω,F,P).•Random variable,random vector,σ-field.◦σ-filed F is a class of subsets ofΩsatisfying(1)includesΩ;(2)closure for complement;(3)close for countable union.◦σ-filed sounds like a class of events we cantell.•Probability space(Ω,F,P).•Random variable,random vector,σ-field.◦σ-filed F is a class of subsets ofΩsatisfying(1)includesΩ;(2)closure for complement;(3)close for countable union.◦σ-filed sounds like a class of events we cantell.•Information implied by a random variable:σ(X)={A={ω:X(ω)∈B}:B is a Borel set of IR}.•Probability space(Ω,F,P).•Random variable,random vector,σ-field.◦σ-filed F is a class of subsets ofΩsatisfying(1)includesΩ;(2)closure for complement;(3)close for countable union.◦σ-filed sounds like a class of events we cantell.•Information implied by a random variable:σ(X)={A={ω:X(ω)∈B}:B is a Borel set of IR}.•Distribution,expectation.•Given D∈F with P(D)>0,define conditional probability on F by P(A|D):=P(A∩D)•Given D∈F with P(D)>0,define conditional probability on F by P(A|D):=P(A∩D)•Given D∈F with P(D)>0,define conditional probability on F by P(A|D):=P(A∩D)•Given D∈F with P(D)>0,define conditional probability on F by P(A|D):=P(A∩D)•Given D∈F with P(D)>0,define conditional probability on F by P(A|D):=P(A∩D)•Given D∈F with P(D)>0,define conditional probability on F by P(A|D):=P(A∩D)•For generalσ-field G,any integrable r.v.X, E(X|G)is defined as a random variable Z, which satisfies•For generalσ-field G,any integrable r.v.X, E(X|G)is defined as a random variable Z, which satisfies(i)Z is integrable and G-measurable,•For generalσ-field G,any integrable r.v.X, E(X|G)is defined as a random variable Z, which satisfies(i)Z is integrable and G-measurable, (ii)For any B∈G,E[1B X]=E[1B Z].•For generalσ-field G,any integrable r.v.X, E(X|G)is defined as a random variable Z, which satisfies(i)Z is integrable and G-measurable, (ii)For any B∈G,E[1B X]=E[1B Z].•If Z1,Z2satisfy the definition,thenZ1=Z2a.s..•For generalσ-field G,any integrable r.v.X,E(X|G)is defined as a random variable Z, which satisfies(i)Z is integrable and G-measurable,(ii)For any B∈G,E[1B X]=E[1B Z].•If Z1,Z2satisfy the definition,then Z1=Z2a.s..•P(A|G):=E[1A|G].•For generalσ-field G,any integrable r.v.X,E(X|G)is defined as a random variable Z, which satisfies(i)Z is integrable and G-measurable,(ii)For any B∈G,E[1B X]=E[1B Z].•If Z1,Z2satisfy the definition,then Z1=Z2a.s..•P(A|G):=E[1A|G].•For any r.v.’s Y,any integrable r.v.X,E[X|Y]:=E[X|F Y].•If G={∅,Ω},then E(X|G)=E X.•If G={∅,Ω},then E(X|G)=E X.•If X is G-measurable,Y is integrable,thenE[XY|G]=X E[Y|G].•If G={∅,Ω},then E(X|G)=E X.•If X is G-measurable,Y is integrable,thenE[XY|G]=X E[Y|G].•If G1⊂G2,thenE[E[X|G2]|G1]=E[X|G1].•If G={∅,Ω},then E(X|G)=E X.•If X is G-measurable,Y is integrable,thenE[XY|G]=X E[Y|G].•If G1⊂G2,thenE[E[X|G2]|G1]=E[X|G1].•If F X and G are independent(also say X is independent with G),then E[X|G]=E X.•If G={∅,Ω},then E(X|G)=E X.•If X is G-measurable,Y is integrable,thenE[XY|G]=X E[Y|G].•If G1⊂G2,thenE[E[X|G2]|G1]=E[X|G1].•If F X and G are independent(also say X is independent with G),then E[X|G]=E X.•Denote F∨G as the smallestσ-field containing F and G.If X is independent with F,thenE[X|F∨G]=E[X|G].•In the multi-period binomial model,denote X n∆t=log S n∆t,R n:=X n∆t−X(n−1)∆t•In the multi-period binomial model,denote X n∆t=log S n∆t,R n:=X n∆t−X(n−1)∆t◦R n:a sequence of i.i.d.random variables,•In the multi-period binomial model,denote X n∆t=log S n∆t,R n:=X n∆t−X(n−1)∆t◦R n:a sequence of i.i.d.random variables,◦X n∆t=X0+ n i=1R i,a random walk.•In the multi-period binomial model,denoteX n∆t=log S n∆t,R n:=X n∆t−X(n−1)∆t◦R n:a sequence of i.i.d.random variables,◦X n∆t=X0+ n i=1R i,a random walk.•T ake X0=0.If wefix T,∆t=T/N,{ξn}i.i.d, P(ξn=1)=P(ξn=−1)=1/2,•In the multi-period binomial model,denoteX n∆t=log S n∆t,R n:=X n∆t−X(n−1)∆t◦R n:a sequence of i.i.d.random variables,◦X n∆t=X0+ n i=1R i,a random walk.•T ake X0=0.If wefix T,∆t=T/N,{ξn}i.i.d, P(ξn=1)=P(ξn=−1)=1/2,◦X n∆t=X0+ n k=1ξn/√•In the multi-period binomial model,denoteX n∆t=log S n∆t,R n:=X n∆t−X(n−1)∆t◦R n:a sequence of i.i.d.random variables,◦X n∆t=X0+ n i=1R i,a random walk.•T ake X0=0.If wefix T,∆t=T/N,{ξn}i.i.d, P(ξn=1)=P(ξn=−1)=1/2,◦X n∆t=X0+ n k=1ξn/√√)for0<s<t<T.T◦X t→a continuous-time random process.Random process and its information•A discrete random process X is a randomvariable sequence{X n(ω):n=0,1,···}•A continuous-time random process X is a random function{X t(ω):t∈IR+}•A continuous-time random process X is a random function{X t(ω):t∈IR+}◦Fixω,X t(ω)is a function on IR+•A continuous-time random process X is a random function{X t(ω):t∈IR+}◦Fixω,X t(ω)is a function on IR+◦Fix t,X t(ω)is a random variable.•A continuous-time random process X is a random function{X t(ω):t∈IR+}◦Fixω,X t(ω)is a function on IR+◦Fix t,X t(ω)is a random variable.◦Given t1<t2<···<t n,F t1,···,t n(x1,···,x n)=P(X t1≤x1,···,X t n≤x n)denotes the joint distribution of X t1,···,X tn.•A continuous-time random process X is a random function{X t(ω):t∈IR+}◦Fixω,X t(ω)is a function on IR+◦Fix t,X t(ω)is a random variable.◦Given t1<t2<···<t n,F t1,···,t n(x1,···,x n)=P(X t1≤x1,···,X t n≤x n)denotes the joint distribution of X t1,···,X tn.•Information in a r.v.Y:σ(Y):={{ω:Y(ω)∈Γ}:Γ∈B}•A continuous-time random process X is a random function{X t(ω):t∈IR+}◦Fixω,X t(ω)is a function on IR+◦Fix t,X t(ω)is a random variable.◦Given t1<t2<···<t n,F t1,···,t n(x1,···,x n)=P(X t1≤x1,···,X t n≤x n)denotes the joint distribution of X t1,···,X tn.•Information in a r.v.Y:σ(Y):={Y−1(Γ):Γ∈B}.•Informationflow in a random process X up to t: F X t:=σ(X s:s≤t).•For any t1<t2<···<t n•For any t1<t2<···<t n◦W t i+1−W t i are independent normal random variables.•For any t1<t2<···<t n◦W t i+1−W t i are independent normal random variables.◦(W t1,···,W t n)is a normal random vector.•For any t1<t2<···<t n◦W t i+1−W t i are independent normal random variables.◦(W t1,···,W t n)is a normal random vector.◦E W t i=0,V ar(W t i)=t i.•For any t1<t2<···<t n◦W t i+1−W t i are independent normal random variables.◦(W t1,···,W t n)is a normal random vector.◦E W t i=0,V ar(W t i)=t i.◦For t i<t j,E[W t i W t j]=t i.•For any t1<t2<···<t n◦W t i+1−W t i are independent normal random variables.◦(W t1,···,W t n)is a normal random vector.◦E W t i=0,V ar(W t i)=t i.◦Cov(W t i,W t j)=E[W t i W t j]=min{t i,t j}.•For any t1<t2<···<t n◦W t i+1−W t i are independent normal random variables.◦(W t1,···,W t n)is a normal random vector.◦E W t i=0,V ar(W t i)=t i.◦Cov(W t i,W t j)=E[W t i W t j]=min{t i,t j}.(W t1,···,W tn)∼N(0,(σi,j)n×n)withσi,j=min{t i,t j}.•A general Brownian motion(BM in short)is generalized from the standard one•A general Brownian motion(BM in short)is generalized from the standard one◦W0=w0for some constant w0.•A general Brownian motion(BM in short)is generalized from the standard one◦W0=w0for some constant w0.◦W t−W s∼N(µ(t−s),σ2(t−s))for some constantµandσ.。
