Girsanov's Type Transformation Of Measures For Generalized Brownian Motion And Generalized Brownian Sheet And Their Applications

As 1997 Merton. R and Scholes.M obtained the Nobel Prize for Economics byBlack- Scholes option pricing model,the financial circles have paid close attentionto Black-Scholes option pricing theory. It is known as the secnod revolution of theWall Street. Girsanov's type transformation of measures theorem plays a key role inthe process of derivation of Black-Scholes option pricing model. Now, Girsanovtheorem has been an indispensable tool in mathematical and financial theory.Oneparameter stochastic processes is not enough to describe changing stochastic phe-nomenon. In many cases, only multi-parameter stochastic processes can describestochastic phenomenon accurately. In both theory and practical application,it fre-quently refer to Generalized Brownian Sheet and Generalized Brownian Motion. It isnatural to ask:What is Girsanov's type transformation of measures for GeneralizedBrownian Sheet and Generalized Brownian Motion ? What has it application ? Thispaper will explore these issues in depth. It reveals Girsanov's type transformation ofmeasures for Generalized Brownian Sheet and Generalized Brownian Motion,and itis applied to obtain the probability distribution of maximum of Generalized Brown-ian Sheet with drift on increasing path and passage times of Generalized BrownianMotion with drift.The main results are as follows:(â… )Girsanov's type transformation of measures for Generalized Brownian Sheettheorem:Theorem 1 Let {F_z}satisfies usual conditions, and {W_z,F_z; z∈R_z0} beGBS-F on (Ω,F, P), define Z_z(θ)=exp{integral from n=R_z(θ_udW_u)-1/2integral from n=R_z(θ_u²dF(u))}, whichθ∈H₀.If|θ_u| is bounded, define (?)_z=W_z-integral from n=R_z(θ_udF(u)). Under probability measures (?): (?)(A)=E(I_AZ_z0),A∈F_z0,{(?)_z,F_z;z∈R_z0}is GBS-F on (Ω,F,(?)).(â…¡)Girsanov's type transformation of measures for Generalized Brownian mo-tion theorem: Theorem 2 Let {F_t} satisfies usual conditions, and B={(B_t⁽¹⁾),...,B_t^(d)),F_t;0≤t＜∞} be d dimensional GBM-F on(Ω,F, P), where F=(F₁,.., F_d).Let X={X_t=(X_t⁽¹⁾,...,X_t^(d)), F_t;0≤t＜∞} be a vector of measurable, adeptedprocess satisfing: P[integral from n=0 to T((X_t⁽ⁱ⁾)²dF_i(t))＜∞]=1, 1≤i≤d,0≤T＜∞.Define (?)_t⁽ⁱ⁾=B_t⁽ⁱ⁾-integral from n=0 to t(X_s⁽ⁱ⁾dF_i(s)),1≤i≤d:0≤t＜∞, (?)={((?)_t⁽¹⁾,...,(?)_t^(d),F_t;0≤t＜∞}and Z_t(X)=exp{sum from i=1 to d integral from n=0 to t(X_s⁽ⁱ⁾dB_s⁽ⁱ⁾)-1/2integral from n=0 to t(|X_s|²dF(s))}.If E[exp{1/2integral from n=0 to T(|X_t|²dF(t))}]＜∞;0≤T＜∞.Under probability measures (?)_T: (?)_T(A)=E[I_AZ_T(X)],(?)A∈F_T,{(?)_t,F_t;0≤t≤T} is d dimensional GBM-F on (Ω,F_T,(?)_T).(â…¢)Two applications:(1)The probability distribution of maximum of Generalized Brownian Sheetwith drift on increasing path:Let {W_z,F_z;z∈R₊²} be GBS-F on(Ω,F,P),and F: t=φ(s),0≤s≤s₀be a increasing path between 0 and z₀=(s₀, t₀), P((?)(W_z-cF(z))≥λ)=|λ|/2Ï€^1/2integral from n=0 to s₀([F(G(t))]^3/2)exp{-cλ-c²/2F(G(t))-λ²/2F(G(t))}dF(G(t)),which G(s)=(s,φ(s)).(2)The probability distribution of passage times of Generalized Brownian Mo-tion with drift: Let {(?)_t,(?)_t;0≤t＜∞} be GBM-F on (Ω,F, P), (?)_b=inf{t≥0,(?)_t-ct=b};b≠0,which is passage times of {(?)_t-ct,(?)_t,t≥0} with respect to b≠0, where F'(t)=α₁I_{[0≤t≤α]}+α₂I_{[t＞α]}.When t∈[0,α], P((?)_b∈dt)=|b|/2πα₁t^31/2exp{-cb/α₁-c²t/2α₁-b²/2α₁t}dt.When t∈[α,∞], P((?)_b∈dt)=(integral from n=-∞to +∞(g(x)1/2πα₁α^1/2e^{-x2/2α₁αdx-integral from n=-∞to +∞integral from n=0 to a g(b+x)1/2πα₁2s(a-s)s1/2exp{-x2/2α₁(a-s)-b2/2α₁s}dsdx}dt,where g(x)=|b-x|/[2πα₂(t-a)]^31/2exp{(b-x)²/2α₂(t-a)-1/α₁-1/α₂)(cx+ac²/2)-c/2α₂(2b+tc)}.