Weak Duality Of Ito Diffusion And Monte Carlo Simulation Of The Fundamental Solution

The Cauchy problem asks for the solution of a partial differential equation with initial or boundary conditions, which has many applications on many fields. In financial field, a fundamental framework for the pricing theory of options was made by Fisher Black, Myron Scholes and Robert Merton, now known as Black-Scholes-Merton model. They derived the famous Black-Scholes differential equation for European-style options, which is a Cauchy problem with terminal condition. It was Robert Merton who gave a more generalized pricing formula. In1997, the importance of the model was recognized when Robert Merton and Myron Scholes were awarded the Nobel prize for economics.In physical field, a special case of Cauchy problem is heat equation, which is an important partial differential equation, can be traced to the physical problem of how to describe the distribution of heat in a domain over time.In quantum mechanics, one of the most applications of Cauchy problem is Fokker-Planck equation, which describes transition density function of the ve-locity of a particle or other observables with time evolution. By Wick rotation, Fokker-Planck equation is changed to time-dependent Schrodinger equation.In probability theory, this type equation is connected with stochastic pro-cess via Kolmogorov forward equations. The foundation of the solution was represented by the central result in the modern theory of stochastic processes, Feynman-Kac formula.However, the simulation for the statistical representation of Feynman-Kac formula needs many painstaking efforts, especially the annoying frequent change of the initial condition. Then many mathematicians and physicists pay atten-tion to fundamental solution, which plays an important role in the research of PDE. In Friedman (1975), the author discussed the existence, uniqueness and the boundary of the fundamental solution. Gronwall’s inequality, Harnack’s inequal-ity and the maximum principle are the commonly used and effective technology. We will follow the assumptions by Friedman to guarantee these basic properties of fundamental solution. In this dissertation, we dedicate to the duality of Ito diffusion and the factorization and simulation of the fundamental solution for parabolic type equation of Cauchy problem. By this simulation, to get the solu-tion we only need to calculate the integral which is much more easier than other methods. Further more, we must point out that the fundamental solution can be treated as adjusted transitional function by the relationship with the transitional probability density function of diffusion.First, we recall the basic facts about Ito diffusion, fundamental solution and Monte Carlo methods that will be used in the whole thesis, tell the ’mismatch’ between Feynman-Kac functional and discrete Markov chain. Then we study the weak dual transformation to solve this ’mismatch’, and give the happy show for popular Ito diffusions. The special case, Feller population process, is considered next. We discuss the duality for this process with the help of the moment gener-ating function. Then to show the more general cases with non-ordinary killing/branching rate, we give the algorithm for the Monte Carlo simulation of the fun-damental solution, one dimension and multivariate case respectively. At last, we consider the applications in boundary value problem. With the help of the idea of kernel, discuss the ’discounting’ in Feynman-Kac formula and the computation of Greeks for European options.There are6chapters in this dissertation, whose main contents can be de-scribed as follows.Chapter1tell the history, motivation and Organization. Chapter2give the preliminaries for the whole thesis, such as Ito diffusion, Feynman-Kac formula, Girsanov transformation and importance sampling.In chapter3, the main result is the factorization of fundamental solution and the weak duality transformation for Ito diffusion, Theorem3.1. Consider the diffusion Xt,(Xt∈R-1, Bt∈Rn), dXt=μ(Xt)dt+σ(Xt)dBt with killing/branching rate λ(x), then the fundamental solution q(t,x,y) of the partial differential equation where L is the generator of Xt, can be presented as follows, q(t,x,y)=p(t,x,y)·w(t,x,y) where p(t, x,y) is transitional probability density of Xt, and Theorem3.3. Consider the diffusion Xt,(Xt∈Rd, Bt∈Rd) dXt=μ(Xt)dt+σ(Xt)dBt where μ:Rd�'Rd is a vector drift function, a:Rd�'Rd×Rd the diffusion matrix, with killing/branching rate λ(x):Rd�'R, the weak dual of Xt, when existed and denoted by Xt*, dXt*＝μ*(Xt*)dt+σ(Xt*)dBt μ*:Rd�'Rd is the dual vector drift function, the diffusion matrix σ remains the same, with killing/branching rate λ*(x):Rd�'R will have parameters satisfying the following weak duality transformationThis transformation is symmetric with respect to (μ,λ) and (μ*,λ*): Suppose the weighted transition functions q(t,x,y) of Xt and q*(t,y,x) of Xt*are compact supported, the weak duality is unique and symmetric in the sense of transitional function: q(t,x,y)=q*(t,y,x) both of which are the fundamental solutions of the corresponding partial differ-ential equations, the operators are H and H*defined as where L and L*denote the generators of Xt and its dual X? respectively.Theorem3.4. Let H and H*are defined as theorem3.3. Suppose, the funda-mental solutions q(t,x,y) and q*(t,y,x) are compact supported, then we have where Hx means that H operates on the variable x and so do the others.Corollary3.1. With notations in theorem3.3, the weak dual transformation takes following simple form for one dimensional diffusions: Corollary3.2.With notations in theorem3.1and theorem3.3, p(t,x,y)·w(t,x,y)=p*(t,y,x)·w(t,y,x)where p*and w*is the corresponding function for the dual diffusion Xt*In chapter4,the interpretation for the special case,Feller population process is considered.The duality under this case can be given asTheorem4.1.The Feller population process Zt,which satisfies the SDE dZt=(-p0Zt+q0)dt+r(?)ZtdBt Z0*x≥0Can by considered as decomposition of two independent processes by distribution, Zt=dXt(0)+Xtwhere the two variables Xt(0) and Xt independent,Xt(0)～Gamma(δ,β(1-ρ2)), and Xt is compound Poissonwith N～Poisson(ρ2x/β(1-ρ2)),Ui～Exp(ρ(1-ρ2)).similarly to the duality Zt*satisfying dZt*=(-p0*Zt*+q0*)+r*(?)dBt Z0*y≥0where p0*=-p0,q0*=r2-q0,r*=r and branching rate λ*=-P0,can also be decomposed as Zt*=dXt*(0)+Xt* where the two variables Xt*(0) and Xt*are independent, Xt*(0)～Gamma(2-δ,β(1-ρ2)), and Xt*is compound Poisson with N*～Poisson(Therefore for the weighted transition function qX(t,x,y) for Xt and qX**(t,y,x) for Xt*, the duality holds, qX(t,x,y)=qX**(t,y,x)In chapter5, first we consider the relationship between the weighted tran-sition function and transition density, and dedicate to the importance sampling algorithm for the simulation of fundamental solution.Theorem5.1. Suppose function u and v satisfy the parabolic equation driven by H and L with the same initial condition f,then u(t,x)-v(t,x)=G(-λ(x)u(t-s,x)) where Particularly, take f as δ function yields to q-p=G(-λq)Corollary5.1. Substitute q=p·w to the formula5.3, w satisfies the following integral equationThe algorithm for one dimensional case can be summarized as, 1.Repeat steps2-3for i=1to n.2.Generate a Brownian bridge path Zt(i) with Z0=Y0=s(x) and Zt=Yt=s(y).3.Calculate as weight function.4.Estimate of w(xt,x,y)using the weighted averageThe algorithm for multivariate case can be summarized as,1.Calculate the transformation γ▽,γ=σ-1(x)the drift function α(y) withwhere ξ(x)=σ(x)σ(x)T.And the functions2.Repeat steps (a) to (c) for i=1to n.(a)Generate a multivariate Brownian bridge Zt(i)with Z0=γ(x),Zt=γ(y). (b) Generate the event times Tk1,Tk2,…, less than t of the A;th component of the Poisson process with intensity θk for k=1,…, d.(c) Estimate3. Estimate of w(t,x,y) using the weighted averageIn chapter6, the applications are considered. In the part of application in the boundary value problem, Theorem6.2. For any t>0, with some restrictions such that every term of the formula below exists,whereCorollary6.1. For x,z€D×(?)D, we have Theorem6.3. Let D be a bounded Lipschitz domain and A λ∈Jloc, Then for any F∈C((?)D), we have where Px[XTD∈dz] implies the harmonic measure on (?)D, also denoted by H(x,dz).In the second part, the kernel of the discounting in Feynman-Kac formula can be written as Γ(t,x,z)=p(t,x,z)-q(t,x,z)w(t,x)the form of inner product in Wiener space is,At the end, we discuss the computation of Greeks for European options with the help of the kernel’s idea and give some figures for general cases.