MonteCarlo 算法

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详细 MonteCarlo 算法蒙特卡罗方法,又称随机模拟方法,属于计算数学的一个分支,它是在上世纪四十年代中期为了适应当时原子能事业的发展而发展起来的。亦称统计模拟方法,
Generalized wiener process △x=a△t+bE△t where a and b are constants Ax is normally distributed with mean of△x=△t standard deviation of△x=by△t variance of△x=b2△t For Long Period change in value of x in any interval is normally distributed with mean of change in x=aT standard deviation of change in x=vT variance of change in x=b T Ito process dx=a(r, tdt +b(x, t)dz where a and b are functions of the underlying variable, x, and time,t Ito's lemma A function G of x and t follow the process dgdG 1 dG dg dG=a+ b- dt+obd2 dx dt 2 d dx where dz is a wiener process G follows an Ito process with drift rate dg dg 1 dG b X at 2 d and a variance rate of dG b dx 8 Application to In(Stock Price) Stock Price model ds=uSdt +osd G=In s Since dG 1 dg 1 dG =0 ds s ds s dt the process follow by g is 2 O dG=l u 十O 2 Property Change G between current time, t, and some future time T is normallly distributed with mean and variance 10 Simulation model △f+OE S(t+△t)=S(t)e Where e is drawn from a standard normal distribution Generating Random Samples of Distribution Froy Xis a random variable with uniform distribution Generate samples of y with distribution Fry) Y=FY(X) Fy(x) P(Y≤y) P(X≤F(y Fy(y X is uniform distributed F(y) BoX Muller generating two independent standard normals from two independent uniforms no closed form for N(=P(X sx) e √2z a cannot not apply Y=FY(X) 3 Some mathematics X三 2 e dx e C polar coordinates x=rcos 0 y=rsin 6, dxdy= rdrde 2丌 e2rrd6=2丌 丌 0 0 Probabilistic Interpretation C yis a pair of independent standard normals probability density r+ (X,Y) (x,y)= 2兀 radially symmetric polar coordinates(R, 6) B is uniformly distributed in the interval [o, 2I Rhas a distribution function P(RSr=I e 2 udud0= e 2 udu=1 10J02x Box Muller R=F(X)=y-2n(1-X If X is standard uniform, R has distribution of Fr(r) Wecan also replace(1-x) U, where U is standard uniform R 2Inv Summary Box Muller takes 2 independent standard uniform variables u, and u generates 2 independent normals X andY 8=2U,R=-2lnU2, X=Rcos e, Y=Rsin 0 Polar Rejection Method 1. Set V=20-l and V,=202 2. Compute W=V+V, 3. If W>l, return to step 1. Otherwise, set (-2lnW), 2In w v W W 2 17 Relating Box Muller and Polar rejection V, and v, are independent uniform W=V+v2 (vv2) P(W≤w=P(V12+V2)≤w) ∴ W is uniforn R=√2lnW cos 6= V, sIn 6

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