Martingale Theory Problem set 3, with solutions …

MartingaleTheoryProblemset3,withsolution sMartingalesThesolutionsofproblems1,2,3, 4,5,6, j,j= 1,2,..b ( i= +1)=p,P( i= 1)=q:= 1 p,andFn= ( j,0 j n),n 0,theirnatural := nj=1 j,n 0.(a)ProvethatMn:= (q/p)Snisan(Fn)n 0- Martingale .(b)For >0determineC=C( )sothatZ n:=Cn Snb ean(Fn)n :(a)E(Mn+1 Fn)=E(Mn(q/p) n+1 Fn)=MnE((q/p) n+1 Fn)=MnE((q/p) n+1)=Mn(p(q/p) +q(p/q)) =Mn.(b)C=C( ) =(E( )) 1= p+ 'sRuin,1 Agamblerwinsorlo osesonep oundineachroundofb etting,withequalchancesandindep ettingwiththe rmdeterminationthatshewillstopgamblingwh eneithershewonap oundsorshelostbp ounds.(a)Whatistheprobabilitythatshewill b (b)Whatistheexp ectednumb erofherb ettingroundsb :Mo deltheexp j,j= 1,2,..b ( i= +1)=12=P( i= 1),andFn= ( j,0 j n),n 0,theirnatural 0, Sn:=n j=1 j, n nethestoppingtimesTL:= inf{n >0 :Sn= b}, TR:= inf{n >0 :Sn= +a}, T:= min{TL,TR}.Notethat{thegamblerwinsap ounds}={T=TR},{thegamblerlo osesbp ounds}={T=TL}.

3.6HW We place N balls in K urns (in whatever way) and perform the following discrete time process. At each time unit we choose one of the balls uniformly at random (that is : each ball is chosen with probability 1=N) and place it in one of the urns also uniformly chosen at random (that is: each urn is chosen with probability 1=K). Denote by X ...

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