Nonlinear Option Pricing

264 McKean Nonlinear Stochastic Differential Equations

From the independence of the processes {Y

i

s

}

1≤i≤N

and the fact that Law(Y

i

s

)

= P

s

, we get

E[A

s

(h)] =

2 − N

N

2

E[h(Y

1

s

, P

s

)

2

] +

1

N

2

E[h(Y

1

s

, Y

1

s

)

2

]

+

N − 1

N

2

E[h(Y

1

s

, Y

2

s

)

2

] −

2

N

2

E[h(Y

1

s

, P

s

)h(Y

1

s

, Y

1

s

)] ≤

K(t)

N

The four expectations above are finite due to the Lipschitz conditions on b

and σ and the fact that E[sup

0≤s≤t

|Y

s

|

2

] < ∞.

PROOF of Theorem 10.3 Let us denote by µ

t

the law of the solution

X

t

of the McKean SDE (10.2). From Theorem 10.2 (iii), it is enough to show

that for all ϕ ∈ C

b

(R

n

),

1

N

N

X

i=1

ϕ(X

i,N

t

)

L

1

−→

N→∞

Z

ϕ dµ

t

(10.13)

where the X

i,N

t

are defined by (10.6). Now,

E

"







1

N

N

X

i=1

ϕ(X

i,N

t

) −

Z

ϕ dµ

t







#

≤ E

µ

N

"





