2 calculus problem need in 2 hours
marjoee25
lecture_notes_for_reference_only.pdf
Stochastic Calculus, Summer 2014, July 22,
Lecture 7
Connection of the Stochastic Calculus and Partial Differential
Equation
Reading for this lecture:
(1) [1] pp. 125-175 (2) [2] pp. 239-280 (3) Professor R. Kohn’s lecture notes PDE for Finance, in particular Lecture 1
http://www.math.nyu.edu/faculty/kohn/pde_finance.html
Today throughout the lecture we will be using the following lemma.
Lemma 1. Assume we are given a random variable X on (Ω, F, P) and a filtration (Ft)t≥0. Then E(X|Ft) is a martingale with respect to filtration (Ft)t≥0.
Proof. The proof is very easy and follows from the tower property of the conditional expectation. �
Corollary 2. Let Xt be a Markov process and Ft be the natural filtration asso- ciated with this process. Then according to the above lemma for any function V
process E(V (XT )|Ft) is a martingale and applying Markov property we get that E(V (XT )|Xt) is a martingale. In the following we often write E(V (XT )|Xt) as EXt=xV (XT ).
As we will see this corollary together with Itô’s formula yield some powerful results on the connection of partial differential equations and stochastic calculus.
Expected value of payoff V (XT ). Assume that Xt is a stochastic process satis- fying the following stochastic differential equation
dXt = a(t, Xt)dt + σ(t, Xt)dBt, (1)
or in the integral form
Xt − X0 =
t ∫
0
a(s, Xs)ds +
t ∫
0
σ(s, Xs)dBs. (2)
Let u(t, x) = EXt=xV (XT ) (3)
be the expected value of some payoff V at maturity T > t given that Xt = x. Then u(t, x) solves
ut + a(t, x)ux + 1
2 (σ(t, x))2uxx = 0 for t < T, with u(T, x) = V (x). (4)
By Corollary 2 we conclude that u(t, x) defined by (3) is a martingale. Applying Itô’s lemma we obtain
du(t, Xt) = utdt + uxdXt + 1
2 uxx(dXt)
2
= utdt + ux(adt + σdBt) + 1
2 uxxσ
2 dt
= (ut + aux + 1
2 σ
2 uxx)dt + σuxdBt, (5)
1this version July 21, 2014
1
2
Since u(t, x) is a martingale the drift term must be zero and thus u(t, x) solves
ut + aux + 1
2 σ
2 uxx = 0.
Substituting t = T is (3) we get that u(T, x) = EXT =x(V (XT )) = V (x).
Feynman-Kac formula. Suppose that we are interested in a suitably “discounted” final-time payoff of the form
u(t, x) = EXt=x
(
e
−
T ∫
t
b(s,Xs)ds
V (XT ) )
(6)
for some specified function b(t, Xt). We will show that u then solves
ut + a(t, x)ux + 1
2 σ
2 uxx − b(t, x)u = 0 (7)
and final-time condition u(T, x) = V (x). The fact that u(T, x) = V (x) is clear from the definition of function u. Therefore
let us concentrate on the proof of (7). Our strategy is to apply Corollary 2 and thus we have to find some martingale involving u(t, x). For this reason let us consider
e
−
t ∫
0
b(s,Xs)ds
u(t, x) = e −
t ∫
0
b(s,Xs)ds
EXt=x
(
e
−
T ∫
t
b(s,Xs)ds
V (XT ) )
= EXt=x
(
e
−
T ∫
0
b(s,Xs)ds
V (XT ) )
. (8)
According to Corollary 2
EXt=x
(
e
−
T ∫
0
b(s,Xs)ds
V (XT ) )
is a martingale and thus e −
t ∫
0
b(s,Xs)ds
u(t, x) is a martingale. Applying Itô’s lemma we get
d (
e
−
t ∫
0
b(s,Xs)ds
u(t, x) )
= (ut + a(t, x)ux + 1
2 σ
2 uxx − b(t, x)u)e
−
t ∫
0
b(s,Xs)ds
dt
+ e −
t ∫
0
b(s,Xs)ds
uxdBt. (9)
Since the drift must be equal to zero we obtain that u(t, x) satisfies (7).
Example. [Black-Scholes PDE] We assume that the underlying (stock, for in- stance) follows a geometric Brownian motion. That is in the risk-neutral measure it satisfies SDE
dSt = rStdt + σStdBt, (10)
where r is the risk-free rate which we assume to be constant. The payoff of a European option at maturity T is known and is equal to V (ST ). Then to find the value of the option at some earlier time t < T we have to compute
ESt=x
(
e −rt
V (ST ) )
. (11)
3
From the Feynman-Kac formula we conclude that u(t, x) solves partial differential equation
ut + rxux + 1
2 σ
2 x
2 uxx − ru = 0 (12)
with u(T, x) = V (x). Equation (12) is the famous known Black-Scholes PDE.
Running payoff. Now suppose that we are interested in
u(t, x) = EXt=x
(
T ∫
t
b(s, Xs)ds )
(13)
for some specified function b(t, x). First of all, let us find the final-time condition for u(t, x). Clearly, u(T, x) = 0. Our next step is to find a martingale involving
u(t, x) and then use Corollary 2. Add to both parts of (13) t ∫
0
b(s, Xs)ds. Then
u(t, x) +
t ∫
0
b(s, Xs)ds = EXt=x
(
T ∫
0
b(s, Xs)ds )
(14)
is a martingale. Applying Itô’s lemma we obtain that u(t, x) satisfies
ut + a(t, x)ux + 1
2 σ
2 uxx + c(t, x) = 0. (15)
Boundary value problems and exit times. In previous examples we were in- terested in the expectation of form (3), that the expectation of some payoff at specified maturity T.
Now let us assume that we are given a region D ⊂ R and process Xt starts from some point x ∈ D. Let
τ (x) = min{T, inf{t : Xt 6∈ D}}.
That is τ (x) is the first time Xt exits from region D if prior to T, otherwise τ = T. Assume at exit time τ the payoff of an option is given by function V. We are interested in the fair price of such an option at some earlier time t, i.e., in the following quantity
u(t, x) = EXt=xV (τ, Xτ ). (16)
We will see that just like in the previous examples u(t, x) solves partial differential equation, but in contrast the PDE must be solved inside the region D with suitable boundary data. The key to derivation of the PDE is the following lemma.
Lemma 3. Let Eτ < ∞. Then E(V (Xτ )|Ft) is a martingale with respect to filtra- tion (Ft∧τ )t≥0.
Applying Itô’s lemma we get
du(t, x) = (ut + aux + 1
2 σ
2 uxx)dt + uxσdBt. (17)
By lemma 3 function u(t, x) from (16) is a martingales and therefore there is no drift term in (17). Thus u(t, x) solves the following PDE
ut + aux + 1
2 σ
2 uxx = 0 (18)
with boundary conditions u(t, x) = V (t, x) for x ∈ ∂D and u(t, x) = V (T, x) for x ∈ D.
4
Application: distribution of the first arrivals. As the first application let us consider EXt=x1τ <T = PXt=x(τ < T ). According to the above we have to solve PDE
ut + aux + 1
2 σ
2 uxx = 0 (19)
with boundary condition u = 1 at x ∈ ∂D.
References
[1] Richard Durrett, Stochastic Calculus: A Practical Introduction. [2] Ioannis Karatzas, Steven E. Shreve, Brownian Motion and Stochastic Calculus.
