Chapter 13: Connections between SDEs and PDE Flashcards
Consider F(t, x) where t ∈ [0, T] and x ∈ R. We will begin by looking at the partial differential
equation
∂F/∂t (t, x) + α(t, x) ∂F/∂x (t, x) + (1/2)β(t, x)^2 ∂^2F/∂x^2 (t, x) = 0 *
F(T, x) = Φ(x) **
Consider F(t,x), where t in [0,T] and x in R
(t is time and x is space)
which solves
∂F/∂t (t, x) + α(t, x) ∂F/∂x (t, x) + (1/2)β(t, x)^2 ∂^2F/∂x^2 (t, x) = 0 *
F(T, x) = Φ(x) **
TERMINAL BOUNDARY CONDITION
Here, α(t, x) and β(t, x) are (deterministic) continuous functions, and Φ(x) is a function known as the boundary condition.
NOTE
SOLVING
∂F/∂t (t, x) + α(t, x) ∂F/∂x (t, x) + (1/2)β(t, x)^2 ∂^2F/∂x^2 (t, x) = 0 *
F(T, x) = Φ(x) **
TERMINAL BOUNDARY CONDITION
Let (X_t) be a stochastic process satisfying
dX_u= α(u, X_u) + β(u, X_u) dB_u
we will use X to solve *
and want to solve above where we look over [t,T] abd initial condition will be setting X_t=x
notation
E_{t,x}
P_{t,x}
E_{t,x}
P_{t,x}
mean exp or probability where condition imposed at time t(X_t=x)
We will need to consider solutions to this SDE where we vary the initial value of x, and also the time at which the ‘initial’ value occurs. For given x in R and t in [0,T] we write these above to specify X represents a sol of * with initial cond X_t=x
we are e then interested in Xu during time u ∈ [t, T]). We will continue to use P and
E to denote starting at time 0 with unspecified initial value X0. 0 less than t less than T
chapter 13; Notation F(t,x)
t is time in [0,T}
space x in R
interested when this solves
∂F/∂t (t, x) + α(t, x) ∂F/∂x (t, x) + (1/2)β(t, x)^2 ∂^2F/∂x^2 (t, x) = 0 *
with bc
F(T, x) = Φ(x) **
Chapter 13; X
interested in X
X is a stochastic process which satisfies
dX_u= α(u, X_u) + β(u, X_u) dB_u
for the two functions defined in *
We see that X solves *
LEMMA 13.1.2
if F solves * and **
Suppose that F is a solution of
∂F/∂t (t, x) + α(t, x) ∂F/∂x (t, x) + (1/2)β(t, x)^2 ∂^2F/∂x^2 (t, x) = 0 *
and
F(T, x) = Φ(x) **
and also that
β(t, Xt) = ∂F/∂x (t, Xt)
is in H^2
Then, F(t, x) = E_{t,x} [Φ(X_T )]
for all x ∈ R and t ∈ [0, T].
proof by itos formula using f(t,x) = F(t,x) functions α and β
we write in integral form for integrals over t to T:
F(T, XT ) = F(t, X_t) + INTEGRAL_{t,T}
[β(u, X_u) (∂F/∂x) (u, Xu) ] dB_u.
we now take Expectation E_{t,x}:
RHS as ito integrals are 0 mean martingales
LHS
E_{t,x}[F(T,X_T)] = E_{t,x}[F(t,X,t)]
notation rhs means F(T,X_T) =Φ(X_T) by 13.2
lhs means X_t=x
E{t,x} [ Φ(x_T)] = E_{t,x}[F(t,x)] = F(t,x)
as F is a deterministic function
The PDE that will turn out to be important for option pricing is
FEYMAN-KAC
∂F/∂t (t, x) + α(t, x) ∂F/∂x (t, x) + (1/2)β(t, x)^2 ∂^2F/∂x^2 (t, x) − rF(t, x)
= 0 (13.4)
F(T, x) = Φ(x) (13.5)
where α, β, Φ are as before and r is a deterministic constant
Lemma 13.1.3
FEYMAN-KAC
13.4 and 13.5 not * and **
Suppose that F is a solution of
(13.4) and (13.5),
and also that
β(t, Xt) (∂F/∂x)(t, Xt)
is in H^2
Then,
F(t, x) = e^(−r(T −t))E({t,x} [Φ(X_T )] (13.6)
for all x ∈ R and t ∈ [0, T].
Remark 13.1.4 Setting r = 0 in Lemma 13.1.3 gets us back to the statement of Lemma 13.1.2.
Lemma 13.1.3
FEYMAN-KAC
13.4 and 13.5 not * and **
PROOF
Suppose that F is a solution of
(13.4) and (13.5),
and also that
β(t, Xt) (∂F/∂x)(t, Xt)
is in H^2
Then,
F(t, x) = e^(−r(T −t))E({t,x} [Φ(X_T )] (13.6)
for all x ∈ R and t ∈ [0, T].
Apply itos to Z_T=e^(−rt)F(t,X_t)
set f(t,x)=e^(−rt)F(t,X_t)
dZ_t=
= e^{−rt} [−rF + e^{−rt} ∂F/∂t + α(t, Xt) ∂F/∂x + (1/2)β(t, Xt)^2 ∂^2F/∂x^2]dt
+
e^−rt β(t, Xt) ∂F/∂x dB_t
We know that F satisfies * and so first term =0
dZ_t= e^−rt β(t, Xt) ∂F/∂x dB_t
in integral form:
e^{−rT}F(T, XT ) =
e^{−rt}F(t, Xt) + ∫_{t,T} [e^{−ru} (∂F/∂x) (u, X_u) dB_u] .
Z_T- z_t= ∫_{t,T} [β(t, Xt) (∂F/∂x) (u, X_u) dB_u]
2nd term on the right is an ito integral and hence has mean 0 . Taking exp E_{t,x} :
E_{t,x}[e^{−rT}F(T, X_T )]
= E_{t,x}[e^{−rt}F(t, Xt)]
At time T F is bc:
F(T,X_T)=Φ(X_T )
and at time t deterministic:
X_t=x
Under E_{t,x} we have X_t = x, so F(t, X_t) = F(t, x) which is deterministic. We obtain that
e^{−r(T −t)}E_{t,x} [F(T, X_T )] = F(t, x) and using (13.5) then gives us
e^{−r(T −t)}E_{t,x} [Φ(X_T )] = F(t, x)
F(t, x) = e^(−r(T −t))E({t,x} [Φ(X_T )] (13.6) and
isk-neutral valuation formula
We can start to see the connection to finance emerging in (13.6). If we set t = 0, we obtain
F(0, x) = e
−rTE0,x[Φ(XT )]
which bears a resemblance to the risk-neutral valuation formula we found in Chapter 5.
EXAMPLE 13.1.6
In cases where we can solve (13.3), we can use the Feynman-Kac formula to find explicit solutions to PDEs. For example, consider
∂F/∂t (t, x) + (1/2)σ^2 ∂^2F/∂x^2(t, x) = 0
with bc
F(T, x) = x^2
Here, σ ≥ 0 is a deterministic constant and t, x ∈ R.
We have that α(t, x) = 0, β(t, x) = σ (both constants), and Φ(x) = x^2 r=0
From Lemma 13.1.2 the solution is given by
F(t, x) = E_{t,x}[X^2_T]
where
dXu = 0 du + σ dBu
= σ dBu.
This means that
X_T = X_t + σ ∫_{t,T} dBu
= Xt + σ(B_T − B_t).
Therefore
F(t,x) = E_{t,x} [ (X_t + σ(B_T − B_t))^2]
= E[(x + σ(B_T − B_t))^2]
= E[x^2 + σ^2(B_T − B_t)^2 + 2xσ(B_T − B_t)]
= x^2 + σ^2(T − t).
Here, we use that Xt = x under Et,x and that
B_T − B_t ∼ B_T −t ∼ N(0, T − t) by def of BM
See exercises 13.1 and 13.2 for further examples of this method.
MARKOV PROPERTY
want to find the present value distribution of steps don’t need to know the values before the time
giving a process (X_u) we want to make guesses about the properties of X_T - not necessarily values
notation E_{t,x}[…]
(start process (X) at time t with run from time t to T, take expectation)
along with conditional expectation allows us to express the Markov Property
Suppose we have a stochastic process (F_t), and waited until time t so info is visible and given by curly F_t
We want to make a best guesses for some information about X_T where T > t. In symbols this means that we have chosen some (deterministic) function Φ and we are interested in
E[Φ(XT )| Ft].
In principle, we have access to all the information in Ft- information on what happened up to including time t
E_{t,Xt}[Φ(X_T )]. - we only know the value at specific time- value X_t
However, in many cases it holds that
E[Φ(XT )| Ft] =
E_{t,Xt}[Φ(X_T )].
(13.7)
ie replaced by as we only need that info
*we say that the markov property holds for (X_u) if hold for all Φ, t T
lemma 13.2.1
Markov property
Lemma 13.2.1 All Ito processes satisfy the Markov property.
In particular, the formula for GBM satisfies the MP
E[Φ(X_T )| Ft] = E_{t,Xt}[Φ(X_T )].
holds when X is Brownian motion, and when X is geometric Brownian motion.
that is it’s constantly forgetting the past and to predict the future you only need the present for the best guess
