Overview of Stochastic Calculus Foundations

This is a quick refresher/overview of Stochastic Calculus Foundations. This assumes you have done a Stochastic Calculus course previously and now want to review/revise the material to prepare for a course that lists Stochastic Calculus as a pre-req. In these 11 slides, I list the key content you must be familiar with within Stochastic Calculus.

1. Refresher on Stochastic Calculus Foundations
Ashwin Rao
ICME, Stanford University
November 13, 2018
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 1 / 11

2. Continuous and Non-Dierentiable Sample Paths
Sample paths of Brownian motionztare continuous
Sample paths ofztare almost always non-dierentiable, meaning
lim
h!0
zt+hzt
h
is almost always innite
The intuition is that
dzt
dt
has standard deviation of
1
p
dt
, which goes to
1asdtgoes to 0
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 2 / 11

3. Innite Total Variation of Sample Paths
Sample paths of Brownian motion are of innite total variation, i.e.
lim
h!0
n1
X
i=m
jz
(i+1)hzihjis almost always innite
More succinctly, we write
Z
T
S
jdztj=1(almost always)
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 3 / 11

4. Finite Quadratic Variation of Sample Paths
Sample paths of Brownian Motion are of nite quadratic variation, i.e.
lim
h!0
n1
X
i=m
(z
(i+1)hzih)
2
=h(nm)
More succinctly, we write
Z
T
S
(dzt)
2
=TS
This means it's expected value isTSand it's variance is 0
This leads to Ito's Lemma (Taylor series with (dzt)
2
replaced withdt)
This also leads to Ito Isometry (next slide)
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 4 / 11

5. Ito Isometry
LetXt: [0;T]!Rbe a stochastic process adapted to ltration
Fof brownian motionzt.
Then, we know that the Ito integral
R
T
0
Xtdztis a martingale
Ito Isometry tells us about the variance of
R
T
0
Xtdzt
E[(
Z
T
0
Xtdzt)
2
] =E[
Z
T
0
X
2
tdt]
Extending this to two Ito integrals, we have:
E[(
Z
T
0
Xtdzt)(
Z
T
0
Ytdzt)] =E[
Z
T
0
XtYtdt]
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 5 / 11

6. Fokker-Planck equation for PDF of a Stochastic Process
We are given the following stochastic process:
dXt=(Xt;t)dt+(Xt;t)dzt
The Fokker-Planck equation of this process is the PDE:
@p(x;t)
@t
=
@f(x;t)p(x;t)g
@x
+
@
2
f

2
(x;t)
2
p(x;t)g
@x
2
wherep(x;t) is the probability density function ofXt
The Fokker-Planck equation is used for problems where the initial
distribution is known (Kolmogorov forward equation)
However, if the problem is to know the distribution at previous times,
the Feynman-Kac formula can be used (a consequence of the
Kolmogorov backward equation)
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 6 / 11

7. Feynman-Kac Formula (PDE-SDE linkage)
Consider the partial dierential equation foru:R[0;T]!R:
@u(x;t)
@t
+(x;t)
@u(x;t)
@x
+

2
(x;t)
2
@
2
u(x;t)
@x
2
V(x;t)u(x;t) =f(x;t)
subject tou(x;T) = (x), where; ;V;f; are known functions.
Then the Feynman-Kac formula tells us that the solutionu(x;t) can
be written as the following conditional expectation:
E[(
Z
T
t
e

R
u
t
V(Xs;s)ds
f(Xu;u)du) +e

R
T
t
V(Xu;u)du
(XT)jXt=x]
such thatXuis the following Ito process with initial conditionXt=x:
dXu=(Xu;u)du+(Xu;u)dzu
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 7 / 11

8. Stopping Time
Stopping timeis a andom time" (random variable) interpreted as
time at which a given stochastic process exhibits certain behavior
Stopping time often dened by a \stopping policy" to decide whether
to continue/stop a process based on present position and past events
Random variablesuch thatPr[t] is in-algebraFt, for allt
Deciding whethertonly depends on information up to timet
Hitting time of a Borel setAfor a processXtis the rst timeXt
takes a value within the setA
Hitting time is an example of stopping time. Formally,
TX;A= minft2RjXt2Ag
eg: Hitting time of a process to exceed a certain xed level
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 8 / 11

9. Optimal Stopping Problem
Optimal Stopping problem for Stochastic ProcessXt:
V(x) = max

E[G(X)jX0=x]
whereis a set of stopping times ofXt,V() is called the value
function, andGis called the reward (or gain) function.
Note that sometimes we can have several stopping times that
maximizeE[G(X)] and we say that the optimal stopping time is the
smallest stopping time achieving the maximum value.
Example of Optimal Stopping: Optimal Exercise of American Options
Xtis stochastic process for underlying security's price
xis underlying security's current price
is set of exercise times corresponding to various stopping policies
V() is American option price as function of underlying's current price
G() is the option payo function
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 9 / 11

10. Markov Property
Markov property says that theFt-conditional PDF ofXt+hdepends
only on the present stateXt
Strong Markov property says that for every stopping time, the
F-conditional PDF ofX+hdepends only onX
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 10 / 11

11. Innitesimal Generator and Dynkin's Formula
Innitesimal Generator of a time-homogeneousR
n
-valued diusionXt
is the PDE operatorA(operating on functionsf:R
n
!R) dened as
Af(x) = lim
t!0
E[f(Xt)jX0=x]f(x)
t
ForR
n
-valued diusionXtgiven by:dXt=(Xt)dt+(Xt)dzt,
Af(x) =
X
i
i(x)
@f
@xi
(x) +
X
i;j
((x)(x)
T
)i;j
@
2
f
@xi@xj
(x)
Ifis stopping time conditional onX0=x, Dynkin's formula says:
E[f(X)jX0=x] =f(x) +E[
Z

0
Af(Xs)dsjX0=x]
Stochastic generalization of 2nd fundamental theorem of calculus
Ashwin Rao (Stanford) Stochastic Calculus Foundations November 13, 2018 11 / 11