Stochastic Calculus of Standard Deviations: An
Introduction
By
Ahsan Amin
INFINITI DERIVATIVES TECHNOLOGIES,
Lahore, Pakistan
ahsanamin2999@gmail.com
July 17, 2013
ABSTRACT
Every density produced by an SDE which employs normal random variables for its
simulation is a linear or mostly non-linear transformation of normal random
variables. We find this transformation in case of a general SDE by taking into
account how variance evolves in that certain SDE. We find Jacobian of this
transformation with respect to normal density and employ change of variables
formula for densities to get the density of simulated SDE. Briefly in our method,
domain of the SDE is divided into standard deviation fractions that expand or
contract as the variance increases or decreases such that probability mass within
each SD fraction remains constant. Usually 200-400 SD fractions are enough for
desired accuracy. Within each SD fraction stochastic integrals are evolved
independently of other SD fractions. The work for each step is roughly the same
as that of one step in monte carlo but since SD fractions are only a few hundred,
this technique is much faster than monte carlo. Since this technique is very fast,
we are confident that it will be the method of choice to evolve distributions of
SDEs as compared to monte carlo simulations and partial differential equations.
1. Introduction to our setup
In Stochastic Calculus of Standard Deviations, we evolve a tree that branches just
once at the origin. We take a grid of about 2N+1 equidistant points that usually
covers -5 to +5 standard deviations. This should vary according to nature of the
problem. We evolve stochastic integrals on these grids independently of
neighboring points according to the algorithm given later in this work. Once we
reach the final time, we calculate the cross-sectional derivative of these integrals
with respect to simple Gaussian density. This is done simply by finite differences.
This is the only point where interdependence between neighboring points on
standard deviation comes into play. Once we have calculated this derivative, we
use it to calculate the density of stochastic integral as mentioned in section 2.
Suppose, we have an Arithmetic Brownian motion or a random normal variable in
evolution such that density of the variable remains normal though its mean or
variance might continue to change with time. As a start we give the dynamics of
the arithmetic Brownian motion as
( )= ( )
( )
We divide this density into cells that contain a fixed standard deviation fraction in
them such that cell size at any time t is given as
∆ ( ) = ∆ (0) ( )√
Where ∆ (0) is standard deviation fraction.
We divide the density of normal random variable in evolution into 2N+1 standard
deviation fractions ranging from –N to +N. In that case, at any specific time, the
centre of the nth cell can be found at
( , ) = 0 + ∆ ( ) = 0 + ∆ (0) ( )√
In the above ∆ (0) is the standard deviation fraction in which the cells are
divided. We make the observation that the probability mass in each standard
deviation fraction remains constant with time. Since these cells are measured in
SDs, their size increases as variance of evolving normal variable increases and
decreases if the variance of the normal variable decreases. So we have divided
normal variable in evolution into expanding and contracting cells in a way that
probability mass in each cell remains constant since each cell has a constant
fraction of SD in it. This probability mass can be calculated as
Pr
( , ) (0) =
1
√2
( )√
exp (−.5(
( , ) − (0)
) )∆ ( )
( )√
=
1
( )√
√2
exp (−.5( ∆ (0)) )∆ ( )
We can of course, at any time in transition, map these standard deviation
fractions back to absolute value of grid size and also to absolute value of random
variable there.
2. Conditional Expected Value and Densities of Functions of Brownian
Motion
Since we have the exact density of Brownian motion, we can directly find the
conditional expected value of functions of Brownian motion at any time by using
the formula
( , )
( , )=
= ( )Pr ( ( ) = )
Many times, however, we need density of F(z(t)). This can be found by using the
formula
Pr
This derivative
( )
( , )
( )
( , )=
=
( )
Pr ( ( ) = )
is found by finite difference as
=(
( + 1, ) − ( ( − 1), ))/(2 ∆ (0) √
)
3. Substitution of Gaussian Uncertainty by Difference of Square Root
of Variance along Standard Deviation Fractions
We do not have any real or pseudo-uncertainties to mimick the evolution of
random process. We replace the occurrence of noise by change in square root of
variance of noise in case of simple Gaussians since variance of ordinary Brownian
motion is given as √ , we get the following
( )
=
( , )
( )
( )
( ) ∆ (0) √ + ∆ − √
Where n takes values between –N to +N.
4. Calculation of dt-Integrals of Arithmetic Brownian Motion
The algorithm to advance the integral of functions of z along time in each step is
given as
∆
( ( , ))
=
( ( , ))∆ − .125
− .25(.5 )
( )
( )∆
( )
( ) ∆ (0)
(( + ∆ )
.
√3
−
.
)
Now conditional expected value of ∫
( ( , ))
| ( , ) = 0+
Probability Distribution of ∫
( ( , ))
Pr
Where
∫
( ( , ))
( ( , ))
( )√
( ( , ))
( )=
=
is found as
=
( ( , ))
Pr ( ( , ))
can be found as
∫
( ( , ))
Pr ( ( ) = )
is numerically calculated by Finite differences since we have
conditional value of time integral along continuum of Z(n,t). We will want to
mention that numerical coefficients of dt-integrals in this work are only
approximate.
5. Calculation of dz-Integrals of Arithmetic Brownian Motion
The algorithm to advance the integral of functions of z along noise in each step is
given as
∆
( ( , ))
( )
∆
=
( ) ( ) ∆ (0) √ + ∆ − √
−
.5
( )
( )
Where second integral on RHS is calculated as a dt-integral. Please note that half
appears in quadratic variations since the above formula is derived from
( ( , ))
( )=
( ( , + 1)) − ( ( , )) +
( )=
.5
( )
( )
∫ ( )
( )
And since dz according to our logic is given as
= ( ) ∆ (0) √ + ∆ − √
Adding higher order terms of Taylor expansion improves the order of
approximation of dz and greatly improves the accuracy here.
And second term is quadratic variation which must be added. Now conditional
expected value of ∫
( ( , ))
is found as
( ( , ))
( )| ( , ) = 0 +
=
( , )
( ) Pr
( )√
( , )
Probability Distribution of ∫
Pr
( , )
( ( , ))
( )=
( )
=
can be found as
∫
( , )
( )
Pr ( ( ) = )
6. Calculation of Non-Gaussian Noises
Here we discuss noise which is non Gaussian and we deal with SDEs of the form
( )=
( ) ( )
( )
It is V that we are referring as non-Gaussian noise though driving uncertainty is
still a function of Gaussian.
In our set up this is approximated by
( , +∆ )= ( , )+
( , ) ( ) ∆ (0) √ + ∆ − √
∆
−
.5
Where the last integral is taken as a dt-integral and we show in section 7 how to
solve it.
Probability Distribution of ∫
( ) ( )
Pr [ ( , )| ( ) = ] =
( ) can now be found as
( , )
Pr ( ( ) = )
( , )
= ( ( + 1, ) − ( − 1, ))/(2 ∆ (0) √
)
Density of Noise: gamma=.35, sigma=.5,V(0)=1,T=1,deltaX(0) =.06,2N=200
Monte Carlo Density
Analytic Density
0.8
0.7
0.6
Density
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
Noise
Density of Noise: gamma=.95, sigma=.5,V(0)=1,T=1,deltaX(0) =.06,2N=200
Monte Carlo Density
Analytic Density
1
Density
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
Noise
2.5
3
3.5
4
Density of Noise: gamma=.65, sigma=.5,V(0)=1,T=1,deltaX(0) =.06,2N=200
1
Monte Carlo Density
Analytic Density
0.9
0.8
0.7
Density
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
Noise
2.5
3
3.5
Density of Noise:gamma=.65,sigma=.35,V(0)=1,T=4,deltaX0=.02,2N=400
0.8
Monte Carlo Density
Analytic Density
0.7
0.6
Density
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
2.5
Noise
3
3.5
4
4.5
5
Density of Noise:gamma=.95,sigma=.35,V(0)=1,T=4,deltaX0=.02,2N=400
1
Monte Carlo Density
Analytic Density
0.9
0.8
0.7
Density
0.6
0.5
0.4
0.3
0.2
0.1
0
0
1
2
3
Noise
4
5
6
7. Calculation of dt-Integrals of Non-Gaussian Noise
Here we discuss integrals of the form
( )
Where dynamics of V(t) are given as
( )=
( ) ( )
( )
These integrals are calculated according to the equation
∆
( , )
=
( , ) ∆ − .125
( , )
− .25(. 5 ) ( − 1) ( , )
Probability Distribution of ∫
( , )
Pr
∫
( , )
=(
( , )
( )=
( + 1, )
∆ (0)
(( + ∆ )
.
−
.
√3
∆
can now be found as
=
∫
−
( , )
( − 1, )
Pr ( ( ) = )
)/(2 ∆ (0) √
)
)
Density of dt-Integral:gamma=.35,Beta=1,sigma=.5,V(0)=1,T=1,deltaX0=.02,2N=400
1.5
Monte Carlo Density
Analytic Density
Density
1
0.5
0
0
0.5
1
1.5
dt-integral
2
2.5
3
Density of dt-Integral:gamma=.65,Beta=1,sigma=.5,V(0)=1,T=1,deltaX0=.02,2N=400
1.6
Monte Carlo Density
Analytic Density
1.4
1.2
Density
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
dt-integral
Density of dt-Integral:gamma=.95,Beta=1,sigma=.5,V(0)=1,T=1,deltaX0=.02,2N=400
1.6
Monte Carlo Density
Analytic Density
1.4
1.2
Density
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
dt-integral
2
2.5
Density of dt-Integral:gamma=.65,beta=1,sigma=.35,V(0)=1,T=4,deltaX0=.02,2N=400
0.35
Monte Carlo Density
Analytic Density
0.3
Density
0.25
0.2
0.15
0.1
0.05
0
0
2
4
6
dt-integral
8
10
12
Density of dt-integral:gamma=.95,beta=1,sigma=.35,V(0)=1,T=4,deltaX0=.02,2N=400
0.35
Monte Carlo Density
Analytic Density
0.3
Density
0.25
0.2
0.15
0.1
0.05
0
0
5
10
dt-integral
15
8. Calculation of dz-Integrals of Non-Gaussian Noise
Here we discuss integrals of the form
( )=
( )
( )
Where dynamics of V(t) are given as
( )= ( )
( )
( )
To calculate these integrals we evolve V according to algorithm given in section….
And then use the formula
∆
( , )
∆
=
( , ) ∆ (0) √ + ∆ − √
−
.5
( , )
Since the evolution of integral is independent of its value and takes an exogenous
input, it can also be solved by using the formula
∆
( , )
∆
=⎛
( , )
⎝
( , )
⎞ ∆ (0)
⎠
∆
−
−
.5
( , )
Probability Distribution of ∫
Pr
∫
( , )
( , )
( )
( ) ( )=
=(
( + 1, )
( ) can now be found as
∫
=
−
( , )
( − 1, )
Pr ( ( ) = )
)/(2 ∆ (0) √
)
Density of dz-Integral:gamma=.35,alpha=.5,sigma=.5,V(0)=1,T=1,deltaX0=.02,2N=400
0.45
Monte Carlo Density
Analytic Density
0.4
0.35
Density
0.3
0.25
0.2
0.15
0.1
0.05
0
-3
-2
-1
0
1
2
dz-integral
3
4
5
6
7
Density of dz-Integral:gamma=.65,alpha=.5,sigma=.5,V(0)=1,T=1,deltaX0=.02,2N=400
0.45
Monte Carlo Density
Analytic Density
0.4
0.35
Density
0.3
0.25
0.2
0.15
0.1
0.05
0
-3
-2
-1
0
1
2
dz-integral
3
4
5
6
7
Density of dz-Integral:gamma=.95,alpha=.5,sigma=.5,V(0)=1,T=1,deltaX0=.02,2N=400
0.45
Monte Carlo Density
Analytic Density
0.4
0.35
Density
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
-3
-2
-1
0
1
2
dz-integral
3
4
5
6
7
Density of dz-Integral:gamma=.65,alpha=.5,sigma=.35,V(0)=1,T=4,deltaX0=.02,2N=400
0.25
Monte Carlo Density
Analytic Density
0.2
Density
0.15
0.1
0.05
0
-5
0
5
10
15
20
dz-integral
Density of dz-integral:gamma=.95,alpha=.5,sigma=.35,V(0)=1,T=4,deltaX0=.02,2N=400
0.25
Monte Carlo Density
Analytic Density
0.2
Density
0.15
0.1
0.05
0
-6
-4
-2
0
2
dz-integral
4
6
8
10
9. Evolving SDEs with Drift.
Now we want to evolve the SDEs with drift. Let us consider an SDE of the form
( )=
( )
+
( )
( )
This SDE can simply be solved by following the equation below
∆
Q(n, t + ∆t) = Q(n, t) + μ
∆
Q(n, t) dt + σ
Q(n, t) dz
Here the second integral is taken as a dz-integral and first integral is taken as a dtintegral and they are solved accordingly using methods mentioned in previous
sections.