Arnold Sommerfeld Center for Theoretical Physics
Ludwig-Maximilians-Universität München
Prof. Dr. E. Frey
M. Weber, C. Weig
Stochastic Processes in Physics and Biology – Exercise sheet 5
Winter term 2011/2012
Exercise 10 Discrete-time dynamics
a) Consider the discrete-time dynamics of a stochastic variable x(t). We assume that the time interval [t0 , t] can
be split up into n steps of equal length ∆t = ti+1 − ti , 0 ≤ i ≤ n − 1, such that t0 < t1 < . . . tn = t. Using Itō’s
prescription for SDEs the dynamics of x reads:
xi+1 = xi + a(xi )∆t + b(xi )∆Wi
with
xi = x(ti )
Time increment: ∆t = ti+1 − ti
Stochastic Wiener increment:
∆Wi = W (ti+1 ) − W (ti )
with h∆Wi iW = 0 and h∆Wi ∆Wj iW = δij ∆t .
We assume that a(x(t)) and b(x(t)) are twice differentiable, non-anticipatory1 functions. Furthermore, let f (x(t))
be an arbitrary function with the same properties. Derive a discrete-time update rule for the average hf (xi+1 )iW
by omitting all terms of order higher than ∆t.
b) Instead of focusing on averages hf (x(t))iW over realizations of the Wiener process W , we can alternatively
consider x as a random variable with conditional probability distribution p(x, t|x0 , t0 ). We may then rewrite all
averages as:
Z
f (x(t)) W = dx p(x, t|x0 , t0 )f (x) .
Apply this relation to the update rule of hf (xi+1 )iW to derive a Fokker-Planck equation for p(x, t|·). You may
assume that p(x, t|·) vanishes at some given boundary.
c) In the previous parts of this exercise we assumed that the dynamics of x(t) follows Itō dynamics. We now
want to see how the update rule changes if we assume Stratonovich dynamics for the evolution of x(t):
x
x
i+1 + xi
i+1 + xi
xi+1 = xi + a
∆t + b
∆Wi
2
2
Derive a new update rule for f (xi+1 ) W . Your result should read:
f (xi+1 )
W
0
1
= f (xi ) W + f 0 (xi )a(xi ) + f 0 (xi )b(xi ) b(xi ) W ∆t
2
d) Derive a discrete-time Fokker-Planck equation for the Stratonovich evolution of p(x, t|·).
e) Show that the Stratonovich update of x approximates an Itō update with parameters aI = a + 12 bb0 and bI = b.
Exercise 11 Simulation of the Ornstein-Uhlenbeck Process
Recall the Ornstein Uhlenbeck Process from the lecture – an overdamped particle in a potential with fluctuating
Gaussian noise term:
√
1 dU
dt + 2D dW (t) ,
dx(t) = −
γ dx
with hdW (t)i = 0 and hdW (t1 ) dW (t2 )i = δ(t1 − t2 ) dt .
We will simulate this process numerically in MATLAB by using an explicit Euler algorithm for the harmonic
potential U = k2 x2 .
a) Write a function that takes D, k, γ, the initial position of the paricle, the time step and the total duration
of the simulation as input and returns two vectors, the time values T and the position vector X of the particle.
The time vector is useful for plotting the evolution of x(t). You can generate Gaussian random numbers with unit
variance by using the function randn. Note, however, that dW does not have unit variance.
1 non-anticipatory
means that f (x(ti )) is independent from ∆Wj for i ≤ j
b) Run the simulation for D = γ = 1, k = 0.1, x(0) = 0, a time step of 0.01 and a simulation duration of 15.
Plot your results using the plot command.
c) Compare the simulation results to the analytic results for hx(t)i and hx(t)2 i as derived in the lecture. For
this purpose average the observables over 500 runs of your simulation. You can start with OUP script.m which is
available from the lecture’s website. Plot the results and compare with analytical calculations.
Exercise 12 Accumulated wealth (parts d) - g) may be handed in later, see below)
As physicists trained in stochastic processes you have the best opportunities to cause the next financial crisis. But
in order to do so you first have to gain the trust of some financial mathematicians by following the path of Itō.
Assume that you own a portfolio of total value x(t), which earns an interest at a constant rate r. Furthermore,
you raise your investment by a single unit of money per unit time. Since your portfolio may also contain shares of
stock its value fluctuates over time with volatility σ, which we assume to be constant. The portfolio’s value can
be modeled by an Itō stochastic differential equation:
dx(t) = (1 + rx(t)) dt + σx(t) dW (t)
a) Assume that x(t0 ) > 0 and argue that x(t) ≥ 0 for all times t ≥ t0 and that the probability to find x(t) = 0
is zero (you may argue this as a physicist, i.e. you do not have to prove it).
b) Derive a Fokker-Planck equation for the conditional probability p = p(x, t|x0 , t0 ) from the Itō SDE by using
the results from part a) and p(∞, t|x0 , t0 ) = 0.
c) Derive equations for the evolution of the first and second moments of x and solve these equations. You may
find it helpful to use the method of variation of constants to solve the equation for hx2 i. Under which conditions
on the interest rate r does the mean value hxi of the portfolio remain finite in the limit t → ∞? For which values
of r does the second moment of x remain finite as well? (Sure you do not want interest rates like this.)
d) As a next step, we intend to derive the stationary probability distribution ps = ps (x). For the purpose, it
proves useful to consider the logarithmic, or continuously compounded return:
R(t) = log
x(t)
x(t0 )
Its definition follows from the behavior of the relative growth of x(t) over arbitrarily small time periods τ :
x(t + τ ) − x(t) /τ
ẋ(t)
d
lim
=
= R(t)
τ →∞
x(t)
x(t)
dt
Use Itō’s lemma to derive a stochastic differential equation for R(t). Sketch the deterministic drift of the equation
with respect to R(t) and argue for which interest rates the logarithmic return remains finite as t → ∞.
e) Show that the mean and variance of the logarithmic return evolve linearly in the regime R(t) 1. In
particular, show that:
hR(t)i ' hR(t0 )i + (r −
1 2
σ )(t − t0 ) ,
2
Var (R(t)) ' σ 2 (t − t0 ) .
f ) Derive a Fokker-Planck equation for the logarithmic return. By demanding that the probability flux is zero
at stationarity, show that the stationary distribution qs (R) fulfills:
1
1 −R 1
d
− r − σ2 +
e
qs + σ 2
qs = 0
2
x(t0 )
2 dR
Solve this equation without caring about integration constants. They will be fixed in the next part of the exercise
by normalization.
g) Use the relation:
1 x qs log
x
x(t0 )
to derive the stationary distribution of the portfolio’s value. Show that the distribution has a power-law tail for
x 1. Use ps to check your previous results on the existence of ps (i.e. of R), hxi and hx2 i for different interest
rates. You may do this part of the exercise either for the regime R 1 or by using the full form of qs . In the
latter case your result should read:
ps (x) =
2
− 2
ps (x) =
e
σ x
2r
1−
( 12 σ 2 ) σ2
Γ(1 −
2r
)
σ2
1
·
x
2− 2r
2
σ
.
Exercises 10, 11 and 12 a-c are to be handed by Monday, 12th of
December, at 12 o’clock in the boxes in front of office 333. The
remaining part of exercise 12 (d-g) may be handed in until Monday, 19th of December at 12 o’clock and will be discussed in
the week before Christmas.