Slides for DN2281, KTH1

January 28, 2014

1Based on the lecture notes Stochastic and Partial DifferentialEquations with Adapted Numerics, by J. Carlsson, K.-S. Moon, A.Szepessy, R. Tempone, G. Zouraris.

DN2281, Computational Methods for Stochastic DifferentialEquations. Spring 2014.Brownian motion, stochastic integrals, and diffusions assolutions of stochastic differential equations. Functionals ofdiffusions and their connection with partial differentialequations. Weak and strong approximation, efficient numericalmethods and error estimates, variance reduction techniques.Prerequisite: Familiarity with stochastic processes, ordinarydifferential equations, numerical methods.Instructor: Erik von Schwerin (schwerin@csc.kth.se)

January 22, 2014 - Class contents:

1. Course Introduction, Admin details

2. Motivating examples

Admin detailsI email listI 15 lectures, link to schedule on the course web pageI Reading: Lecture notes on home page. Please print as

you go, not all at once!I Examination:

I 5 sets of homeworks, written reports, one group presentsI 1 final project/paper presentationI Final written exam, mostly consisting of questions

randomly selected from a list of “study questions”

I Homeworks, presentations, to be done in groups (of 2?).I First homework due in 2 weeks, form groups next lectureI The group that makes the presentation is encouraged to

hand in a draft of their solution a couple of days inadvance.

I “office hours”, for now email me at schwerin@kth.se toset a time to meet

The goal of this course is to give useful understanding forsolving problems formulated by stochastic differential equationsmodels in science, engineering, and mathematical finance.

Motivating examples (Chapter 1)

Noisy Evolution of Stock Values

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Noisy Evolution of Stock Values

Denote stock value by S(t). Assume that S(t) satisfies thedifferential equation

dS

dt= a(t)S(t),

which has the solution

S(t) = e∫ t0 a(u)duS(0).

Since we do not know precisely how S(t) evolves we would liketo generalize the model to a stochastic setting

a(t) = r(t) + ”noise”.

For instance,

dS(t) = r(t)S(t)dt + σS(t)dW (t), (1)

where dW (t) will introduce noise in the evolution.What is the meaning of (1)? The answer is not as direct as inthe deterministic ODE case.

One way to give meaning to (1) is to use the Forward Eulerdiscretization,

Sn+1 − Sn = rnSn∆tn + σnSn∆Wn. (2)

Here ∆W n are independent normally distributed randomvariables . . .

. . . with zero mean and variance ∆tn, i.e.

E [∆W n] = 0

andVar [∆W n] = ∆tn = tn+1 − tn.

Then (1) is understood as a limit of (2) when max ∆t → 0.

Noisy Evolution of Stock Values

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

time

S

N = 64

0 0.2 0.4 0.6 0.8 1

10−0.03

10−0.02

10−0.01

100

100.01

100.02

100.03

100.04

100.05

100.06

time

log(

S)

One realization with N = 64 steps, σ = 0.15 and r = 0.05

Noisy Evolution of Stock Values

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

time

S

N = 128

0 0.2 0.4 0.6 0.8 1

10−0.03

10−0.01

100.01

100.03

100.05

100.07

time

log(

S)

One realization with N = 128 steps, σ = 0.15 and r = 0.05

Noisy Evolution of Stock Values

0 0.2 0.4 0.6 0.8 10.9

0.95

1

1.05

1.1

1.15

1.2

1.25

time

S

N = 256

0 0.2 0.4 0.6 0.8 1

10−0.03

10−0.01

100.01

100.03

100.05

100.07

time

log(

S)

One realization with N = 256 steps, σ = 0.15 and r = 0.05

Applications to Option pricing

European call option: is a contract signed at time t whichgives the right, but not the obligation, to buy a stock (orother financial instrument) for a fixed price K at a fixed futuretime T > t.At time t the buyer pays the seller the amount f (s, t; T ) forthe option contract.What is a fair price for f (s, t; T )?

The Black-Scholes model for the valuef : (0,T )× (0,∞)→ R of a European call option is thepartial differential equation

∂tf + rs∂s f +σ2s2

2∂2s f = rf , 0 < t < T ,

f (s,T ) = max(s − K , 0), (3)

where the constants r and σ denote the riskless interest rateand the volatility, respectively.

Stochastic representation of f (s, t)

The Feynmann-Kac formula gives the alternative probabilityrepresentation of the option price

f (s, t) = E [e−r(T−t) max(S(T )− K , 0))|S(t) = s], (4)

where the underlying stock value S is modeled by thestochastic differential equation (1) satisfying S(t) = s.Thus, f (s, t) is both the solution of a PDE (3) and theexpected value of the solution of a SDE (4)!

Which one should we choose to discretize?

p(x)g(x)

time t

x

x0

Sample paths for the approximation of a put option.

Stochastic Particle SimulationsMolecular dynamics simulation of particles, with positions X t

in a potential, V (X ).

Standard method to simulate MD in the microcanonicalensemble of constant number of particles, volume, and energy,(N,V,E), is to solve Newton’s equations (deterministic)

dX ti = v t

i dt,

Mdv ti = −∂Xi

V (X t) dt(5)

Stochastic Particle Simulations

To simulate a system with constant temperature instead ofenergy, (N,V,T), one often simulate (5) but add a regularrescaling of the kinetic energy to keep T constant,“thermostats”.An alternative is to simulate the Langevin dynamics

dX ti = v t

i dt,

Mdv ti = −∂Xi

V (X t) dt − v ti

τdt +

√2γ

τdW t

i ,(6)

where Wi are independent Brownian motions, τ is a relaxationtime parameter, and γ := kBT .

Stochastic Particle Simulations

Under some assumptions on the potential V , one can samplethe same invariant measure ,

e−V (X )/γ dX∫R3N e−V (X )/γ dX

, (7)

using overdamped Langevin dynamics: SmoluchowskiDynamics at constant temperature, T ,

dX si = −∂Xi

V (X s)dt +√

2γ dW si . (8)

Optimal Control of Investments

Suppose that we invest in a risky asset, whose value S(t)evolves according to the stochastic differential equation

dS(t) = µS(t)dt + σS(t)dW (t),

and in a riskless asset Q(t) that evolves with

dQ(t) = rQ(t)dt.

It is reasonable to assume r < µ, why?Our total wealth is then X (t) = Q(t) + S(t).

Goal: determine an optimal instantaneous policy ofinvestment to maximize the expected value of our wealth at agiven final time T .

Let the time dependent proportion,

α(t) ∈ [0, 1],

be defined byα(t)X (t) = S(t),

so that(1− α(t))X (t) = Q(t).

Then our optimal control problem can be stated as

maxα∈A

E [g(X (T ))|X (t) = x ] ≡ u(t, x), (9)

where g is a given function.How can we determine α?

The solution to (9) can be obtained by means of a HamiltonJacobi equation, which is in general a nonlinear partialdifferential equation satisfied by u(t, x) of the form

ut + H(u, ux , uxx) = 0.

Part of our work is to study the theory of Hamilton Jacobiequations and numerical methods for control problems todetermine the Hamiltonian H and the control α.

Towards a definition of SDEs: Ito Integrals