Optimal portfolio allocation for long-term growth of wealth in the presence of

transaction costs

by

Ricardo A. Rodriguez-Pedraza

A thesis submitted to the graduate faculty

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Major: Applied Mathematics

Program of Study Committee:Ananda Weerasinghe, Major Professor

Justin PetersJianwei Qiu

Iowa State University

Ames, Iowa

2005

Copyright c© Ricardo A. Rodriguez-Pedraza, 2005. All rights reserved.

ii

Graduate CollegeIowa State University

This is to certify that the Master’s thesis of

Ricardo A. Rodriguez-Pedraza

has met the thesis requirements of Iowa State University

Major Professor

For the Major Program

iii

DEDICATION

Ad Santum Ioseph Sponsum Beatae Mariae Vırginis

I would also like to dedicate this work to my wife Carolina, and my daughter Camila.

Without their support and loving guidance I would have not been able to complete this work.

iv

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

CHAPTER 1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . 1

CHAPTER 2. THE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 State and Control Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 First Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Second Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Non Triviality Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Potential Performance and Hamilton-Jacobi-Bellman Equations . . . . . . . . . 11

2.2.1 Classical Bellman Equation . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2.2 Hamilton-Jacobi-Bellman Inequalities . . . . . . . . . . . . . . . . . . . 14

2.3 Control Limit Policies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

CHAPTER 3. VERIFICATION THEOREM . . . . . . . . . . . . . . . . . . 18

CHAPTER 4. SOLUTION TO THE H-J-B EQUATIONS . . . . . . . . . . 20

4.1 Necessary and Sufficient Conditions for Optimality . . . . . . . . . . . . . . . . 20

4.2 Solution of the H-J-B Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 22

CHAPTER 5. APPLICATION AND CONCLUSIONS . . . . . . . . . . . . 26

5.1 Stock Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1.1 Estimation of µ and σ2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

5.1.2 Finding A and B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

5.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

v

APPENDIX

ITO’S LEMMA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vi

LIST OF FIGURES

Figure 5.1 Monthly evolution of stock prices . . . . . . . . . . . . . . . . . . . . . 27

Figure 5.2 Exponential regression of the evolution of stock prices . . . . . . . . . 28

Figure 5.3 V ′(x) and −G(x) between [A, B] . . . . . . . . . . . . . . . . . . . . . 30

vii

ABSTRACT

We study the classical problem of allocation of funds between a bank account which grows

with a deterministic rate and of a risky asset such as a stock whose value follows a geometric

Brownian motion with a drift. We maximize the expected rate of growth of the net wealth

in the presence of proportional transaction costs when transactions are made between the two

assets.

Our optimal strategy strategy keeps the ratio of the values of these assets in an interval with

minimum control. Finally an application of the model to a real stock is presented.

1

CHAPTER 1. INTRODUCTION

We study a portfolio allocation problem in continuous time. An investor needs to decide

how to distribute his money between two available assets. In the first asset, which we will call

the bank, money deposited grows with a deterministic interest rate r, r > 0 and it is essentially

risk free. For the second asset, the stock, the money invested will have a return rate that

follows a (µ, σ) Brownian motion.

It is possible for the investor at any given time transfer funds from bank to stock and

vice versa however, there are fees associated with each type of transaction. We will consider

here, proportional transaction costs. The investor would like to transfer the funds between the

two assets in order to maximize the long-term growth rate. As a consequence, the Hamilton-

Jacobi-Bellman equation becomes a variational inequality.

There is no limit on the rates of transfers between assets; thus it is always possible to

withdraw instantaneously any finite amount of money from the bank or stock and this will

lead to a problem of singular control of the Brownian motion.

A similar problem in continuous time without transaction costs was first solved by Merton

[21],[22]. His optimal strategy is to keep a fixed fraction of the risky asset. As time passes it is

assumed that the portfolio is continuously rebalanced to keep this fraction constant, moreover

this fraction is independent of the investor’s horizon. In the presence of transaction costs this

strategy will drive the investor to an almost sure ruin.

We have not discussed yet, the criteria of optimality. Following the classification of Akian

et al. [1], we have three major possibilities. We can choose to maximize the cumulative

expected utility of consumption over an infinite horizon [10], [21],[22]. The second possibility

is to consider a model without consumption and choose to maximize the utility function of the

2

wealth over a finite time horizon [30]. The third alternative, this is the criteria that we have

chosen, is to maximize the long-run average growth of wealth with a log utility function, this

problem is equivalent to maximize the almost sure long-run growth rate [1], [27]. Since the

time horizon is infinite the optimal strategy will be stationary. For an alternative approach

to the optimality criteria see [29], where the probability of reaching certain level of wealth is

maximized.

The quantity of interest is the net wealth. This quantity follows a diffusion process with

state-dependent coefficients. The optimal strategy consists in keeping this ratio within an

interval of no trade. Now, when the ratio exits the borders, i.e. the ratio drops or exceeds

certain constants A < B, we must transfer a minimal amount of funds from bank to stock or

from stock to bank in order to keep the ratio between A and B. A drawback of this strategy

is the continuous trade at the boundary.

This thesis generalizes the work done by Taksar et al. [27], where the authors consider the

same problem with transaction costs imposed only on transactions from the bank to the stock.

We will consider transaction costs when there is transfer from the bank to the stock and vice

versa and we obtain a close form solution for the differential equation of the diffusion process

coming from variational inequalities. Akian et al. [1] moves along the same lines, but our work

differs from theirs in two aspects. First we model the dynamics of the stock using a geometric

Brownian motion and they use a logarithmic Brownian motion. Second we employ different

methods to solve and prove uniqueness of the variational inequality: viscosity techniques in

their case and by direct verification in ours.

We will study here only the presence of proportional costs. But there also exists the

possibility of fixed transaction costs. Korn [17], for example, does that in presence of finite

time horizon, Bielecki and Pliska [5] analyze the two types of fees combined with risk sensitive

considerations with a infinite horizon time. As a consequence of the fixed costs the optimal

strategy involves impulse control. Leland [18] includes also capital gains taxes and focus

himself in the implementation of the optimal strategy with exogenous goals. See Liu [19] for

an interesting study of the effects of the transaction costs in the optimal trading policies.

3

This thesis is organized as follows. Chapter 2 defines the processes that define the allo-

cation problem together with the Hamilton-Jacobi-Bellman inequalities to solve optimally the

stochastic control problem. Chapter 3 contains the verification theorems necessary to link

solutions of such inequalities with the optimization problem. In Chapter 4, we exhibit an

analytical solution of the conditions found in Chapter 2 and finally we have an application and

conclusions in Chapter 5.

4

CHAPTER 2. THE MODEL

A canonical stochastic control problem with infinite horizon consists of a performance

functional to be maximized or minimized:

∫ ∞

0f (t, X (t, ω) , u (t, ω)) dt

subject to the following constrains:

u (t, ω) ∈ U ⊆ R

X ′ (t, ω) = g (t, X (t, ω) , u (t, ω))

X (0, ω) = X0

Where X (0, ω) is the state variable, u (t, ω) is the control function, U denotes the control

set, X ′(t) is the derivative with respect to t and g (t, X (t, ω) , u (t, ω)) is the law of motion.

This frame will be used to model the optimal allocation problem. We will need also to take

into account the uncertainty generated by the fluctuations of price of the risky asset so g will

include a term proportional to white noise which a formal derivative of the standard Wiener

process. We will begin with the description of the state variables and control processes.

2.1 State and Control Processes

2.1.1 First Formulation

In probabilistic modeling the first object presented is the probability space (Ω,F,P) a triple

composed by Ω ,the set of all possible outcomes of an experiment (experiment here is defined

quite broadly), by F the σ-algebra which specifies the set of all possible events. Events are

understood as subsets of Ω to which we can assign probability numbers using the probability

5

function P. In finance and economics P quantifies the subjective uncertainty of a rational

agent. To model the continuous growth of information, a filtration Ft; t ≥ 0 of (Ω,F) is

introduced. Ft is a family of σ-algebras, such that Ft ⊆ F for all t ≥ 0 and Fs ⊆ Ft if t ≥ s.

Therefore the filtration represents the sets of events that have or have not occur until time t.

Equivalently, Ft can also be understood as the gain of knowledge as time flows. Finally we

define F as equal to F∞, the smallest σ-algebra containing Ft for all t ≥ 0.

Now, how can we introduce the uncertainty faced by a long-run investor like a pension fund?

Let W (t, ω) ; t ≥ 0 a stochastic process on (Ω,F) such that under P the process W (t, ω) is

a standard Brownian motion with initial state X01. Since uncertainty desappears as soon as

information is made available we will also require that the original filtration coincides with

the one generated from the Brownian Motion, formally: Ft = σ (W (t) ; s ≤ t). For technical

reasons we will require in addition right-continuity of the filtration therefore we define a new

filtration F+t = σ (Ft ∪ N), where N is the collection of null sets of P. It is clear that this will

have no affect in our calculations since F+t and Ft differ only in sets of zero measure. It is

worth to notice at this point that W (t + s) − W (s) is independent of F+s for 0 < t, this fact

will imply the Markov property [6], [11],[14]. One consequence of this property is that the

future outcomes are independent of what had happened in the past. Thus it is impossible for

the controller to deterministically predict the future

Let B(t) and S(t) the amount of money the investor has at time t invested in the bank and

the stock respectively. In absence of any transaction costs between these assets the process

B(t) grows deterministically at exponential rate r , this means that one dollar deposited in the

bank at time 0 will become ert dollars at time t. Similarly with no transaction costs the stock

follows a geometric Brownian motion: dollar invested becomes eX(t) after a time t where we

represent X(t) as µt + σW (t) with W (t) the standard Brownian motion defined above.

If we want to buy or sell the stock we need to transfer funds from the bank to stock or vice

versa,this transactions involve costs ( like commission for buying or commission for sales, etc).

The costs that we will consider are called proportional costs. So if we withdrew m dollars from

1For an example of how to construct a Brownian motion see [14] and references within.

6

the bank we only buy mλ worth of stock where 0 < λ < 1 and similarly if we sale n dollars of

stock we only get αn in cash. with α ≥ 0.

Let U(t) be the total amount of money transfered from the stock to the bank until time

t and let Z(t) the amount transferred from bank to stock. The dynamics of the controlled

system is therefore governed by:

dS(t) = S(t)

[(

µ +σ2

2

)

dt + σdW (t)

]

+ dZ(t) − dU(t) (2.1)

dB(t) = rB(t)dt − (1 + α)dZ(t) + (1 − λ)dU(t) (2.2)

with initial conditions S(0−) = S0 and B(0−) = B0. Note that we are not considering

dividends coming from the stock and there is no consumption. We are not assuming that U(0)

or Z(0) are 0, thus S(0+) and B(0+) are not equal to S0 and B0. This just means that we are

allowing for rebalance of the portfolio at time t = 0+.

In the context of stochastic control U(t) and Z(t) are the control processes. They represent

the cumulative amount of money withdrawn or deposited into the stock asset; our objective

is to obtain the optimal strategy. Therefore we are looking for a pair of processes of bounded

variation U(t), L(t), t ≥ 0 such that

• U(·), Z(·) are nonnegative, nondecreasing, right-continuous processes, since they they

represent the total amount of money transfered from and to each asset.

• U(t), Z(t) are Ft-adapted. The controller only has the past and the present information

to make his decisions.

Controls U(·), Z(·) are feasible if

• Ex U(t) , Ex Z(t) < ∞ for each t.

• The system of equations (2.1) and (2.2) has a unique nonnegative solution for B(t) and

S(t) for all t ≥ 0.

These requirements deserve a commentary. The condition U(t) ≥ 0, Z(t) ≥ 0 for all t ≥ 0

implies that we are looking for controls where shortselling and borrowing are not allowed. Note

7

also that from (2.1) and (2.2) it follows that the buying of the stock is totally financed by the

money in the bank account and all proceeds from the stock go to the bank.

So far controls (U(t), Z(t)) can have discontinuities at the same points, which means si-

multaneous transactions between the assets in both directions. This type of policies can be

improved without difficulty.

In order to define the performance functional we need to define a new process Y (t) which

we call the net wealth process, defined as:

Y (t) ≡ (1 − λ) S(t) + B(t) (2.3)

Thus Y (t) is the money in the bank plus the net amount of cash that the investor would

received if the stock is sold; for this reason we called Y (t) the net wealth. In differential terms

we have

dY (t) = (1 − λ) S(t)

[(

µ +σ2

2

)

dt + σdW (t)

]

+ rB(t)dt + (λ + α) dZ(t) (2.4)

or in integral form:

Y (t) = Y0 +

∫ t

0

[

rB(t) + (1 − λ)

(

µ +σ2

2

)]

dt + (1 − λ)σ

∫ t

0dW (t) + (λ + α) Z(t) (2.5)

The processes U(·) and Z(·) and hence Y (·) are right-continuous with left limit at each t ≥ 0.

Now, for each available strategy (Z, U), we can associate a feasible set of controls of the long-

term performance functional

Jc(Z, U) = lim inft→∞

1

tEx [ln(Y (t))] (2.6)

With c = (B, S), c ∈ R2+. Our objective is to optimize the long-run rate of growth:

sup(Z,U)

Jc(Z, U) (2.7)

Where the supremun is taken over all feasible policies (Z, U). The functional Jc(Z, U) requires

some explanation. We are assuming that the net wealth Y (t) is also equal to the value of the

investors portfolio and is of the order of magnitude ekt for all t > 0, where is the expected

growth rate. Thus the goal is the optimization problem is maximize k. In the literature of

stochastic game theory this known as Kelly’s criteria [16],[20].

8

Let (Z, U) any feasible policy. This set of controls can be approximated by a sequence of

continuous processes (Zn, Un) such that for Yn the net wealth corresponding to them, we have:

lim inft→∞

1

t[lnY (t)] ≤ lim

nlim inft→∞

1

t[lnYn(t)]

Thus we can safely assume that (Z, U) are continuous, nondecreasing processes such that

Z(0) = U(0) = 0.

Using a generalized Ito’s Lemma see the appendix or [9],[15] [25], we can find how the

process ln[Y (t)] is related with the processes B(t) and S(t).

ln[Y (t)] = ln[Y0] +

∫ t

0

1

Y (s)

[

rB(s) + (1 − λ)

(

µ +σ2

2

)

S(s)

]

ds

+ (1 − λ) σ

∫ t

0

S(s)

Y (s)dW (s) − (λ + α)

∫ t

0

dZ(s)

Y (s)

−1

2σ2(1 − λ)2

∫ t

0

S(s)2

Y (s)2ds (2.8)

2.1.2 Second Formulation

Here we follow the formulation of the problem in Taksar et al. [26] Now we will define a

new state variable:

X(t) = (1 − λ)S(t)

B(t)(2.9)

In terms of this new process we find,

B(t)

Y (t)=

1

X(t) + 1(2.10)

S(t)

Y (t)=

1

(1 − λ)

X(t)

X(t) + 1(2.11)

We can also redefine the controls as:

dR(t) =dZ(t)

B(t)(2.12)

dL(t) =dU(t)

S(t)(2.13)

The controls R(t) and L(t) are the cumulative percentage of money withdrawn from the

bank and from the stock respectively. With these new definitions we can rewrite (2.8) us-

9

ing (2.10),(2.11),(2.12),(2.13) as:

ln[Y (t)] = ln[Y0] + rt −

∫ t

0

[

1

2σ2

(

X(s)

X(s) + 1

)2

− aX(s)

X(s) + 1

]

ds

+σ

∫ t

0

X(s)

X(s) + 1dW (s) − (λ + α)

∫ t

0

1

X(s) + 1dR(s) (2.14)

where

a =

(

µ +σ2

2− r

)

(2.15)

Dividing by t and taking expectations we obtain

1

tEx [ln[Y (t)]] =

ln[Y0]

t+ r −

1

tEx

[∫ t

0h(X(s))ds +

∫ t

0k(X(s))dR(s)

]

(2.16)

The integral involving dW (s) is a local martingale therefore its expectation is equal to zero.

h(x) and k(x) are equal to:

h(x) =1

2σ2

(

x

x + 1

)2

− ax

x + 1(2.17)

k(x) = (λ + α)1

x + 1(2.18)

Therefore the objective now is to minimize the new functional Jx with

Jx(L, R) = lim supt↔∞

1

tEx

[∫ t

0h(X(s))ds +

∫ t

0k(X(s))dR(s)

]

(2.19)

Everything has been expressed in terms of the process X(t) but we still need to find its law of

motion. With this goal in mind, we use integration by parts [9] to obtain from (2.9):

−(1 − λ)S(t)

B(t)2dB(t) + (1 − λ)

dS(t)

B(t)= (1 − λ)

S(t)

B(t)+ (1 − λ)

S(0)

B(0)(2.20)

using (2.1),(2.2),(2.10),(2.11),(2.12),(2.13), to replace in (2.20) we get:

dX(t) =

(

µ +σ2

2− r

)

X(t)dt + [(1 − λ) + (1 + α)X(t)] dR(t)

+X(t)σdW (t) −(

X(t) + X2(t))

dL(t) (2.21)

where we have used a as defined in (2.15). In integral form (2.21) looks:

X(t) = x +

∫ t

0aX(s)ds +

∫ t

0m(X(s))dR(s) −

∫ t

0j(X(s))dL(s) +

∫ t

0σX(s)dW (s) (2.22)

10

with

m(x) = (1 − λ) + (1 + α)x (2.23)

j(x) = x + x2 (2.24)

For completeness we exhibit the inverse relations between the old processes Y (t), S(t) and B(t)

with our new state variable X(t):

Y (t) = Y0 exp

[

rt + σ

∫ t

0

X(s)

X(s) + 1dW (s) −

∫ t

0

λ + α

X(s) + 1dR(s) −

∫ t

0h(X(s))dt

]

S(X(t)) =X(t)Y (t)

(1 − λ)(1 + X(t))

B(X(t)) =Y (t)

(1 + X(t))

To summarize. A policy is a pair (L, R) of two increasing, continuous and adapted processes.

A policiy is feasible if

•

dX = σXdW + aXdt − j(X)dL + m(X)dR (2.25)

X(0) = x (2.26)

has a unique solution,

• and

Ex

[∫ t

0S(s)dL(s)

]

+ Ex

[∫ t

0B(s)dR(s)

]

< ∞ (2.27)

with the new performance functional which needs to be minimized given by:

Jx(L, R) = Ex

[∫ t

0h(X(s))ds +

∫ t

0k(X(s))dR(s)

]

(2.28)

The quantities in the first integral and second integral sometimes are called holding cost and

cost of control respectively.

2.1.3 Non Triviality Conditions

Before we solve the stochastic control problem we need to study when it becomes trivial. If

a < 0 we have h′(x) ≥ 0 and likewise for a > σ2, h′(x) ≤ 0 which imply that h(x) is increasing

11

(decreasing) in [0,∞) if and only if a ≤ 0 (a ≥ σ2), where a has been defined in (2.15). In

these cases the optimal policy consists to move the process X(t) to zero or to +∞ respectively.

Therefore the problem at hand is non trivial when

0 < a < σ2 (2.29)

condition that we will always assume for now on. In this case h(x) attains its minimun at

point

x∗ =a

σ2 − a(2.30)

with value:

h(x∗) = −a

2σ2(2.31)

If λ → 0 and α → 0, i.e no transaction costs of any kind, X(t) = x∗ and the expected average

rate of growth of funds per unit of time will be r + a2

2σ2 ≡ µ +(

r − µ + σ2

2

)2which is greater

than r or µ alone. This result coincides con Merton’s analysis [20], [21] with no transaction

costs and no consumption.

2.2 Potential Performance and Hamilton-Jacobi-Bellman Equations

We have now a well defined problem. But before anything else some remarks are in order.

Note first that in selecting the controls at any time, only is important the state X(t) and not

the particular time t because we are looking for the long-term behavior of the system.

Let J(x, t) the functional defined in (2.28), and suppose that we have already found the

optimal controls (L∗, R∗) were as always x is the initial state. For t < ∞:

J(x, t) =(

J(x, t) − J(0, t))

−(

J(0, t) − J(0, 0))

(2.32)

since we are assuming

J(0, 0) = 0 (2.33)

The time-homogenueity implies that the first term cannot depend on t and similarly the second

term should be proportional to t [3]. Therefore if we set:

V (x) = J(x, 0) and d = J(0, 1)

12

we find:

J(x, t) = V (x) + td (2.34)

the constant that appears in (2.34) also has another interpretation:

d = limt→∞

1

tJ(x, t) = Jx(L

∗, R∗) (2.35)

this means that d is the long-term expected cost per unit of time t and therefore V (x) + td

is the cumulative expected cost up to time t. So we need now to find V (x) and d; with this

objective in mind we employ the Dynamic Programming Principle [2] or Bellman’s Principle

[4], for short. This is the way as Bellman formulated his principle:

“An optimal policy has the property that whatever the initial state and the

initial decision are, the remaining decisions must constitute an optimal policy with

regard to the state resulting from the first decision.”

Bellman of course referred to a discrete time process. In our case, the “initial decision” is the

choice of controls. Formally this means that if v(x) is the value function and δ is an initial

period of time, with δ < ∞, we have the relation

v(x) = infβ

E

[∫ δ

0f (β, xs) ds + v(δ)

]

(2.36)

where f is the return function and β is the set of feasible controls. Equation (2.36) translated

to our particular problem becomes:

V (x) + td = inf(L,R)

Ex

[∫ δ

0h(X(s))ds +

∫ δ

0k((X(s))dR(s) + (t − δ)d + V (δ)

]

(2.37)

since we are asumming that (L∗, R∗) have already been found, we have:

V (x) + td = Ex

[∫ δ

0h(X(s))ds +

∫ δ

0k((X(s))dR∗(s) + (t − δ)d + V (δ)

]

(2.38)

2.2.1 Classical Bellman Equation

Suppose that in the stochastic control problem above formulated the long-run investor can

only transfer (or push in the standard terminology) funds from the bank to the stock or vice

13

versa at a rate which cannot exceed θ < ∞. Therefore a policy is a pair of processes (L,R)

having the form

R(t) =

∫ t

0r(s)ds (2.39)

L(t) =

∫ t

0l(s)ds (2.40)

where r(t) and l(t) are Ft-adapted and 0 ≤ r(t), l(t) ≤ θ. Using Bellman’s principle we can

write:

V (x) + td = inf0≤r(t),l(t)≤θ

Ex

[∫ δ

0h(X(s))ds +

∫ δ

0k((X(s))dR(s) + (t − δ)d + V (δ)

]

(2.41)

Ito’s formula [9],[15] [25] allows to express V (X(δ)) as:

V (X(δ)) = V (x) +

∫ δ

0V ′(X(s))dX(s) +

1

2

∫ δ

0V ′′(X(s))σ2X(s)2ds (2.42)

Replacing (2.25) in (2.42) we obtain:

V (X(δ)) = V (x)

+

∫ δ

0V ′(X(s)) [σX(s)dW (s) + aX(s) − j(X(s))dL(s) + m(X(s))dR(s)]

+1

2

∫ δ

0V ′′(X(s))σ2X2(s)ds (2.43)

therefore (2.41) becomes:

0 = inf0≤r(t),l(t)≤θ

[

h(x) + k(x)r − d + axV ′(x) − V ′(x)j(x)l + V ′(x)m(x)r +1

2V ′′(x)σ2x2

]

which can be rewritten,

0 = inf0≤r(t),l(t)≤θ

[

ΓV (x) + h(x) − d + rm(x)(

V ′(x) − G(x))

− lj(x)V ′(x)]

(2.44)

where the differential Γ = 12σ2x2 d2

dx2 +ax ddx

is the generator of an Ito diffusion [24] with variance

σ = σx and drift a(x) = ax and

G(x) =k(x)

m(x)=

λ + α

((1 − λ) + (1 + α)x)(1 + x)(2.45)

Equation (2.44) is known as the classical Bellman equation [26]. Let Ck(I) the set of all k-

continuously differentiable functions on the interval I and let R+ the positive half of the real

14

line. From now on we will assume that V (x) ∈ C2(R+) Therefore the infimum is attained and

the minimizing values for r and l are:

r(t) = θ1A(x(t)) A =

x ∈ R : V ′(x) − G′(x) ≥ 0

l(t) = θ1B(x(t)) B =

x ∈ R : −V ′(x) ≥ 0

This is a bang-bang policy. In each state x, each available control r,l is used at maximun

possible intensity or else at minimun intensity of zero. The appearance of this type of policy

is not totally unexpected as is well familiar in the stock control literature [3], [10], [14], [27],

[29].

2.2.2 Hamilton-Jacobi-Bellman Inequalities

We have walk a long way with the Bellman equation (2.44) but we still have not found

V (x). So we will go back to Bellman’s principle (2.38) and consider three situations:

First we do not exercise any control for a δ-period and then we proceed optimally.The

performance then will be suboptimal, therefore we have:

V (x) + td ≤ Ex

[∫ δ

0h(X(s))ds + (t − δ)d + V (δ)

]

subtracting V (x) + td from both sides, diving by δ and taking the limit we get:

0 ≤ limδ→0

1

δ

Ex

[∫ δ

0h(X(s))ds

]

+ (Ex[V (X(s))] − V (x)) − dδ

0 ≤ h(x) + limδ→0

1

δEx[V (X(δ))] − V (x) − d (2.46)

to finally obtain,

0 ≤ h(x) + ΓV (x) − d (2.47)

where the proof of the equality between the generator Γ of the Ito diffusion and the second

term of (2.46) can be found in [24].

Second, at time 0+ we make an instantaneous transfer of γ dollars from the stock to the

bank:

X(

0+)

− x = −∆Lj(x)

γ

j(x)= ∆L

15

the last quantity is just the amount of control necessary to perform the transfer, but since

there is no associated cost,2 we simply find:

V (x) + td ≤ V (x − γ) + td

0 ≤V (x − γ) − V (x)

γ

0 ≤ −V ′(x) (2.48)

Likewise at time 0+ if we make an instantaneous transfer of γ dollars from the bank to the

stock:

X(

0+)

− x = ∆Rm(x)

γ

m(x)= ∆R∗

where again the last quantity is just the amount of control necessary to perform the transfer.

The associated cost is

γ(λ + α)

m(x)

to finally obtain:

V (x) + td ≤ V (x − γ) + td + γk(x)

m(x)

0 ≤V (x + γ) − V (x)

γ+ G(x)

−G(x) ≤ V ′(x) (2.49)

Notice that we have now three inequalities (2.47),(2.48),(2.49) instead of the single Bellman

equation (2.44). This last relation was deduced with the extra hypothesis (2.39) and (2.40).

Now we conjecture that the bang-bang policy is optimal in the more general situation. If this

is the case one of the inequalities must be tight which means:

min

ΓV (x) + h(x) − d, V ′(x), V ′(x) + G(x)

= 0 (2.50)

2Note that ∆L does not appear in (2.28)

16

Since the solution of (2.50) is invariant if V (x) is changed by a constant we can add the

condition3:

V (0) = 0 (2.51)

2.3 Control Limit Policies

As consequence of our bang-bang controls the process X(t) is now kept between two barriers

defined by the sets A and B, and this is done with minimal cost. We will further assume that

these sets consists each one of a single point A = A, B = B with 0 < A < B < ∞. This

means that X fluctuates freely if X ∈ [A, B] but it is reflected when X(t) = A or X(t) = B.

From the chapter 23 of [12], it follows that for each 0 < A < B and for each A ≤ X0 ≤ B

exits two continuous Ft-adapted functionals l and r and a unique process X(t) such that

A ≤ X(t) ≤ B for all t ≥ 0

X(t) = X0 +

∫ t

0σX(s)dW (s) +

∫ t

0aX(s) + r(t) − l(t) (2.52)

with:

r(t) =

∫ t

01A(X(s))dr(s)

l(t) =

∫ t

01B(X(s))dl(s)

these last two conditions only mean that r(t) and l(t) only increase when X(t) = A or X(t) = B

respectively. X(·) is known as the reflecting diffusion process on the state space [A, B] with

generator Γ and with instantantaneous reflection at the points A and B.

Take R(t) = r(t)m(A) and L(t) = l(t)

j(B) to substitute in (2.52) to obtain:

X(t) = X0 +

∫ t

0σX(s)dW (s) +

∫ t

0aX(s) +

∫ t

0m(X(s))dR(s) −

∫ t

0j(X(s))dL(s) (2.53)

the policy (L, R) for which we have (2.53) is defined as control limit policy with control limits

A and B [13].

Let us find the limiting expected cost per unit of time for a control limit policy.

3this condition was already used in (2.33) to heuristically deduce the form of the potential function.

17

Theorem 1. Let fe ∈ C2([A, B]) and e constant such that

Γfe(x) + h(x) − e = 0 A ≤ x ≤ B (2.54)

f ′e(A) = −G(A) (2.55)

f ′e(B) = 0 (2.56)

and (L, R) be a control limit policy with control limits A and B. Then for this policy

e = lim supt↔∞

1

tEx

[∫ t

0h(X(s))ds +

∫ t

0k(X(s))dR(s)

]

Proof. Since fe ∈ C2([A, B]) we can apply Ito’s lemma in this interval:

fe(X(t)) − fe(X0) =

∫ t

0f ′

e(X(s)) [σX(s)dW (s) + aX(s)ds]

+

∫ t

0f ′

e(X(s)) [m(X(s))dR(s) − j(X(s))dL(s)] +1

2

∫ t

0f ′′

e (X(s))X(s)2σ2ds

=

∫ t

0f ′

eσX(s)dW (s) +

∫ t

0

[

f ′e(X(s))aX(s) +

1

2f ′′

e (X(s))X(s)2σ2

]

+

∫ t

0f ′

e(X(s))m(X(s))dR(s) −

∫ t

0f ′

e(X(s))j(X(s))dL(s)

Notice that the last term is zero by (2.56). Taking expectations and using (2.55) we obtain:

Ex [fe(X(t)) − fe(X0)] = Ex

[∫ t

0(e − h(X))ds −

∫ t

0m(A)G(A)dR(s)

]

= et − Ex

[∫ t

0h(X(s))ds −

∫ t

0k(X(s))dR(s)

]

where we have used the fact that the expectation of the term proportional to dW is zero

because it is a local martingale. Dividing by t:

1

tEx [fe(X(t)) − fe(X0)] = e −

1

tEx

[∫ t

0h(X(s))ds −

∫ t

0k(X(s))dR(s)

]

(2.57)

and realizing that since X(t) is bounded and fe is continuous, fe is bounded in t, the first term

is zero when t → ∞. Therefore letting t → ∞, we obtain the statement of the theorem.

18

CHAPTER 3. VERIFICATION THEOREM

In this chapter we establish the connection between smooth solutions of the Hamilton-

Jacobi-Bellman inequalities (2.47),(2.48),(2.49) and the optimal stochastic control problem

defined in (2.28). The connection will be established in the next theorem.

With the solution (V (x), d) of (2.50) and (2.51) we just have found the solution to the

particular problem where the policy is a control limit one. But we want to show also that they

solve the general problem. This means we want to prove that d really minimizes the expected

cost per unit of time for any feasible policy (L, R) and the optimal policy is indeed bang-bang.

Lemma 1. If (V (x), d) is a solution of (2.50) and (2.51) such that V (x) ∈ C2(R+) and V (x)

bounded, then for any feasible policy (L, R)

d ≤ lim inft→∞

1

tEx

[∫ t

0h(X(s))ds +

∫ t

0k(X(s))dR(s)

]

Proof. The proof is very similar to the Theorem (4.7) of [26]. We write it here for completeness.

Since V (x) ∈ C2(R+) we can apply Ito’s lemma.

V (X(t)) − V (X0) =

∫ t

0V ′(X(s)) [σX(s)dW (s) + aX(s)ds + m(X)dR(s) − j(X)dL(s)]

+1

2

∫ t

0V ′′(X(s))X2(s)σ2ds

=

∫ t

0V ′(X)σX(s)dW (s) +

∫ t

0

[

V ′(X(s))aX(s) +1

2V ′′(X(s))X2(s)σ2

]

ds

+

∫ t

0V ′(X(s))m(X(s))dR(s) −

∫ t

0V ′(X(s))j(X(s))dL(s)

taking expectation in both sides and employing (2.47),(2.48),(2.49)1,we get:

Ex [V (X(t)) − V (X0)] = Ex

[

td −

∫ t

0h(X(s))ds −

∫ t

0k(X(s))dR(s)

]

1notice that condition (2.50) implies inequalities (2.47),(2.48),(2.49).

19

divide the inequality by t and take the lim inf to obtain:

lim inft→∞

1

tEx [V (X(t)) − V (X0)] = d − lim inf

t→∞

1

tEx

[∫ t

0h(X)ds +

∫ t

0k(X)dR(s)

]

because V (x) is bounded the left side goes to zero and then we have proved the lemma.

Verification Theorem. Let (V (x), d) as defined in Lemma above. If there exists an interval

[A, B] such that:

ΓV (x) + h(x) − d = 0 A ≤ x ≤ B (3.1)

V ′(x) = −G(x) x ≤ A (3.2)

V ′(x) = 0 B ≤ x (3.3)

then the optimal policy is a control limit with control limits A and B.

Proof. Because V (x) ∈ C2(R+), in particular V (x) ∈ C2([A, B]). By (3.2) and (3.3) we have

V ′(A) = G(A) and V ′(B) = 0. Thus we can use the Theorem (1) applied to (V (x), d) with

x in [A, B]. This theorem implies that for the control limit policy with control limits A and

B, d is equal to the limit (2.19). By the Lemma, d minimizes the limit for any feasible policy,

therefore the policy (V (x), d) is optimal.

20

CHAPTER 4. SOLUTION TO THE H-J-B EQUATIONS

In this chapter we will exhibit the analytic solution of the H-J-B equations (2.50) with their

respective control limit policy.

4.1 Necessary and Sufficient Conditions for Optimality

Let us suppose for a moment that we have found a function V (x) such that (3.1),(3.2),(3.3)

are satisfied. Note that this does not guarantee that this function is optimal, because V can

be a piecewise C2 function in R+ but the verification theorem requires that f(x) ∈ C2(R+) for

optimality. Hence to meet the H-J-B relations the next theorem requires V ′′(A) = −G′(A)

and V ′′(B) = 0.

Theorem 2. If (V (x), d) is a solution of (2.50),(2.51) subject to (3.1),(3.2),(3.3) if and only

if:

V ′′(A) = −G′(A) (4.1)

V ′′(B) = 0 (4.2)

Proof. • Necessary Conditions This part is very similar to the necessary condition of the

Theorem (4.7) of [26], it is here for completeness. Assume that ((V (x), d) is a solution

of (2.50),(2.51) subject to (3.1),(3.2),(3.3) but, for example, (4.2) is not true. Suppose

21

first that V ′′(B−) > 0. By the Verification Theorem:

1

2σ2B2V ′′(B+) + aBV ′(B+) + h(B) − d

= h(B) − d

=1

2σ2B2V ′′(B−) + aBV ′(B−) + h(B) − d −

1

2σ2B2V ′′(B−)

= −1

2σ2B2V ′′(B−)

Since we are assuming that V ′′(B−) > 0, the last term is less that zero which contradicts

(2.50). Now if V ′′(B−) < 0 , there exists a point y, y < B, such V ′(y) < V ′(B) = 0

which again contradicts (2.50). The case for (4.1) is similar.

• Sufficient Conditions

Direct calculation shows for x ≤ A

ΓV (x) + h(x) − d ≡1

2σ2x2G′(x) + aG(x) + h(x) − d

=1

2σ2

[(

1+α1−λ

)

x]2

[

1 +(

1+α1−λ

)

x]2 − a

(

1+α1−λ

)

x

1 +(

1+α1−λ

)

x− d

= h

[(

1 + α

1 − λ

)

x

]

evaluating at x = A, we find

h

[(

1 + α

1 − λ

)

A

]

= 0 (4.3)

Now, h(x) is decreasing for 0 ≤ x ≤ x∗, where x∗ was defined in (2.30). Notice also that

A ≤(

1+α1−λ

)

A since α ≥ 0 and 0 ≤ λ ≤ 1. Therefore for x ≤ A ≤(

1+α1−λ

)

A we get:

h(x) ≥ h(A) ≥ h

(

1 + α

1 − λA

)

= d

For x ≥ B, ΓV (x) + h(x) − d ≡ h(x) − d by (3.3) and (4.2). For x ≥ B ≥ x∗, h(x) is an

increasing non positive function1 and therefore h(x) ≥ d for x ≥ B

1We are assuming that µ > r, i.e the drift of the stock is strictly bigger that the interest rate of the bank.

22

An important consequence of the previous theorem is that we now know how to find the

value of d. We observe that

d = h(B) (4.4)

To summarize: We want to find a function V (x) ∈ C2(R+) such that:

V ′(x) = −G(x), 0 ≤ x ≤ A (4.5)

for A ≤ x ≤ B solves the differential equation

ΓV (x) + h(x) − d = 0 (4.6)

subject to:

V ′(A) = −G(A) V ′′(A) = −G′(A)

V ′(B) = 0 V ′′(B) = 0(4.7)

and finally,

V ′(x) = 0, B ≤ x (4.8)

This is one dimensional free boundary problem or a boundary value problem with overde-

termined boundary data.

4.2 Solution of the H-J-B Equations

Using elementary methods [31] the solution for (4.6) can be found:

V (x) =2d

2a − σ2lnx + ln(x + 1) +

c1

1 − 2σ2 a

x1− 2

σ2

a + c2 (4.9)

where a was defined in (2.15), d in (2.35) and (4.4) and c1, c2 are constants of integration. We

can fix the value of c1 using the boundary condition V ′(B) = 0. From

V ′(x) =2d

2a − σ2

1

x+

1

1 + x+ c1x

− 2

σ2

a (4.10)

we find:

c1 =

[

−2d

2a − σ2

1

B−

1

1 + B

]

B2

σ2

a (4.11)

23

which allows to rewrite (4.10) as:

V ′(x) =2h(B)

2a − σ2

[

1

x−

1

B

(

B

x

)2

σ2

a]

+

[

1

x + 1−

1

B + 1

(

B

x

)2

σ2

a]

(4.12)

Direct calculation shows that V ′(x) complies with the lower conditions of (4.7) but we are still

in need to find B. Once this is done we can find d using (4.4) and A through (4.3) which tell

us that(

1+α1−λ

)

A is the other root of the equation h(x) = d. In order to find B we can use the

condition −G(A) = V ′(A) which reads:

−λ + α

[(1 − λ) + (1 + α)A](1 + A)=

2h(B)

2a − σ2

[

1

A−

1

B

(

B

A

)2

σ2

a]

+

[

1

A + 1+

1

B + 1

(

B

A

)2

σ2

a]

Hence

0 =λ + α

[(1 − λ) + (1 + α)A]+

2h(B)

2a − σ2

[

1

A−

1

B

(

B

A

)2

σ2

a]

+

[

1

A + 1+

1

B + 1

(

B

A

)2

σ2

a]

(4.13)

Now from (4.3):

h

(

1 + α

1 − λA

)

= h(B)

σ2

2

(

1+α1−λ

A)2

(

1+α1−λ

A + 1)2 − a

1+α1−λ

A

1+α1−λ

A + 1=

σ2

2

B2

(B + 1)2− a

B

B + 1(4.14)

defining a new set of variables:

A =(1 + α)A

(1 − λ) + (1 + α)A(4.15)

B =B

B + 1(4.16)

equation (4.14) becomes,

σ2

2A2 − aA =

σ2

2B2 − aB

since A 6= B because A < x∗ < B we obtain the relation

A + B =2a

σ2(4.17)

Now let us go back to (4.13). Using (4.17) and h(B) = σ2

2 B2 − aB we get:

(

B

A

)a 2

σ2−1[

B2 − B]

= B2 −4a − σ2

σ2B +

2a

σ2

2a − σ2

σ2(4.18)

24

employing now

B

1 − B= B

(

1 − λ

1 + α

)

A

1 − A= A

is easy to find

B

A=

(

1 + α

1 − λ

)

(

1 − A

1 − B

)(

B

A

)

(4.19)

and using (4.17) together with (4.18) and (4.19) we finally obtain an equation for B:

[

(

1 + α

1 − λ

)

1 + B − 2aσ2

1 − B

(

B2aσ2 − B

)]2a

σ2−1(

B2 − B)

= B2 −4a − σ2

σ2B +

(

2a

σ2

)(

2a − σ2

σ2

)

(4.20)

What is now left is to prove that this relation has at least one root for B in the interval (x∗,∞).

If this is the case, we can use numerical methods to find such root (in case of many we can

take the smaller one). Let

θ ≡2a

σ2(4.21)

by the non triviality condition (2.29), we have that 1 < θ < 2. Using (4.21) we can rewrite

(4.20) to obtain

(

1 + α

1 − λ

)θ−1

B2 − (θ − 1)B(

B − 1)(

B − θ)

θ−1(

B2 − B)

= B2 − (2θ − 1)B + θ(θ − 1)

Define R(

B)

as

R(

B)

≡ B2 − (2θ− 1)B + θ(θ− 1)−

(

1 + α

1 − λ

)θ−1

B2 − (θ − 1)B(

B − 1)(

B − θ)

θ−1(

B2 − B)

(4.22)

Since x∗ < B < ∞ we have θ2 < B < 1, thus the problem now is to show that R

(

B)

has at

least one root in this interval. The next lemma proves precisely that.

Lemma 2. R(

B)

has at least one root in the interval θ2 < B < 1.

Proof. First notice that R(

B)

is a continuous function of B. So if R(

θ2

)

is positive and R (1)

is negative we are done. Direct calculation shows:

R

(

θ

2

)

=θ

4(θ − 2)

[

1 −

(

1 + α

1 − λ

)θ−1]

25

The first parenthesis of (4.2) is negative since 1 < θ < 2 and the second parenthesis is also

negative for the same reason. Therefore R(

θ2

)

> 0. Let us proceed with R(1). It is not difficult

to prove, using for example L’Hopital’s rule, that:

R(1) = 2 − 3θ + θ2

R(1) = (θ − 1)(θ − 2)

Thus R(1) is negative. Hence R(

B)

= 0 has a solution in the interval ( θ2 , 1).

26

CHAPTER 5. APPLICATION AND CONCLUSIONS

In this chapter we apply the method developed in this thesis to a real stock. We will estimate

the drift µ and the diffusion coefficient σ and with a conservative average of the bank’s interest

rates we will find the floor and the ceiling of our investing strategy.As a example we will use

the monthly close prices of Procter & Gamble. At the end we will write the conclusions.

5.1 Stock Analysis

Our model has three external parameters: µ, σ and r, which are respectively the drift and

the volatility of the stock and the interest rate of the bank account. The first two quantities

need be extracted from the particular stock that we want to analyze. For the interest rate

we have chosen the average inflation rate per year for the same period which is equal to 5.5%

[7]. This annual rate corresponds to a rate of 0.46% monthly average and to r = 0.004. This

choice of rate may seen arbitrary but we are looking to compare with a well defined standard1,

besides banks generally pay below the rate of inflation. For the stock we have chosen Procter

& Gamble, which is traded in the New York Stock Exchange, symbol PG.

5.1.1 Estimation of µ and σ2

As can be seen in Fig.5.1 we will use the close prices as reported in [23] from January 1980

to January 2000 to estimate µ and σ2. Since it is not in our goals to make a detailed analysis

of a particular stock we have used only a very crude methods to estimate µ and σ2. Our

values will be only approximate because we have assumed among other things that µ and σ2

1The interest rates that banks offer to its costumers vary from bank to bank, from type of account, etc

27

0

10

20

30

40

50

60

Jun-8

2

Dec-8

4

Jun-8

7

Dec-8

9

Jun-9

2

Dec-9

4

Jun-9

7

Dec-9

9

Proctor and Gamble

Clo

se P

rice (

US

$)

Time (Months)

Figure 5.1 Monthly evolution of stock prices

are constant in time, and this is not completely true for real stocks. For a more refined study

see for example [8],[28].

In order to estimate µ, the drift, we use an exponential regression of the stock prices. From

Fig.5.2 we have:

B(t) ≈ 1.7 exp(0.013t) (5.1)

therefore we will take µ ≈ 0.013. For the volatility σ2 we will assume that the average price

of the stock is given by the regression (5.1). Then if we want to find the variance σ2P of the

prices P (t) we use:

σ2P =

⟨

(P (t) − 〈P (t)〉)2⟩

σ2P =

⟨

(P (t) − 1.7 exp(0.013t))2⟩

σ2P = 19.02

where the brackets 〈〉 denote temporal average. To approximate the value of σ2 we will use

the price reached at the end of the observation period that is equal to US$50.53. Therefore

28

0

10

20

30

40

50

60

0 50 100 150 200 250

Proctor and Gamble (PG)

y = 1.7 * e^(0.013x) R= 0.98

Clo

se P

rices (

US

$)

Time (Months)

Figure 5.2 Exponential regression of the evolution of stock prices

σ2 ≈ 19.0250.53 = 38%.

5.1.2 Finding A and B

We are ready now to find A and B. Using (4.21) with parameters µ = 0.013, σ2 = 0.38

and r = 0.004 we find:

θ =2a

σ2

θ = 2

(

µ + σ2

2 − r)

σ2

θ = 1.047

where we have used (2.15). Now the constants related with the transaction fees are set to 2:

α = 0.01

λ = 0.02

2This values are typical [19].

29

Solving numerically (4.20) with the help of Maxima we find the root

B = 0.6335

which corresponds to:

B = 1.7288

Using A + B = θ we obtain:

A = 0.4138

this result gives :

A = 0.6850

The limits A, B, mean, using (2.9), that we need to keep the ratio of stock to bank between

0.69 and 1.76. Finally, x∗ is equal to 1.0994 and d = h(B) = −0.0498. Therefore by using

(2.16) we can conclude that the exponential monthly rate of growth will be 0.054 which is

equivalent to an annual rate of around 200%!. The skeptical reader should note how the price

of the stock rose from US$2.25 to US$50.53 during the observation period. To finish we exhibit

the V ′(x) and −G(x) in Fig.5.3.

5.2 Conclusions

• We have generalized the work of Taskar et al. [27] through the inclusion of proportional

transaction costs when the stock is sold. We have also found an analytical solution for the

H-J-B inequalities and we can determine one end of the interval of our optimal control

policy using numerical methods for a transcendental equation. The other quantities

depend in an algebraic way from this value.

• Our optimal strategy keeps the ratio of the value of the assets in an interval exerting

control only when this ratio reaches the extremes of the interval of control. In between we

do nothing. An advantage of this model is the possibility to know a priori the boundaries

of the interval since they depend from quantities that can be extracted from the behavior

of the stock.

30

-0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

V'(x) & -G(x)

V'(x)-G(x)

x

Figure 5.3 V ′(x) and −G(x) between [A, B]

• All these facts make this model suitable for applications. As an example we have ex-

tracted the drift and variance from the data of an actual stock and with a generous

estimate for the interest rate paid by the banks we have determined when to sale or buy

the stock. With this information we have also obtain the long-term rate of growth which,

as was expected, is less than Merton’s findings [20] and less than the one found by Taskar

et al. [27]

• Because the model is analytical it is easy to study the effects of transaction costs in

optimal strategies and their impact for the long-run investor. Further directions of re-

search are generalization to more than one risk asset and not assuming that the drift and

volatility are constants in time.

31

APPENDIX

ITO’S LEMMA

Ito’s Lemma. Let Xt be an Ito process given by

dX(t) = udt + vdW (t)

Let g(t, x) ∈ C2 ([0,∞) × R+), i.e. g is twice continuously differentiable on ([0,∞) × R

+).

Then

Y (t) = g(t, X(t))

is again an Ito process, and

dY (t) =∂g

∂t(t, X(t))dt +

∂g

∂x(t, X(t))dX(t) +

1

2

∂2g

∂x2(t, X(t))(dX(t))2

where (dX(t))2 = (dX(t)) · (dX(t)) is computed according to the rules

dt · dt = 0

dt · dW (t) = 0

dW (t) · dt = 0

dW (t) · dW (t) = dt

Proof. See for example [9] or [24].

32

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35

ACKNOWLEDGEMENTS

I would like to take this opportunity to express my gratitude to those who helped me with

various aspects this thesis. First and foremost, Dr. Ananda Weerasinghe for his guidance,

patience and support throughout the research and the writing of this thesis. His insights have

often inspired and helped me to to see beyond what was on paper. I would also like to thank my

committee members for their efforts and contributions to this work: Dr. Jianwei Qiu and Dr.

Justin Peters. Dr. Qiu whose support made possible for me to study this master’s degree, his

patience and generosity allow me to take hours away from my research in physics. His teaching

style opened for me new horizons personally and professionally. Dr. Peters you received me in

the Mathematics Department. I thank your for your help when I first took classes.

To all of you once again, I thank you.