Existence, Uniqueness and Continuity
of Solutions to Stochastic Differential Equations by the
Localisation Method
Nassir Mohammad
Department of Mathematics, University of Wales Swansea Singleton Park, Swansea SA2 8PP, UK
University of Wales, Swansea 2006
Submitted to the University of Wales in fulfillment of the requirements for the Degree of Master of Sciences
DECLARATION
This work has not previously been accepted in substance for any degree and is not concurrently submitted in candidature for any degree.
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STATEMENT 1
This dissertation is the result of my own independent work/investigation, except where otherwise stated. Other sources are acknowledged by notes giving explicit references. A bibliography is appended.
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STATEMENT 2
I hereby give my consent for my dissertation, if accepted, to be available for photocopying and for inter-library loan, and for the title and summary to be made available to outside organisations.
Abstract
The aim of this thesis is the exposition of multi-dimensional Ito integrals and the first presentation with Zeev Sobol of a newly developed localisation method for stochastic differential equations in several dimensions. The for-mer follows the classic book by Friedman “Stochastic Differential Equations and Applications”, with more attention to spaces of processes.
The main tool of the localisation technique is a nest (increasing sequence) of bounded relatively open sets, and respective exit times. In contrast to a classic approach (see e.g. the mentioned work by Friedman) it requires nei-ther global assumption on the coefficients of the SDE nor even the Lipschitz continuity of the drift. The presented method allows for random coefficients of the SDE, for a locally quasi-monotone drift and locally Lipschitz con-tinuous diffusion matrix with possible degeneracy, for both explosion and non-explosion cases.
Acknowledgements
I would like to express my gratitude to Dr. Zeev Sobol for all his patience, guidance and expertise in writing the thesis. Many Thanks to Prof. N. Jacob, Prof. A. Truman, Dr. J.L. Wu, Dr. C. Yuan and Dr. I.M Davies for providing a highly informative course in financial mathematics, enabling me to appreciate the beauty of the subject. I’d also like to thank my brother Shahjan for his help during the MSC course and my nieces Hafsa and Hanifah for their priceless entertainment.
In precisely built mathematical structures, mathematicians find the same sort of beauty others find in enchanting pieces of music, or in magnificent architecture. There is, however, one great difference between the beauty of mathematical structures and that of great art. Music by Mozart, for instance, impresses greatly even those who do not know musical theory; the cathedral in Cologne overwhelms spectators even if they know nothing about Christian-ity. The beauty in mathematical structures, however, cannot be appreciated without understanding of a group of numerical formulae that express laws of logic. Only mathematicians can read “musical scores” containing many numerical formulae, and play that “music” in their hearts...
Contents
1 Introduction 5 2 Background Material 10 2.1 Stochastic Processes . . . 10 2.2 Brownian Motion . . . 17 3 Stochastic Integrals 21 3.1 Spaces of Processes . . . 21 3.2 Stochastic Integrals in Multi-dimensions . . . 26 3.3 The Multi-dimensional Ito Formula . . . 344 Uniqueness, Existence & Continuity Theorems 37
4.1 Uniqueness, Existence & Continuity
Chapter 1
Introduction
Consider the following dynamical system: ˙
xt=Random+P redicted,
with the change rate ˙xt involving random and predicted components.
The Itˆo interpretation of the above is thatxtsatisfies the following
stochas-tic integral equation:
xt =x0+ Z t 0 µ(xs, s)ds+ Z t 0 σ(xs, s)dωs, (1.0.1) where µ(x, t) ∈ RM, σ(x, t)∈
RM×N and wt is an N-dimensional Brownian
motion. In the differential form we write this as follows: dxt=µ(t, xt)dt+σ(t, xt)dwt.
The latter is referred to as a stochastic differential equation (SDE) with the drift coefficient µ modelling the predicted component, and the diffusion
coefficient σ modelling the random component of the change, respectively. This thesis presents an exposition of the stochastic integration with re-spect to the Brownian motion wt, and brand new results of the joint research
with Dr Z.Sobol on the well-posedness of an SDE, that is, on the existence and uniqueness of solutions to an SDE, and their continuity with respect to the initial data.
During the past twenty years, there has been an increasing demand for tools and methods of stochastic calculus in a variety of scientific fields. It has become indispensable for the study of financial markets, reflected in the unpredictable phenomena of daily changes in stock and share prices. In particular, it is used for pricing and hedging financial derivatives such
as options and forward contracts. In Engineering, stochastic calculus has applications in filtering and control theory, whilst in Physics it is used to study the effects of random excitations on various physical phenonmena. In Biology, it is used to model the effects of stochastic variability in reproduction on populations and for birth-death processes. These are just a few examples of the many applications of stochastic analysis.
We give the reader a few problems to consider, whose solutions lie within stochastic calculus.
Problem 1. Population Growth Model.
Consider the simple population growth model. dN
dt =a(t)N(t), N(0) =N0 (constant)
where N(t) is the size of the population at time t, and a(t) is the relative rate of growth at time t. It might happen thata(t) is not completely known, but subject to some random environmental effects, so that we have
a(t) =r(t) + “noise”,
where we do not know the exact behaviour of the noise term, only its proba-bility distribution. The function r(t) is assumed to be non-random. How do we solve problem 1 in this case?
Problem 2. Optimal Stopping
Suppose a person has an asset or resource (e.g. a house, stocks, oil etc) that she is planning to sell. The price xt at time t of her asset on the open
market varies according to a stochastic differential equation of the same type as in Problem 1:
dxt
dt =rxt+αxt· “noise”,
where r and α are known constants. The discount rate is a known constant ρ. At what time should she decide to sell?
We assume that she knows the behaviour of xs up to the present time t,
but because of the noise in the system she can of course never be sure at the time of the sale if her choice of time will turn out to be the best. So what we are searching for is a stopping strategy that gives the best result in the long run, i.e. maximizes the expected profit when the inflation is taken into account. This is an optimal stopping problem.
Problem 3. Pricing of options
Suppose that at time t = 0 a person is offered the right (but not the obligation) to buy one unit of a risky asset at a specified price K and at a specified future time t =T. Such a right is called an European call option. How much should the person be willing to pay for such an option?
This problem was solved by Fischer Black and Myron Scholes (1973) using stochastic analysis and an equlibrium argument to compute a theoretical value for the price, the now famous Black-Scholes option price formula. This theoretical value agreed well with the prices that had already been established as an equilibrium price on the free market. Thus it represented a triumph for mathematical modelling in finance. It has become an indispensable tool in the trading of options and other financial derivatives. In 1997 Myron Scholes and Robert Merton were awarded the Nobel Prize in Economics for their work related to this formula. (Fischer Black died in 1995.)
The first problem in analysing of the stochastic integral equation (1.0.1) is the meaning of the second (stochastic) integral. The latter cannot be under-stood in the usual Lebesgue-Stieltjes sense since wt is nowhere differentiable
with probability 1(see Theorem 2.2.5 below). Therefore one returns to the antique method of quadratures. First we define the stochastic integral I(f),
I(f) =
Z b
a
f(t)dwt,
for an a.s. bounded piecewise constant (step) function f: given a partition a = t0 < t1 < t2 < . . . tn = b such that f(t) := f(ti−1 for t ∈ [ti−1, ti),
i= 1,2, . . . n, I(f) := n X i=1 f(ti−1)(ωti −ωti−1).
Then one proves (see Corollary 3.2.5 below) that f 7→ I(f) is a continuous operator from the spaceL1
w[a, b] of non-anticipating a.s. integrable processes
(see Definition 3.1.1) into the space L0 of a.s. finite random variables (see
Definition 2.1.9). Thus, the definition of a stochastic integral is extended to all stochastic functions from L1
w[a, b]. The exposition follows [2, Ch.1-5] and
[3].
The main result in stochastic differential calculus, is the chain rule for stochastic differentials, called the Itˆo formula. Given a stochastic differen-tial dxt = µ(t)dt+σ(t)dwt, µt ∈ RM, σ(t) ∈ RM×N, wt an N-dimensional
Brownian motion, and a functionu(z, t), continuous inRM×[0,∞) together with its derivatives ∂u∂t, ∂z∂ui,
∂u
∂zi∂zj, the stochastic differentialdu(xt, t) is
Kolmogorov operator Lis defined as follows: Lu= 1 2 M X i,j=1 N X k=1 σikσjk ∂2u ∂zi∂zj + M X i=1 µi ∂u ∂zi + ∂u ∂t.
The result holds for any a.s. integrable µ and a.s. square integrable σ (see Theorem 3.3.2 below). We do not provide the proof of the theorem due to the space restrictions. A reader can look for a one, e.g., in [2, Theorem 7.1]. The classical approach to (1.0.1) is the successive approximation scheme (Picard’s method). That is, the r.h.s. of (1.0.1) is considered as a non-linear map F in a space of processes, and the solution to the equation x := F(x) is sought as the limit of the iteration xn
t := F(xn
−1). The well-posedness
is established under the global Lipschitz continuity assumptions on µand σ and their sub-linear growth at infinity (see e.g. [2][Ch.5]).
This result was substantially improved by Krylov [4] by the localization method and theory of quasi-monotone maps. The idea of the localization, first developed in [6], is that the state space is considered as a union of relatively open sets {Sm}∞m=1, such that the solution of (1.0.1) is well-posed
till the exit of Sm, m = 1,2, . . . Then a careful analysis of the exits allows
for a global well-posedness of the problem.
A map x 7→ µ(x) is called monotone if an error x−y in the coordinate makes an obtuse angle with the respective error µ(x)−µ(y) of the map, that is, hx−y,µ|x(−xy)|−2µ(y)i ≤ 0 (a multi-dimensional analog of a decreasing function). If the latter quantity is bounded above, the map is called quasi-monotone. In particular, every Lipschitz continuous map is quasi-monotone. Quasi-monotone maps are notorious in ODE since they allow for a control of an error during an approximation procedure (see e.g. [7]).
Following the ideas of [4], we consider locally quasi-monotone driftµand locally Lipschitz continuous diffusion σ (see Hypothesis 4.1.5 below). Our main advantage is that we do not require the solution of (1.0.1) to exist globally in time but allow for the explosion. Also, we allow for the solution existing in a sub-space of RM and obtain less restrictive conditions of
non-explosion (see Theorem 4.1.9).
The thesis is organised as follows: Chapter 2 begins detailing some basic concepts from probability theory and stochastic processes. We also give a brief exposition of Brownian motion and develop the theory into the multi-dimensional case. It helps also for understanding the notation of the note. Chapter 3 starts by defining the spaces of processes and gives some necessary definitions of step-functions to facilitate our understanding of the stochastic integral. Then we present definitions and properties of the stochastic inte-gral and extend them to its multi-dimensional form. We also give several
notations of a multi-dimensional version of Itˆo’s formula. Our treatment of Chapters 2 and 3 generally follows Friedman [2, Ch.1-5] and Oksendal [3]. Chapter 4 devoted for the well-posedness SDE starts from the detailed state-ment of the problem and formulation of the main results, with the proofs split into numerous lemmas at the end of the chapter, so as to benefit the readers more interested in the results and not in the detailed proofs.
Chapter 2
Background Material
2.1
Stochastic Processes
Definition 2.1.1. (σ-field) Let Ω be a set and F ⊂ P(Ω) (the power set). We call F a σ-field in Ω if
1. Ω∈ F
2. A∈ F ⇒Ac ∈ F
3. An ∈ F, n ∈N⇒S∞n=1An ∈ F
Definition 2.1.2. Letξ⊂ P(Ω)be any family of subsets. Thenσ(ξ)denotes the σ-field generated by ξ defined by
σ(ξ) = ∩{F;ξ ⊂ F and F is a σ-field }.
We call ξ a generator of σ(ξ)
Definition 2.1.3. (Borel σ-field) In Rn a set U ⊂
Rn is called open if for
every x∈U there exists r >0 such that Br(x)⊂U where
Br(x) ={y∈Rn;|x−y|< r}.
Denote by O(n) the set of all open sets in
Rn. The σ-field generated by O(n)
is called the Borel σ-field in Rn and denoted by B(n)(
Rn) or just B(n), i.e.
B(n)=σ(O(n))6=P( Rn).
Furthermore, we note that since the complements of the open sets are the
closed sets, i.e.
C(n) ={C ⊂
Rn;Cc ∈O(n)}
we have
Definition 2.1.4. (Measure)
Let Ω be a set and F be a σ-field in Ω. We call µ : F 7−→ [0,∞] a
measure on (Ω,F) if
µ(∅) = 0
and if for a sequence {An} of disjoint sets
An∈ F, An∩Am =∅ for n 6=m⇒µ ∞ [ n=1 An ! = ∞ X n=1 µ(An). (2.1.1)
We note that a σ-field defines the set of events that can be measured, which in a probability context is equivalent to events that can be discrimi-nated. Intuitively a measure on Ω is a function which assigns a real number to subsets of Ω; this can be thought of as making precise a notion of ‘size’ or ‘volume’ for sets. One might like to assign such a size to every subset of Ω, but when for example the size under consideration is standard length for subsets of the real line, then there exist sets known as Vitali sets for which no size exists (the Vitali set being an elementary example of a set of real numbers that is not Lebesgue measurable). For this reason, one considers in-stead a smaller collection of privileged subsets of Ω whose measure is defined; these sets constitute the σ-field.
(The first indication that there might be a problem in defining length for an arbitrary set, say Ω, came from Vitali’s theorem which basically states that you can take an interval of length 1, disect it into pieces, move the pieces around and get an interval of length 2. Sometimes this result is called the Hausdorff paradox.)
Definition 2.1.5. (Measurable sets, Measure Spaces, Complete Measure Spaces and Probability spaces)
• A set Ω together with a σ-field in Ω is called a measurable space (Ω,F), and elements in F are called measurable sets.
• If Ωis a set, F a σ-field in Ωandµa measure on(Ω,F)then(Ω,F, µ)
is called a measure space.
• A measure space is called complete if any subset of a measurable set of measure zero is measurable (hence of measure zero). A σ-field F of a non-complete measure space can be completed by adding to F all the subsets of all sets of measure zero.
• A complete measure space (Ω,F,P) is called a probability space if
P(Ω) = 1. In this case P is called a probability measure or
some-times a probability.
Lemma 2.1.6. Let (Ω,F, µ) be a measure space. Let Sn ⊂ F be a sequence
of measurable sets. Let lim infSn and lim supSn be as follows:
lim inf n→∞ Sn = ∞ [ n=1 ∞ \ m=n Sm !
and lim sup
n→∞ Sn= ∞ \ n=1 ∞ [ m=n Sm ! .
Then the following identities hold:
µ(lim inf
n Sn) = lim infn µ(Sn) and µ(lim supn Sn) = lim supn µ(Sn)
The Lemma follows from (2.1.1).
Definition 2.1.7. Let (Ω,F,P) be a probability space and (E,E) be a mea-surable space (called state space). Then
• a function X: Ω→ E is called E/F-measurable if
X−1(U) :={ω ∈Ω;X(ω)∈U} ∈ F for all U ∈ E. (2.1.2) • two E/F-measurable maps X and Y are called indistinguishable if
P{X 6= Y} = 0. Indistinguishability is an equivalence relation
sep-arating the set of E/F-measurable maps into indistinguishable classes called random variables in E.
If we let (E,E) = (R,B1) then we talk about real-valued random
variables, if (E,E) = (Rn,Bn) then we talk about n-dimensional or
Rn-valued random variables.
• the σ(X) = σ{X−1(U)|U ∈ E}, where X is a random variable (i.e.
runs through the respective class of indistinguishable maps), is called
σ-fieldgeneratedbyX. Note that σ(X) is complete. In particular for
(E,E) = (R,L1), σ(X) =σ{X−1{−∞, t}|t∈ R}.
Definition 2.1.8. (Convergence of random variables)
A random variable Xn →X
• in probability ⇔P{|Xn−X|> } →0 as n→ ∞, ∀ >0.
• has p-mean convergence if E|Xn−X|p →0 as n→ ∞.
If Xn → X a.s. or if Xn → X in p-mean sense where 1 ≤ p ≤ ∞, then
Xn → X in probability. On the other hand if Xn → X in probability, then
there existsXnk (a sequence) such thatXnk →Xa.s. If in addition|Xn| ≤Y a.s. with EYp <∞, then Xn→X inp-mean sense ([1, Ch.III]).
Definition 2.1.9. (Spaces of random variables)
The space L0 of all a.s. finite random variables, equipped with the topology of convergence in probability, is a locally convex Hausdorff complete metric space with the metric d(X, Y),
d(X, Y) := inf
>0{+P(|X−Y|> )}.
Note that d is a translation invariant metric.
The space Lp = {X ∈ L0 : E|X|p < ∞}, p ≥ 1 is a separable Banach
space, equipped with the norm kXkp := (E|X|p)1/p. Convergence in Lp is the
p-mean convergence.
The space L∞ of all bounded random variables is a Banach space equipped with the norm kXk∞ := inf{α :P{|X|> α}= 0}. ([1, Ch.IV])
Definition 2.1.10. (Independence)
Let (Ω,F,P) be a probability space and (Aj)j∈I ⊂ F be a family of
events. Then we call this family independent if for every finite subset
{j1,· · ·, jm} ⊂I the following holds
P(Aj1 ∩. . .∩Ajm) = P(Aj1)·. . .·P(Ajm). (2.1.3)
This definition extends to families (ξj)j∈I, ξj ⊂ F for j ∈ I, as
fol-lows. The families (ξ)j∈I are called independent if, for every finite subset
{j1,· · ·, jm} ⊂I and every choice Ajv ∈ ξjv, v = 1,· · · , m, equality (2.1.3)
holds. Random variables (ξj)j∈I are called independent if such σ-fields
σ(Xj)j∈I are independent.
Now we are ready to state what a sochastic process is:
Definition 2.1.11. (Stochastic Process)
A stochastic process with state space (Rn,Bn) is a parameterised
collection of random variables
defined on a probability space (Ω,F,P) and assuming values in Rn. The parameter space I is usually taken to be R+ = [0,∞), however I may be an interval.
We note that for each t∈I fixed, we have a random variable
ω 7→Xt(ω), ω ∈Ω.
On the other hand, fixing ω∈Ω we can consider the function
t7→Xt(ω), t∈I
which is called a sample path of Xt.
Definition 2.1.12. (Continuous Stochastic Process)
If for a.a. ω the sample paths are continuous functions for all t∈I, then we say that the stochastic process is continuous. If for a.a. ω the sample paths are right (left) continuous, then we say that the stochastic process is right (left) continuous.
A stochastic process {Xt, t ∈ I} is said to be continuous in probability if
for any s∈I and >0,
P[|Xt−Xs|> ]→0 if t∈I, t→s.
Definition 2.1.13. (Filtration)
A filtration is an increasing family of σ-fields on a measurable space.
Given a measurable space (Ω,F), a filtration is a family of σ-fields
{Ft|0 ≤ t < ∞} with Ft ⊆ F for each t, and Ft1 ⊆ Ft2 for t1 ≤ t2. We
define F∞ as the σ-field generated by ∪Ft’s.
A filtration is often used to represent the change in the set of events that can be measured, through gain or loss of information. A typical example is in mathematical finance, where a filtration represents the information available at each time t, and is more and more precise (the set of measurable events is staying the same or increasing) as information from the present becomes available.
Definition 2.1.14. (Adapted Process, Measurable Process and Separable Process)
Let (Ω,F,P) be a probability space, the index set I ⊆ R+ = [0,∞) and
{Ft|t ∈ I} be a filtration. Let {Xt}t∈I be a stochastic process taking values
in a state space (E,E).
• The process {Xt} is said to be adapted to the filtration {Ft}
• The stochastic process X is called measurableif, for every A∈ E, the set {(t, ω);Xt(ω)∈A} belongs to the product σ-field B([0,∞))⊗ F; in
other words, if the mapping
(t, ω)→Xt(ω) : ([0,∞)×Ω,B([0,∞))⊗ F)→(E,E)
is measurable.
• For (E,E) = (Rn,B(n)), an n-dimensional stochastic process
{Xt, t∈I } is called separable if there exists a sequence {tj}dense in
I, and a subset N of Ω with P(N) = 0 such that, if ω /∈N,
{X(t, ω)∈F ∀t∈J}={X(tj, ω)∈F ∀tj ∈J}
for any open subset J of I and for any closed subset F of Rn. The sequence {tj} is called a set of separability.
It is easy to see, that a continuous process is separable with the set of separability Q∩I, and that a separable process is measurable.
We note that intuitvely, an adapted process (or non-anticipating process) is one that cannot “see into the future”. In the subsequent chapters we will see that this notion is essential for the definition of the Ito Integral which only makes sense if the integrand is an adapted process.
Example 2.1.15. An indicator process of the interval [α, β] is defined as
11[α,β](t) = ( 1, α≤t < β 0 otherwise Note that 11[α,β](t) = 1 = α≤t < β ∩ α≤t ∩ β > t
proving 11[α,β](t) is an adapted process.
Definition 2.1.16. (Stopping Time)
Let {Ft}t≥0 be a filtration. A R+-valued random variable τ is called a
stopping time with respect to the filtration {Ft} if {τ ≤ t} ∈ Ft for all
t ≥0.
Notice that the sets {τ > t}, {s < τ ≤ t}, {s < τ < t}, {s ≤ τ ≤ t},
{s ≤τ < t} with s ≤t also belong to Ft.
In this thesis note that stopping times are usually denoted by lowercase greek letters. Given two stopping times α and β, α ≤ β a.s., we denote by
(α, β), (α, β], [α, β), [α, β], the set of (Ft)-stopping times τ satisfying a.s.
Lemma 2.1.17. Let {Ft}t≥0 be a filtration, yt be a continuous nonnegative
Ft-adapted process and let γ be an {Ft}-stopping time.
Assume E(yτ)≤N, for any {Ft}-stopping time τ ≤γ.
Then
P{sup
t≤γ
yt≥} ≤N/ (2.1.4)
for any >0.
Proof. Set τ =γ∧inf{t ≥0 :yt≥}. Note that τ is an {Ft}-stopping time
and τ ≤γ. The following equality holds:
ω : sup t≤γ yt≥ ={ω :yτ ≥}. (2.1.5)
Indeed, the following two cases are possible: either γ(ω) is larger than inf{t≥0 :yt(ω)≥}or it is smaller then this.
Ifγ(ω) is the largest of the two then supt≤γyt(ω)≥ and
τ(ω) = inf{t≥0 :yt(ω)≥}.Hence yτ(ω)≥, by continuity of the process.
Alternatively, ifτ(ω) =γ(ω)≤inf{t ≥0 :yt(ω)≥} then
supt≤γyt(ω)≤ and yτ(ω) = yγ(ω)≤.
Finally we use Chebyshev’s inequality to show that (2.1.4) holds, i.e.
P{w:yτ ≥} ≤ E
{yτ}
≤
N .
Corollary 2.1.18. Let Yt(n) be a sequence of (Ft)-adapted processes, γ be a
bounded (Ft)-stopping time.
Assume that sup
τ∈[0,γ)
E|Yτ(n)| →0 as n → ∞.
Then sup
0<t<γ
|Yt(n)| →0 as n→ ∞, in probability. (2.1.6)
Proof. By Lemma 2.1.17, the following estimate holds for all >0:
P sup 0<t<γ |Yt(n)| ≥ ≤ sup τ∈[0,γ)E |Yτ(n)| . (2.1.7) Then the numerator of the r.h.s. of (2.1.7) vanishes as n → ∞, by the assumption.
2.2
Brownian Motion
Robert Brown, an English botanist first observed the phenomenon of the erratic motion of grains of pollen suspended in a liquid in the summer of 1827. He further observed that pollen grains suspended in water performed a swarming motion, hence the motion was named after Robert Brown. He reported all this in R. Brown (1828)
In 1900 L. Bachelier tried to establish the framework for a mathematical theory of Brownian motion and used it as a model for the stock market. However, Bachlier’s work was largely ignored by academics, but later his work was to be highlighted as the first step in the mathematical theory of stock markets.
Many years following the discovery of Brownian motion, Albert Einstein (1905) gave the first theoretical approach to explain the phenomenon. He asserted that the Brownian motion originates from the continuous bombard-ment of pollen grains by the surrounding water, hitting the pollen grains from different angles with different intensities.
However, Brownian motion had more to offer and it took a few more years until a clear picture of the Brownian stochastic motion could be built with a rigorous mathematical foundation. It was in 1923 that Norbert Wiener gave this rigorous treatment and proved its existence. Hence, the name Wiener process. In this text we will interchangably use the term Brownian motion and Wiener process.
Definition 2.2.1. (Brownian Motion)
Brownian motion, Bt, is a stochastic process (or guassian process)
start-ing at the origin 0 at time t = 0 with independent increments which are guassian, centered at 0 and with variance (t−s), where (t−s) denotes the increment in time. It is also homogeneous in time, i.e. it starts afresh at each time with no memory of the past.
In stricter mathematical terms its properties are thus summarised as: 1. B0 = 0.
2. for 0≤s < t, Bt−Bs∼N(0, t−s).
3. Independence of increments: The increments of Bt are independent,
i.e. for any finite set of times 0≤t1 ≤t2 ≤ · · · ≤tn < T, the random
variables Bt2 −Bt1, Bt3 − Bt2,· · · , Btn −Btn−1 are independent. i.e. Bt−Bs is independent of the past, Bu where 0≤u≤s.
We note that Bt(t ≥ 0) is a continuous time stochastic process which is
a function of t(For a proof see Friedman [2, Theorems 1.1 and 1.2]. We also have E(Bt) = 0 and E(Bt2) =t.
Proposition 2.2.2.
E{BtBs}=min(t, s) = (t∧s).
Proof. We find that for t > s and by writing Bt= (Bt−Bs) +Bs
E(BtBs) = E((Bt−Bs) +Bs)Bs)
= E((Bt−Bs)Bs) +E(Bs2).
By independence we see that
E(BtBs) =E(Bt−Bs) | {z } 0 E(Bs) +s =s = (t∧s).
Proposition 2.2.3. Let I = [a, b] and J = [c, d] be two intervals, with
I ⊂R+, J ⊂
R+. Also Let ∆B(I) =Bb−Ba and ∆B(J) =Bd−Bc.
Then we have the remarkable formula:
E(∆B(I)∆B(J)) =|I∩J| (2.2.1)
where |E| denotes the Lebesgue measure (or length) of a set E.
Proof. Assume we have the numbers a, b, c, d, then we can assume without loss of generality thatd=max(b, d). Now we will have four cases to consider: Either c = min(a, c) or a = min(a, c) and either c = min(b, c) or b = min(b, c).
We will consider one case in the following order on the real line: a < c < b < d. The other cases follow similarly to this proof.
Note that the diagram below illustrates a clear understanding of where the intervals are situated and how they overlap each other.
a c b d J I
So expanding out the formula: E{∆B(I)∆B(J)}we have
E{∆B(I)∆B(J)} = E{Bb−Ba}{Bd−Bc}
= E{BbBd} − {BbBc} −E{BaBd}+E{BaBc}.
Using the earlier resultE{BtBs}= (t∧s), we find that the r.h.s is equal
to b−c−a+a =b−c=|I∩J|.
Similarly proving the other cases completes our proof. An important consequence of the above is the formula (dBt)2 =dt.
With the preliminary notions of Brownian motion covered, we move on to look at Brownian motion in n-dimensions where we generalise our theory.
Definition 2.2.4. An n-dimensional process wt = (w
(1)
t ,· · · , w
(n)
t ) is called
an n-dimensional Brownian motion (or Wiener process) if each of the pro-cesses wt(i) is a Brownian motion and if the σ-fields Fi := σ(w
(i)
t , t ≥0) are
independent for all 1≤i≤n.
Theorem 2.2.5. ∀T >0, wt is almost surely nowhere Holder C1/2
continu-ous on [0, T).
For a proof see Friedman A. [2, Thrm 2.1].
Theorem 2.2.6. Let wt, t ≥ 0 be a continuous process and let {Ft}t≥0 be
a filtration such that wt is adapted and the following identities hold for all
0≤s < t,
E[(wt−ws)|Fs] = 0 a.s.
and
E[(wt−ws)2|Fs] =t−s a.s.
Then wt is a Brownian motion.
For a proof see Friedman A. [2, Thrm 5.1].
Theorem 2.2.7. Letwt= (w
(1)
t ,· · · , w
(n)
t ), t≥0be a continuous,n-dimensional
process and let Ft(t ≥ 0) be an increasing family of σ-fields such that wt is
Ft measurable and, for all 0≤s≤t≤ ∞,
E[wt−ws|Fs] = 0 a.s.
E[(wt(i)−ws(i))(w
(j)
t −ws(j))] =δij(t−s) a.s.
Proof. For anyγ ∈Rn,|γ|= 1, the conditions of Theorem (2.2.6) are satisfied
forγ·wt. Hence,γ·wtis a Brownian motion. Thus we havew
(i)
t as a Brownian
motion for each i.
It remains to show that each of these motions are mutually independent. To verify the independence we show that σ(w(ti), t ≥ 0) is independent of σ(wt(j), t ≥0) ifi6=j. Takingi= 1 and j = 2 we see that γ1w
(1)
t +γ2w (2)
t is
a Brownian motion if γ12+γ22 = 1, we have
E[γ1w (1)
t +γ2w (2)
t ]2 = 1.
This is easily seen since;E{(w(ti))2}=tand E{w(1)t w(2)t = 0}, due to the fact that each wt(i) is a Brownian motion.
Now we see that
E{w(1)t+sw (2) t }=E{w (2) t E[w (1) t+s|Ft]}=Ew (2) t w (1) t = 0.
Here we see that w(1)t+s is independent of w(2)t . Similarly wt(2)+s is independent of w(1)t . We chose t and s as arbitary non-negative numbers, and so
Chapter 3
Stochastic Integrals
In the following let us assume that we have a probability space (Ω,F,P) and we have Brownian motion wt in Rn adapted to a filtration {Ft}t≥0.
Further assume greek letters, in particular α and β, denote {Ft}-adapted
random variables (Markov stopping times). One can take for instance, Ft=
σ(ws,0≤s≤t).
3.1
Spaces of Processes
Definition 3.1.1. A Rn-valued separable (F
t)-adapted measurable processes
f(t) will be called non-anticipating. Two non-anticipating processes f and
g areindistinguishable on (α, β) if the following holds:
dt×P{(t, ω) :α(ω)< t < β(ω), f(t, ω)6=g(t, ω)}= 0.
We denote by L0
w[α, β] the set of classes of non-anticipating processes
indis-tinguishable on (α, β). We denote by Lp
w[α, β]⊂L0w[α, β] the set consisting of classes of all
sep-arable (Ft)-adapted measurable processes f(t) with (1≤p≤ ∞) satisfying:
P Z β α |f(t)|pdt <∞ = 1, 1≤p < ∞ P ess sup α<t<β |f(t)|<∞ = 1, p=∞.
Let Cw[α, β]⊂L∞w[α, β] denote the sub-space of continuous processes.
We denote by Mp
w[α, β] the subset of Lpw[α, β] consisting of all processes
kfkp :=E Z β α |f(t)|pdt <∞, 1≤p <∞ kfk∞ := ess sup ω∈Ω,α(ω)<t<β(ω) |f(t, ω)|<∞, p=∞.
Definition 3.1.2. (Step function)
Let a and b denote non-random stopping times. A non-anticipating pro-cess f(t) on [0,∞] is called a step function if there exists a partition
0 = t0 < t1 <· · ·< tr <∞ of [a, b] such that
f(t) =
(
f(tk) if tk≤t < tk+1, 0≤k ≤r−1
0 if t ≥tr
Note that a step function belongs to L∞w and that f(ti) is a bounded
random variable.
Proposition 3.1.3. The space Lp
w[α, β]is a complete, locally convex,
Haus-dorff vector space equipped with the metric d(f, g),
d(f, g) := inf γ>0 γ +P Z β α |f −g|pdt > γp , with d(fn, f)→0 iff Z β α |f −fn|pdt →0 in probability as n→ ∞.
Proof. Assume that there exists γ0 >0 such that
P Z β α |fn−f|pdt > γ0p ≥σ ∀n, then inf γ>0 γ+P Z β α |fn−f|pdt > γp ≥γ0∧σ90. Therefore, d(f, g)→0⇒fn−f →0.
Now assume thatRαβ|fn−f|pdt→0, in probability,
then inf γ>0 γ+P Z β α |fn−f|pdt ≥γp ≤σ+P Z β α |fn−f|pdt≥Σp ,
this →σ as n→ ∞ for any σ >0, hence distance lim sup n→∞ d(fn, f)≤σ, ∀σ >0⇒lim sup n→∞ d(fn, f) = 0.
The rest of the assertion follows from [1][Ch.IV.11].
Proposition 3.1.4. The space Mp
w[α, β]is a Banach space equipped with the
norm EnRαβ|f(t)|pdto
The proposition readily follows from [1][Ch.III.6].
Lemma 3.1.5. Let f ∈Lpw[α, β]. Then:
1) there exists a sequence of continuous functionsgn such that
lim
n→∞ Z β
α
|f(t)−gn(t)|pdt = 0 a.s. (3.1.1)
2) there exists a sequence of step functionsfn such that
lim n→∞ Z β α |f(t)−fn(t)|pdt = 0 a.s. (3.1.2) Proof. Let ρ(t) = ( c exp[−1/(1−t2)] if |t|<1, 0 if |t| ≥1;
where cis a positive constant such that R−∞∞ ρ(t)dt= 1. Now definef(t) = 0 if t < α and let
(Jf)(t) = 1 Z β α−1 ρ t−s− f(s)ds, (2 <1). (3.1.3)
Then clearly (Jf)(t) is continuous by definition since as t → u, (Jf)(t)→
(Jf)(u).
Let us present an alternative representation to (3.1.3). Lettingβ =t and (α−1) = (t−2) we have (Jf)(t) = 1 Z t t−2 ρ t−s− f(s)ds. (3.1.4)
Substituting using z =t−s− where s=t−z−, we have dzds = 1 giving −dz =ds. The limits of the integral also change to give t7→(t−t−) =− and t−27→t−(t−2)− =.
So we have 1 Z − ρ(z/)f(t−z−)−dz = 1 Z − ρ(z/)f(t−z−)dz = Z − ρ(z)f(t−z−)dz. Thus giving us (Jf)(t) = Z − ρ(z)f(t−z−)dz. (3.1.5)
Using Schwartz’s inequality and (3.1.5) we find
Z β α (Jf)2 = Z β α Z − ρ1/2(z)·ρ1/2(z)f(t−z−)dz dt ≤ Z β α Z − ρ(z)dz Z − ρ(z)f2(t−z−)dz dt ≤ Z − ρ(z) Z β α−1 f2(t)dt dz.
Following from this, since was arbitary, R−ρ(z)dz ≤ R∞
−∞ρ(z)dz = 1 and
using the fact that f(t) = 0 if t < α, we have
Z β α (Jf)2dt≤ Z β α f2(t)dt. (3.1.6)
Now consider for a fixed w such that Rαβf2(t, w)dt < ∞, let un be a
non-random continuous function such that un(t) = 0 if t < α and
Z β
α
|un(t)−f(t, w)|2dt→0 if n→ ∞. (3.1.7)
Due to the fact that un is continuous (Jun)(t)→un(t) uniformly in
t ∈[α, β], as →0. We now write Z β α |(Jf)(t, w)−f(t, w)|2dt ≤ Z β α |J(f(·, w)−un(·)(t)|2dt + Z β α |(Jun)(t)−un(t)|2dt+ Z β α |un(t)−f(t, w)|2dt.
Using (3.1.6) we see with f replaced by f−un
Z β α (J(f−un))2dt ≤ Z β α (f −un)2(t)dt,
whilst taking →0 we find Z β α |(Jun)(t)−un(t)|2dt →0. Hence we obtain lim sup →0 Z β α |(Jf)(t, w)−f(t, w)|2dt≤2 Z β α |un(t)−f(t, w)|2dt.
Finally using (3.1.7) and taking n → ∞we obtain
lim sup
→0
Z β
α
|(Jf)(t)−f(t)|2dt = 0 a.s.
Since the integrand on the r.h.s. of (3.1.4) is a seperable process that is Ft
measurable, the integral is also Ft measurable. Hence, (3.1.1) holds with
gn=J1/nf.
To prove the second part of (3.1.5) let
hn,m(t) = (−M)∨gn k m ∧(M) if α+k m ≤t < α+ k+ 1 m (0≤k < m(β−α)). Then lim m→∞ Z β α |hn,m(t)−gn(t)|2dt= 0 a.s. (3.1.8)
Now for any δ >0,
P Z β α |f(t)−gn(t)|2dt > δ/2 < δ/2, for some n =n0.
taking n =n0 and using (3.1.8) we find
P Z β α |gn0(t)−hn0,m(t)| 2dt > δ/2 < δ/2, for some m=m0. Hence, P Z β α |f(t)−hn0,m0(t)| 2dt > δ < δ.
Taking δ= 1/k and denoting the correspondinghn0,m0 byψk, it follows that ψk∈L2w[α, β] and
Z β
α
|f(t)−ψk(t)|2dt →0 as n→ ∞, in probability.
3.2
Stochastic Integrals in Multi-dimensions
Lettingaandbdenote non-random variables we define the stochastic integral in this section to be,
I(T) =
Z b
a
f(t)dwt
where wt is a Brownian motion and f(t) ∈ L1w[a, b] is a stochastic function,
giving a clear interpretation of it whilst stating some of its important prop-erties. Note that we cannot define it in the usual Lebesgue sense due to the fact thatwtis nowhere differentiable with probability 1 (See Theorem 2.2.5).
Before proceeding let us adopt the following notation:
∆tk= (tk+1∧b∨a)−(tk∧b∨a) and ∆ωk =ωtk+1∧b∨a−ωtk∧b∨a where we note that
t∧b∨a = t if a≤t≤b a if t≤a b if t≥b
Definition 3.2.1. (The stochastic integral)
Letf(t)be a step function and sayf(t) =fkiftk ≤t < tk+1,0≤k ≤r−1
where 0 =t0 < t1 <· · ·< tr <∞.
The random variable
Z b a f(t)dwt = r−1 X k=0 f(tk)[∆ωk].
is called the stochastic integral of f with respect to the Brownian motion
w or theIto integral on [a, b].
We need to further our definition of the stochastic integral and define it for any function f in L1w[a, b], however, first we require some additional results which we present in the following lemmas.
Lemma 3.2.2. Letf1, f2be two step functions, and letλ1, λ2 be real numbers.
Then λ1f1+λ2f2 is in L1w[a, b] and Z b a [λ1f1+λ2f2]dwt =λ1 Z b a f1(t)dwt+λ2 Z b a f2(t)dwt.
The assertion directly follows from Definition 3.2.1.
Lemma 3.2.3. For a step function f, the following holds:
E Z b a f(t)dwt= 0, (3.2.1) E| Z b a f(t)dwt|2 =E Z b a f2(t)dt. (3.2.2)
Proof. Note that f(tk) is Ftk measurable where as ∆ωk is independent of Ftk. Hence,
Ef(tk)∆ωk=Ef(tk)E∆ωk= 0. (3.2.3)
Summing over k we have the result (3.2.1). Next, we expand the l.h.s. of (3.2.2) to find
E| Z b a f(t)dwt|2 = E ( r−1 X i=0 f(ti)∆wi ! r−1 X k=0 f(tk)∆wk !) = E (r−1 X i,k=0 f(ti)f(tk)∆wi∆wk ) = r−1 X i=k=0 Ef2(tk)(∆wk)2 + r−1 X i<k,k<i E{f(ti)f(tk)∆wi∆wk}.
Ifi < k, (f(ti)f(tk)∆wi) is in the past of ∆wk and so ∆wk is independent of
(f(ti)f(tk)∆wi). A similar result follows for i > k.
In view of the identity (3.2.3) and the finiteness ofE|∆ωk|, we have that r−1 X i<k,k<i E{f(ti)f(tk)∆wi∆wk} = r−1 X i<k E{f(ti)f(tk)∆wi} ·E{∆wk} + r−1 X k<i E{f(ti)f(tk)∆wk} ·E{∆wi} = 0
Hence E| Z b a f(t)dwt|2 = r−1 X k=0 Ef2(tk)(∆ωk)2 = r−1 X k=0 Ef2(tk)E(∆ωk)2 = r−1 X k=0 Ef2(tk)(∆tk) = E Z b a f2(t)dt and (3.2.2) is proved.
Lemma 3.2.4. For any step function f, >0, N >0 and p≥1,
P | Z b a f(t)dwt|> ≤P Z b a fp(t)dt > N +N/p. Proof. Let φN(t)= ( f(t) if tk≤t < tk+1 and Pkj=0|f(a∨b∧tj)|p∆tj ≤N, 0 if tk≤t < tk+1 and Pkj=0|f(a∨b∧tj)|p∆tj > N. where f(t) =f(a∨b∧tj) if (a∨b∧tj)≤t <(a∨b∧tj+1); 0 = t0 < t1 <· · ·< tr <∞. Then φN ∈Lpw[a, b] and Z b a |φN(t)|pdt= v X α≤tj≤tv |f(a∨b∧tj)|p∆tj,
where v is the largest integer such that
v X j=0 |f(a∨b∧tj)|p(∆tj)≤N, v ≤r−1. Hence E Z b a |φN(t)|pdt≤N.
Further, f(t)−φN(t) = 0 for all t ∈[α, β) if Rβ α f p(t)dt < N. Therefore, P Z b a f(t)dwt > ≤P Z b a φN(t)dwt > +P Z b a fp(t)dt > N . Since, by Chebyshev’s inequality, the first integral on the right is bounded by 1/pE Z b a φN(t)dwt 2 ≤N/p, the assertion (3.2.4) follows.
Corollary 3.2.5. The map f 7→Rb
a f(t)dwt is a continuous map
Lpw[a, b] → L0w. i.e. if a sequence of step functions fn is a cauchy sequence
in Lpw, then Rabfn(t)dwt form a cauchy sequence in Lpw,
lim
n,m→∞ Z b
a
|fn(t)−fm(t)|pdt →0, in probability.
Hence the map can be uniquely extended from step functions to whole Lpw. Proof. By Lemma (3.2.4) for any >0, ρ >0
P | Z b a fn(t)dwt− Z b a fm(t)dwt|> ≤P Z b a |fn(t)−fm(t)|pdt > pρ +ρ. Therefore, lim sup n,m→∞ P | Z b a fn(t)dwt− Z b a fm(t)dwt|> ≤lim sup n,m→∞ P Z b a |fn(t)−fm(t)|pdt > pρ +ρ =ρ.
Since ρ >0 is an arbitary number, it can be made as small as we like and so it follows that
Z b a
fn(t)dwt
is a Cauchy sequence in probability. By Lemma (3.1.5) for all f ∈Lp
w[α, β] there exists a sequencefn of step
Z b a
|f(t)−fn(t)|pdt →0 as n→ ∞, in probability.
Thenfnis a cauchy sequence inLpw[a, b] hence
Rb
a fn(t)dwtconverges in
prob-ability to the limit denoted by
Z b
a
f(t)dwt,
called the stochastic integral (or the Ito integral ) off(t) w.r.t. the Brownian motion wt. The above definition is independent of the particular sequence
{fn}.
Now we show some important results which can be used for the evaluation and manipulation of integrals.
Theorem 3.2.6. Let f1, f2 be any functions from Lwp[a, b] and let λ1, λ2 be
real numbers. Then
λ1f1+λ2f2 is in Lpw[a, b] and Z b a [λ1f1(t) +λ2f2(t)]dwt =λ1 Z b a f1(t)dwt+λ2 Z b a f2(t)dwt.
The proof follows from Lemma (3.2.2) and Corollary (3.2.5).
Theorem 3.2.7. If f is a function in Mp w[a, b] then, E Z b a f(t)dwt= 0, E| Z b a f(t)dwt|p =E Z b a fp(t)dt. The proof follows from Lemma (3.2.3).
Theorem 3.2.8. If f is a function from Lp
w[a, b], then, for any >0, N >0,
P | Z b a f(t)dwt|> ≤P Z b a fp(t)dt > N +N/p. (3.2.4)
The proof follows from Lemma (3.2.4).
Lemma 3.2.9. If f ∈Lp
w[a, b] and ξ is a bounded and Fα measurable
func-tion, then ξf is in Lp w[a, b] and Z b a ξf(t)dwt=ξ Z b a f(t)dwt. (3.2.5)
Proof. It is clear that ξf is in Lp
w[a, b]. If f is a step function, then (3.2.5)
follows from the definition of the stochastic integral. For generalf inLpw[a, b], let fn be step functions in Lpw[a, b] satisfying
Z b
a
|fn(t)−f(t)|pdt →0, in probability as n→ ∞.
Applying (3.2.5) to each fn we have
Z b a ξfn(t)dwt=ξ Z b a fn(t)dwt,
and taking n→ ∞ we find using Corollary (3.2.5)
ξ Z b a fn(t)dwt →ξ Z b a f(t)dwt in probability.
The next theorem improves Theorem (3.2.7).
Theorem 3.2.10. Let f ∈Mwp[a, b] , then
E Z b a f(t)dwt Fa = 0, (3.2.6) E | Z b a f(t)dwt|p Fa =E Z b a fp(t)dt Fa =E Z b a E[fp(t)Fa]dt , ∀p >2. (3.2.7)
Proof. Letξ be a bounded and Fameasurable function. Then ξf belongs to
Mwp[a, b] and by Lemma (3.2.7)
E
Z b
a
Hence, by (3.2.5) E ξ Z b a f(t)dwt = 0, i.e. E ξE Z b a f(t)dwt Fa = 0.
This implies (3.2.6). The proof of (3.2.7) is similar.
Definition 3.2.11. (Random Limits Integral) We define
Z β α f(t)dwt= Z ∞ 0 f(t)11[α,β](t)dwt,
where 11[α,β]is defined as in Example (2.1.15).
Definition 3.2.12. (The indefinite integral)
Let f ∈L2
w[0, T] and consider the integral
I(t) =
Z t
0
f(s)dws, 0≤t ≤T
where, by definition, R0
0 f(s)dws = 0. We refer to I(t) as the indefinite
intgral of f. Notice that I(t) isFt measurable.
Theorem 3.2.13. If f ∈L2w[0, T], then the indefinite integral
I(t) (0≤t ≤T) has a continuous version.
For a proof see Oksendal [3, Theorem 3.2.5]
Now that we have the necessary definitions of stochastic integrals we can extend the notions to look at the multi-dimension situation.
Definition 3.2.14. We say that a matrix of functions belongs to Lp w[α, β]
(or to Mwp[α, β]) if each of its elements belongs to Lpw[α, β] (or to Mwp[α, β]).
Definition 3.2.15. Let σ=σ(ij) =σ(t, x) = σ11(x, t) σ12(x, t) · · · σ1N(x, t) σ21(x, t) σ22(x, t) · · · σ2N(x, t) · · · · σM1(x, t) σM2(x, t) · · · σM N(x, t) (3.2.8)
be an M ×N matrix that belongs to Lpw[α, β], then the stochastic integral Rβ α σ(t)dwt is an M-vector defined by Z β α σ(t)dwt= ( N X j=1 Z β α σij(t)dw (j) t ) i=1,···,M
where each integral on the right is defined as in Definition (3.2.11).
Proposition 3.2.16. Let σ ={σij}i=1,...,M,j=1,...,N ∈Mw2[t1, t2] then E| Z t2 t1 σ(t)dwt|2 =E Z t2 t1 |σ(t)|2dt, where |σ|2 = M X i=1 N X j=1 (σij)2. Proof. If substitute A=Rt2 t1 f dwi, B = Rt2 t1 gdwi in the identity 4AB= (A+B)2−(A−B)2, and use Theorem (3.2.3), we find that
E Z t2 t1 f(t)dw(ti) Z t2 t1 g(t)dw(ti) =E Z t2 t1 f(t)g(t)dt (3.2.9)
provided f and g belong to Mp
w[t1, t2].
We also observe that
E Z t2 t1 f(t)dwt(i) Z t2 t1 g(t)dw(tj) = 0 ifi6=j. (3.2.10) due to the fact that the integrals are independent and with zero expectation.
3.3
The Multi-dimensional Ito Formula
We now state Ito’s Multi-dimensional formula which is extended from that of the 1-dimensional Ito formula. This is itself a great result and we will make use of it when we show the existence and uniqueness of solutions to SDE’s.
Let us begin with a definition
Definition 3.3.1. Let xt be an M-dimensional process for 0 ≤ t ≤ T, and
suppose that, for any 0≤t1 < t2 ≤T,
xt2 −xt1 = Z t2 t1 µ(t)dt+ Z t2 t1 σ(t)dwt
where µ= (µ1,· · · , µM)and the M×N matrixσ = (σij) as in (3.2.8) belong
to L1w[0, T] and L2w[0, T] respectively. Then we say that xt has a stochastic
differential given by
dxt=µ(t)dt+σ(t)dwt. (3.3.1)
Theorem 3.3.2. (Multi-Dimensional Ito formula)
Letu(z, t)be a continuous function inRM×[0,∞)together with its deriva-tives ∂u∂t,∂z∂ui,
∂u
∂zi∂zj. Let xt be an M-dimensional process having a stochastic
differential
dxt=µ(t)dt+σ(t)dwt
where µ = (µ1,· · · , µM) and σ = (σij),(1 ≤ i ≤ M,1 ≤ j ≤ N), belong to
L1w[0, T] and L2w[0, T] respectively. Then u(xt, t) has a stochastic differential
equation du(xt, t) = ∂u ∂t(xt, t) + M X i=1 ∂u ∂xi(xt, t)µi(t) + 1/2 N X l=1 M X i,j=1 ∂u ∂xi∂xj(xt, t)σil(t)σjl(t) dt + N X l=1 M X i=1 ∂u ∂xi(xt, t)σil(t)dwl(t). (3.3.2)
For a proof see Friedman [2, Theorem 7.1]. If we let qij = N X l=1 σilσjl, i, j = 1, . . . M
and Lu= 1 2 M X i,j=1 qij ∂2u ∂yi∂yj + M X i=1 qi ∂u ∂yi + ∂u ∂t, (3.3.3) whereLis the Kolmogorov operator, we can write Ito’s formula in the short-hand form
du(xt, t) =Lu(xt, t)dt+∇u(xt, t)σ(t)dwt.
Furthermore, if we define formally a multiplication table:
dwitdt = 0, dtdt= 0, dwitdwtj = 0 ifi6=j, dwitdwtj =dt, so that dxidxj = N X l=1 σilσjldt.
Then Ito’s formula (3.3.2) takes the form
du(xt, t) = ∂u ∂t(xt, t)dt+ M X i=1 ∂u ∂xi(xt, t)dx i t+ 1 2 M X i,j=1 ∂2u ∂xi∂xj(xt, t)dx i dxj. From Ito’s formula we also obtain
u(xτ, τ)−u(x0,0) = Z τ 0 (Lu)(xs, s)ds+ Z τ 0 ∂u ∂x(xs, s)σ(s)dw(s) for any random variableτ, 0≤τ ≤T. Ifτ is a stopping time andLu, ∂u∂x·σ are in M1
w[0, T] andMw2[0, T] respectively, then
Eu(xτ, τ)−Eu(x0,0) =E
Z τ
0
Theorem 3.3.3. Let f = (f1,· · · , fn) belong to L2w[0,∞] and suppose that
R∞
0 |f|
2dt =∞ with probability 1, and define
τ(t) = inf s; Z s 0 |f(λ)|2 dλ =t .
Then the process
u(t) =
Z τ(t)
0
f(s)dws
is a Brownian motion.
Chapter 4
Uniqueness, Existence &
Continuity Theorems
4.1
Uniqueness, Existence & Continuity
Theorems on SDE’s
Consider some possible solutionsxt(ω) to the stochastic differential equation
dxt
dt =σ(t, xt)dwt+µ(t, xt), µ(t, x)∈R, σ(t, x)∈R
where wt is 1-dimensional Brownian motion. The Ito interpretation of the
above is that xt satisfies the stochastic integral equation
xt=x0 + Z t 0 µ(s, xs)ds+ Z t 0 σ(s, xs)dws.
In differential form we have
dxt=µ(t, xt)dt+σ(t, xt)dwt. (4.1.1)
This chapter presents joint research with Z. Sobol on new results exploring the existence, uniqueness and continuity with respect to the inital data of solutions to stochastic differential equations. The main novelty of the result is that we require neither Lipschitz continuity of the coeffecients nor a sub-linear growth of µ, in spite of the classical approach (see e.g. [2]). Indeed, we develop a localisation technique originated in Stroock and Varadhan [6] and then developed by Krylov [4]. Then we assumeσ to be locally Lipschitz and µ to be quasi-monotone (see Hypothesis 4.1.5 below). Moreover, we generalise our approach to consider multi-dimensional stochastic differential
equations, thus not only showing solutions in 1-dimensions. The solution is constructed by the Euler method.
Let (Ω,F,P) be a complete probability space and let (wt,Ft) be a N
-dimensional Wiener process on this space defined for t ∈ [0,∞), with the complete filtration (Ft) andσ-fields being complete with respect toF,P. All
the random processes and stopping times are assumed (Ft)-adapted.
The next definition contains the main tools of the localisation technique: an increasing sequence of bounded relatively open sets (a nest), and respec-tive exit times.
Definition 4.1.1. (Nest and relative stopping times)
Let S = (Sm)m∈N be a nest (an increasing sequence) of bounded subsets
of RN such that S
m ⊂Sm+1, dist(Sm, Smc+1)>0, m= 1,2,3, . . .,
S∞ =∪∞m=1Sm, Sm are relatively open in S.
Given a nest and an(Ft)-adapted processx= (xt), we define the following
stopping times:
τm(x) := sup{t >0 :xs ∈Sm for all s∈[0, t]}, m= 1,2,3, . . . ,
τ∞(x) := sup{t >0 :∃m∈N such that xs ∈Sm for all s∈[0, t]}.
(4.1.2)
Obviously, τm(x) ↑ τ∞(x) as m → ∞ a.s. Note also that xτm 6∈ Sm for an (Ft)-adapted right continuous processx. Some important properties of a
nest are given in several lemmas below.
Lemma 4.1.2. Let x, y be continuous (Ft)-adapted processes such that
P{xt =yt,0< t < τm(x)∧τm(y)}= 1 for all m= 1,2, . . .
Then τm(x) =τm(y) for m= 1,2, . . . ,∞ and sup
0<t<τ∞
|xt−yt|= 0 a.s.
Proof. Let γm := τm(x)∧ τm(y), m = 1,2, . . .. By continuity xγm = yγm hence both are out of Sm. So τm(x) =τm(y) andτ∞(x) =τ∞(y) as limits of
τm(x) and τm(y), respectively.
Lemma 4.1.3 (Convergence). Let x, xn, n = 1,2, . . . be continuous (F t)
-adapted processes, γn
m :=τm(xn)∧τm(x), m, n= 1,2, . . ., and let T ≤τ∞(x)
be a stopping time such that sup
0<t<T∧γn m |xn t −xt| →0as n → ∞in probability. Then sup 0<t<T |xn t −xt| →0 as n → ∞ in probability.
Proof. Denote ρ(x, y;τ) := sup
0<t<τ
|yt−xt|. Note the following: for all >0,
In its turn, the last summand enjoys the following estimate:
P{γmn < T ≤τ∞(x)} ≤P{τm−1(x)< T ≤τ∞(x)}+P{τm(xn)< T ≤τm−1(x)}
≤P{τm−1(x)< T ≤τ∞(x)}+P{ρ(x, xn;T ∧γmn)≥dist{Sm−1, Smc }.
Hence, lim sup
n→∞ P
{ρ(x, xn;T) ≥ } ≤
P{τm−1(x) < T ≤ τ∞(x)} for all
m = 1,2, . . .. Since τm−1(x) ↑ τ∞(x) a.s. as m → ∞, we conclude that
P{τm−1(x)< T ≤τ∞(x)} →0 and so lim sup
n→∞ P
{ρ(x, xn;T)≥}= 0.
Lemma 4.1.4 (Fundamental sequence). Let xn, n = 1,2, . . . be continuous
(Ft)-adapted processes, γmn,l := τm(xn) ∧ τm(xl), m, n = 1,2, . . ., and let
T ≤ lim infτ∞(xn) be a stopping time such that sup
0<t<T∧γmn,l |xn
t −xlt| → 0
as l, n→ ∞ in probability.
Then there exists a continuous (Ft)-adapted processx such that
sup
0<t<T∧τ∞(x)
|xn
t −xt| →0 as n→ ∞ in probability.
Proof. As in the preceding lemma, we denote ρ(x, y;τ) := sup
0<t<τ
|yt −xt|.
Note that lim
m lim infn τm(x
n)≤lim infτ
∞(xn), so
lim
m lim supn P{τm−1(x)< T}= 0.
Then ρ(xn, xl;T) → 0 in probability as l, n → ∞, by the argument similar
to one of the previous lemma.
Now assume that, for allt≥0, and x∈RM, we are given the coefficients
of a multi-dimensional version of (4.1.1): a random M × N-dimensional matrix σ(t, x) as defined in (3.2.8), and a random M-dimensional vector µ(t, x),
µ(t, x) = (µ1(x, t), µ2(x, t),· · · , µM(x, t)).
Hypothesis 4.1.5. Assume that σ andµare continuous in xforP×dt-a.a.
ω, t, measurable and (Ft)-adapted, and that there exists a nest S =Sm such
that µ(z, t) and σ(z, t) are defined for all z ∈ S∞ and t ≥ 0. Furthermore,
the following functions belong to M1
w[0, t] for all t >0 and m ∈N,
s7→ sup z∈Sm |µ(z, s)|, s7→ sup z∈Sm |σ(z, s)|2 where |µ|=pPkµ2 k, and kσk= q P ijσ 2 ij.
Our main assumption on µandσ is the following: combining local quasi-monotonicity of µ and local Lipschitz continuity of σ, we assume that func-tions s7→Mm(s) belong to Mw1[0, t] for all t >0 and m∈N,
Mm(s) := sup x,y∈Sm, x6=y
2hx−y, µ(x, s)−µ(y, s)i+|σ(x, s)−σ(y, s)|2
|x−y|2 ∨0.
(4.1.3)
Let Km(t) denote their maximum, i.e.
Km(s) := max{sup z∈Sm |µ(z, s)|, sup z∈Sm |σ(z, s)|2, M m(s)}, s >0, m= 1,2,3, . . .
Then Km ∈Mw1[0, t] for all t >0 and m ∈N.
Definition 4.1.6. (Solution to SDE)
Let xt be an adpated, a.s. continuous process. Let τm be an increasing
sequence of stopping times such that P{xt ∈ Sm,∀t ∈ [0, τm]} = 1, and
let τ∞ = limτm. We call xt a solution to (a multi-dimensional version of )
(4.1.1) with the given initial data x0, if the following identity holds:
xt∧τ∞ =x0+ Z t∧τ∞ 0 µ(xs, s)ds+ Z t∧τ∞ 0 σ(xs, s)dws.
The integrals in the latter are understood in the improper sense. The equiv-alent definition is that for all m= 1,2,3, . . .,
xt∧τm =x0+ Z t∧τm 0 µ(xs, s)ds+ Z t∧τm 0 σ(xs, s)dws.
Theorem 4.1.7 (Uniqueness and Existence of Solutions to SDE’s). Let Hy-pothesis (4.1.5) hold and let x0 ∈ L2 be an F0-measurable, M-dimensional
vector. Then Ito’s equation
dxt=µ(t, xt)dt+σ(t, xt)dwt, t≥0 (4.1.4)
with inital condition x0 has a solution x= (xt)∈Cw[0, τ∞] which is unique.
Theorem 4.1.8 (Continuity w.r.t. initial data). Let Hypothesis (4.1.5)hold,
T be a bounded stopping time, and let
dx=µ(x, s)ds+σ(x, s)ds
and
with E|x0−xn0| 2 →0 as n → ∞. Then sup 0<t<τ∞∧T |xt−xnt| →0 as n→ ∞, in probability.
Theorem 4.1.9 (Solution global in time). Let Hypothesis (4.1.5) hold and let V >0 be such that Sm :={x: V(x) ≤Rm} (our increasing set) and Rm
is an increasing sequence.
Assume that V ∈C2(S) and that LV ≤ λV for some λ ∈ R, where L is the Kolmogorov operator as defined by (3.3.3).
Then
τ∞ =∞ a.s.
The proof of Theorem (4.1.7) is split into several propositions below. The construction of the solution is by the Euler Approximation Scheme.
Definition 4.1.10 (Euler Approximation Scheme). Given the initial value
x0, the Euler Approximation Scheme is a sequence x (n)
t∧τm of processes which
converge as n → ∞, to a solution xt∧τm of (a multi-dimensional version of ) (4.1.1) satisfying the given initial condition, within the set Sm. As m ↑ ∞,
processes xt∧τm and x
(n)
t∧τm converge to a solution xt to (a multi-dimensional
version of ) (4.1.1)within the set S and its Euler approximation x(tn), respec-tively, both stopped when leaving S.
The n-th Euler approximation x(tn∧)τm is constructed as follows. For every
n = 1,2,3, . . ., define a grid tk(n) =k/n, k = 0,1,2,3, . . . as pictured below
0 1/n 2/n 3/n 4/n 5/n ... t(0n) t(1n) t(2n) t(3n) t(4n) t(5n) ...
Thex(tn∧)τm is constructed in an inductive way on intervals from(tk(n−)1, t(kn)], with induction on k= 1,2,3, . . ..
For the base of induction we take x(0n) =x0. Then the step of induction
is x(tn) =x(n) t(n)k−1 + Z t∧τm t(n)k−1 µ(x(n) t(n)k−1 , s)ds+ Z t∧τm t(n)k−1 σ(x(n) t(n)k−1 , s)dws (4.1.5)
for t ≤t(kn)∧τm.
We start with the following lemma.
Lemma 4.1.11. Let Hypothesis (4.1.5)hold. For the n-th Euler approxima-tion denote xˆ(tn) := x(n)
t(n)k−1 for t ∈ [t
(n)
k−1, t (n)
k ), k = 1,2,3, . . .. Then, for any
bounded stopping time T and m= 1,2,3, . . .
(ˆx(tn)−xt(n))→0 as n→ ∞ in L∞w[0, T ∧τm(x(n))], i.e. sup 0<t<T∧τm(x(n)) |xt(n)−xˆt(n)| →0 as n → ∞, in probability. Proof. Denote Ik := [t (n) k−1, t (n) k ]. For t >0, t∈Ik with k =k(t) := [tn]/n+ 1
and a(t) := [tn]/n being in the left end of Ik(t). Let us simply compare the
difference x(tn)−xˆ(tn): x(tn)−xˆ(tn)= Z t a(t) µ(ˆx(sn), s)ds+ Z t a(t) σ(ˆx(sn), s)dws. Therefore sup 0<t<T∧τm |x(tn)−xˆ(tn)| ≤ sup 0<t<T∧τm Z t a(t) |µ(ˆx(sn), s)|ds+ sup 0<t<T∧τm Z t a(t) σ(ˆx(sn), s)dws . (4.1.6) Now we estimate the summands in the r.h.s. of (4.1.6). We show that, as n→ ∞ sup 0<t<T∧τm Z t a(t) |µ(ˆx(sn), s)|ds →0 in L1 and sup τ∈[0T∧τm] E Z τ a(τ) σ(ˆx(sn), s)dws →0.
Then they both tend to 0 in probability: the first summand in (4.1.6) since it vanishes in L1, the second one due to Corollary 2.1.18.
Note that ˆx(sn)∈Sm since s≤τm(x(n)). Hence, as n → ∞,
sup t t Z a(t) |µ(ˆx(sn), s)|ds≤sup t Z Ik(t) sup z∈Sm |µ(z, s)|ds≤sup t Z Ik(t) Km(s)ds→0 in L1 since Km ∈Mw1[0, T] and|Ik(t)|= 1/n→0 as n → ∞.
Now consider the second summand on the r.h.s. of (4.1.6). Let τ be a stopping time satisfying 0 < τ ≤ τm ∧ T. Note that ˆx
(n)
s ≤τm(x(n)). We see the following: E Z τ a(τ) σ(ˆx(sn), s)dws 2 =E Z τ a(τ) |σ(ˆx(sn), s)|2ds ≤E Z Ik(τ) sup |z|≤R |σ(z, s)|2ds≤ E Z Ik(τ) Km(s)ds.
The latter vanishes as n→ ∞ since Km ∈Mw1[0, T] and |Ik(t)|= 1/n→0 as
n → ∞.
Lemma 4.1.12. (The Main Technical Lemma)
Let Hypothesis (4.1.5) hold and let (Ft)-adapted processes xt, xˆt, yt, yˆt,
satisfy the SDE for 0< t < τ∞(ˆx)∧τ∞(ˆy): (
dxt =µ(ˆxt, t)dt+σ(ˆxt, t)dwt,
dyt =µ(ˆyt, t)dt+σ(ˆyt, t)dwt.
For m∈N, let τm := min{τm(x), τm(ˆx), τm(y), τm(ˆy)}.
Then the following estimate holds for any bounded stopping times T and
τ, τ ∈[0, T ∧τm]: E|xτ −yτ|2e−2 Rτ 0 Km(s)ds ≤E|x 0−y0|2 + 4E Z τ 0 Km(s) |xt−xˆt|+|yt−yˆt|+|xˆt−xt|2+|yˆt−yt|2 dt. (4.1.7)
Proof. Let zt := (xt, yt)> ∈ R2M. Then zt satisfies the following SDE
(un-coupled system):
dzt= (µ(xt, t), µ(yt, t))>dt+ (σ(xt, t), σ(yt, t))>dwt.
Since |xt−yt|2 is a function ofzt, the multi-dimensional Ito formula implies
that, with a martingale Mt, the following identity holds:
d|xt−yt|2e−2 Rt 0Km(s)ds =dM t+ 2hxt−yt, µ(ˆxt, t)−µ(ˆyt, t)i +|σ(ˆxt, t)−σ(ˆyt, t)|2−2Km(t)|xt−yt|2 e−2R0tKm(s)dsdt. (4.1.8)
Note the following identity:
2hxt−yt, µ(ˆxt, t)−µ(ˆyt, t)i= 2hxˆt−yˆt, µ(ˆxt, t)−µ(ˆyt, t)i
For t ≤ τm, we have |µ(ˆxt, t)| +|µ(ˆyt, t)| ≤ 2 sup|z|∈Sm|µ(z, t)| ≤ 2Km(t). Hence the following estimate:
2hxt−yt, µ(ˆx (n) t , t)−µ(ˆy (n) t , t)i+|σ(ˆxt, t)−σ(ˆyt, t)|2 ≤2hxˆt−yˆt, µ(ˆx (n) t , t)−µ(ˆy (n) t , t)i+|σ(ˆxt, t)−σ(ˆyt, t)|2 + 4Km(t)(|xt−xˆt|+|yt−yˆt|).
Applyong the montonicity condition (4.1.3) to the first term in the r.h.s we have 2hxt−yt, µ(ˆx (n) t , t)−µ(ˆy (n) t , t)i+|σ(ˆxt, t)−σ(ˆyt, t)|2 ≤Km(t) |xˆt−yˆt|2+ 4(|xt−xˆt|+|yt−yˆt|) . (4.1.9) Note the estimate
2|xt−yt|2 = 2|xˆt−yˆt−(ˆxt−xt) + (ˆyt−yt)|2
≥ |xˆt−yˆt|2−4 |xˆt−xt|2+|yˆt−yt|2
. (4.1.10) Here we used the standard identity 2|u+v+w|2 ≥ |u|2−4 |v|2+|w|2
with u= (ˆxt−yˆt),v =−(ˆxt−xt) andw= (ˆyt−yt). Now we substitute (4.1.9) and
(4.1.10) into (4.1.8), integrate over (0, τ) with a stopping timeτ ∈[0, T∧τm],
and take the expectation:
E|xτ−yτ|2e−2 Rτ 0 Km(s)ds ≤E|x 0−y0|2 +E Z τ 0 Km(t) |xˆt−yˆt|2+ 4 (|xt−xˆt|+|yt−yˆt|)) −Km(t) |xˆt−yˆt|2−4 |xˆt−xt|2+|yˆt−yt)|2 e−2R0τKm(s)dsdt. Finally, note that e−2R0τKm(s)ds ≤1. The assertion follows.
Proposition 4.1.13 (Convergence of the Euler Scheme). Let Hypothesis (4.1.5) hold.
Then, for any bounded stopping time T, and m∈N,
sup
0<t<T∧τm(x(n)t )∧τm(x(l)t )
|xt(n)−xt(l)| →0 as l, n→ ∞ in probability. (4.1.11)
Proof. Fix m ∈ N, let τm := τm(x(n))∧τm(x(l)). Let ˆx
(n)
t be as in Lemma
(4.1.11) and let ˆx(tl) be defined accordingly. We apply Lemma (4.1.12) with xt :=x (n) t , yt:=x (l) t xˆt:= ˆx (n) t and ˆyt= ˆx (l) t :
E|x(τn)−x (l) τ | 2e−2Rτ 0 Km(s)ds ≤4E Z τ 0 |x(tn)−xˆ(tn)|+|x(tl)−xˆ(tl)| +|x(tn)−xˆt(n)|2 +|x(l) t −xˆ (l) t |2 Km(t)dt (4.1.12)
for any stopping time τ ∈[0, T ∧τm].
SinceSm is bounded, |x (n) t −xˆ (n) t |and |x (l) t −xˆ (l)
t |are bounded uniformly
in n, l and t ∈ [0, τm]. Moreover, these quantities vanish in L∞w[0, T ∧τm],
by Lemma 4.1.11. Since Km ∈Mw1[0, T ∧τm], we conclude that the l.h.s. of
(4.1.12) vanishes as l, n→ ∞ uniformly inτ. Hence, by Corollary 2.1.18,
sup
0<t<T∧τm
|xt(n)−xt(l)|2e−2Rt
0Km(s)ds →0 in probability asl, n→ ∞. Since Km ∈Mw1[0, T ∧τm], we have inf0<t<T∧τme
−2Rt
0Km(s)ds >0 a.s. hence we can drop the exponent in the last statement:
sup
0<t<T∧τm
|x(tn)−x(tl)| →0 in probability as l, n→ ∞.
Proposition 4.1.14 (Existence of a Solution). The Euler approximation converges to a solution (of a multi-dimensional version) of (4.1.1) with the initial data x0.
Proof. Let τ∞ := lim inf
n τ∞(x
(n)
t ). By Proposition 4.1.13 and Lemma 4.1.4,
x(tn) →xt uniformly for t∈(0, T ∧τ∞) in probability asn → ∞.
Hence, due to Lemma 4.1.11 we can find a sub-sequence ofx(nk)
t such that ˆ x(nk) t →xt ask → ∞ for all t∈ 0, T ∧τ∞ a.s.
Since z → µ(z, s), z → σ(z, s) are continuous for P ×ds a.a. (ω, s), we conclude that, for ds-a.a. s∈ 0, τ∞
a.s. µ(ˆx(nk) s , s)→µ(xs, s) andσ(ˆxs(nk), s)→σ(xs, s), asn → ∞. Note that E Z t 0 sup z∈Sm |µ(z, s)|+ sup z∈Sm |σ(z, s)|2ds <∞, t >0, m ∈N.
Hence E Z t∧τm 0 µ(ˆx(snk), s)−µ(xs, s) ds→0 and E Z t∧τm 0 kσ(ˆx(nk) s , s)−σ(xs, s)k2ds→0
by the Lebesgue dominated convergence theorem.
So, for allm= 1,2,3, . . ., we can pass to the limit in the following equa-tion, as n→ ∞. x(nk) t∧τm =x0+ Z t∧τm 0 µ(ˆx(nk) s , s)ds+ Z t∧τm 0 σ(ˆx(nk) s , s)dws ↓ xt∧τm =x0+ Z t∧τm 0 µ(xs, s)ds+ Z t∧τm 0 σ(xs, s)dws. (4.1.13)
Proposition 4.1.15 (Uniqueness). Let x and y be two continuous, measur-able Ft-adapted processes such that x0 =y0 a.s. and
• dx=µ(x, s)ds+σ(x, s)ds • dy =µ(y, s)ds+σ(y, s)ds Then τm(x) =τm(y), m = 1,2, . . . ,∞ and sup 0<t<τ∞ |xt−yt|= 0 a.s .
Proof. By Lemma 4.1.2 it suffices to show that, with γm :=τm(x)∧τm(y),
P{ sup
0<t<γm
|xt−yt|= 0}= 1,∀m= 1,2, . . . (4.1.14)
By Lemma (4.1.12) we have the following: for any bounded stopping stopping time T, and any τ ∈[0, T ∧γm],
E|xτ−yτ|2e−
Rτ
0 Km(s)ds = 0. (4.1.15) By Lemma (2.1.17), for all >0.
P sup 0<t<γm∧T |xt−yt|2e− RT 0 Km(s)ds ≥ = 0.
Note that e−R0TKm(s)ds >0 a.s. since Km ∈M1
w[0, T]. Hence P sup 0<t<γm∧T |xt−yt|2 >0 = 0. Finally, as T ↑ ∞, we obtain P sup0<t<γm|xt−yt|2 >0 = 0.
Proof of Theorem (4.1.8). Let γmn :=τm(x)∧τm(xn). By Lemma 4.1.12, for
all stopping times τ ∈[0, γn m∧T], P sup 0<t<τ |xt−xnt| 2 e−Rτ 0 Km(s)ds | {z }
bounded below a.s. > ≤E|x0−x n 0|2 →0, ∀ >0.
Hence, by Corollary (2.1.18), sup0<t<γn
m∧T |xt−x
n
t| →0 in probability. Now
the assertion follows from Lemma 4.1.3.
Proof of Theorem 4.1.9. We prove that, for any boundedT,P{τ∞≤T}= 0.
Note the following:
{τ∞≤T}={τ∞∧T < T}={ sup 0<t<τ∞∧T V(xt) =∞}.= [ m { sup 0<t<τm∧T V(xt) = ∞}
Now differentiating the function e−λt
V(xt) we have
de−λsV(xt) =
e−λtLV−λe−λtV dt+e−λtDV(xt)σ(xt, t)dwt.
Hence, for any stopping time τ ∈[0, τm∧T],
Ee−λτV(xτ)≤EV(x0). By Lemma 2.1.17 we conclude P sup 0<t<τm∧T e−λtV(xt)≥R ≤ EV(x0) R . As R ↑ ∞, we obtain P sup 0<t<τm∧T e−λtV(xt) =∞ = 0. Hence P sup 0<t<τm∧T V(xt) = ∞ = 0, since T is bounded.