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DATE DUE DATE DUE DATE DUE SIOS lthroj/AccaPrelelRCIDateDueJndd ASYMPTOTIC PROPERTIES OF SPOT RATE MODELS AND THEIR CONTROL By Chandni Bhan A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Statistics 2010 ABSTRACT ASYMPTOTIC PROPERTIES OF SPOT RATE MODELS AND THEIR CONTROL By Chandni Bhan The thesis presents the Lyapunov method for studying different forms of stability, such as exponential stability, stability in probability and ultimate boundedness of solutions of stochastic differential equations driven by Lévy noise. This method allows one to study the stability of such stochastic systems without the requirement of explicitly solving the equation. In particular, this technique is applied to various spot rate models to determine their recurrence set. The thesis also presents an application of control theory to explain how suitably chosen cost functions, together with the technique of the Lyapunov function, can make the stochastic systems recurrent. DEDICATION To my family iii ACKNOWLEDGMENT Foremost, I would like to convey my gratitude to my advisor Dr. Mandrekar for his su- pervision, advice, and guidance every step of the way, within and outside the realm of this work. Working with him has been a truly memorable experience, not only is he a great mathematician but also a superb raconteur, full of interesting anecdotes. I gratefully acknowledge Dr. Koul for his crucial advice on how mathematical literature should be presented. Despite his busy schedule as the Chair of the Department, he thor- oughly examined the thesis each time and always patiently helped me improve it further. Words cannot express how much I have learnt and improved because of him. I can only convey my deepest gratitude to him for his toughness as a referee and yet compassionate support as a teacher and a friend. I am also very grateful to my thesis committee members Dr. Levental and Dr. Schroder for their valuable inputs throughout my research and their wonderful support. I would like to specially thank my dear friends Sumit, Sagata and Sue. For without their support, crazy jokes and just their wonderful nature, my life would have had a lot fewer smiles. I cannot even begin to thank my family for the uncountable ways in which they have supported me. Especially my dearest sibling without whose care and encouragement, this would have not been possible. I’d like to make a special mention for Skype for making it possible for me to seek my parent’s invaluable advice for free any time of the day. Last but certainly not the least, my deepest gratitude goes to my darling fiancé for being my rock, my inspiration and my energy. Without his indefatigable faith in me and dedicated efforts in guiding me forward, I would have never reached the summit. iv TABLE OF CONTENTS List of Tables .................................... vi List of Figures ................................... vii Introduction...........................1 1.1 Interest rate models .............................. 1 1.1.1 History of interest rate models .................... 1 1.1.2 Empirical facts for Bond Prices ................... 4 1.2 Asymptotics and applications ......................... 5 1.3 Overview thesis ................................ 5 Preliminaries.......................... 7 2.1 Basic definitions ................................ 7 2.2 The jumps of Lévy processes - Poisson random measures .......... 9 2.3 Lévy stochastic integrals ........................... 11 2.4 SDEs driven by Le’vy processes ........................ 13 Recurrence properties of term structure models driven by Brownian motion 17 3. 1 Introduction .................................. 17 3.2 Ultimate boundedness ............................. 20 3.3 Stability .................................... 24 Stability of stochastic interest rate models driven by general Lévy noise . . 28 4. 1 Introduction .................................. 28 4.2 Exponential p-stability ............................ 30 4.3 Stability in probability ............................ 36 4.4 Exponentially ultimately bounded ...................... 39 4.5 Recurrence .................................. 46 4.6 Invariant measure ............................... 48 4.7 Examples ................................... 50 Stochastic control and dynamic programming . . . . . . . . . . . . 57 5. 1 Introduction .................................. 57 5.2 HJB Equation ................................. 59 5.3 Exponential 2-stabilization .......................... 60 5.4 Exponential 2-ultimate boundedness and control ............... 63 Futurework........................... 68 Bibliography vi LIST OF TABLES 1.1 The statistics for the daily logretums and the absolute changes of the 3 month Libor from December 6, 1984 till May 27, 2010 ............ 5 vii 1.1 3.1 LIST OF FIGURES The figure contains the QQplot for the logretums (left) as well as the changes (right) versus standard normal numbers. ................... 4 The recurrence region for the CIR model (left) and the Vasicek model (right) for a particular choice of the parameters. ............... 27 viii Introduction 1.1 Interest rate models 1.1.1 History of interest rate models The history of modeling interest rates starts with deterministic fixed interest rates in the shape of a yield curve, presenting a different interest rate value for each different tenor. Within this framework, it is possible to price bonds and other very simple financial instru- ments. However, it lacks all dynamic power and ability to forecast future values, thereby making the pricing of complex derivatives impossible. In order to be able to study the time» 1 dependent behaviour of interest rates, more complex models had to be presented that allow for consistency with the current market yield curve, as well as the stochastic or random nature of the various interest rates. In 1976, the idea to treat bond prices as random was introduced in [7] and [28]. Under the assumption that bond prices follow a geometric Brownian motion, it becomes possi— ble to apply the Black-Scholes option pricing formula to fixed income instruments as well. However, this approach is not completely consistent, as on one hand interest rates are de- terministic while discounting, and on the other hand bond prices (and hence interest rates) are treated stochastically. The first step to bring consistency into the stochastic interest rate models, was the con- cept of the short or spot rate model. This unobservable stochastic quantity serves as a stochastic factor driving the entire yield curve. In a first wave of models, the stochastic na- ture of such models was governed by diffusion processes. There exist various implementa- tions of diffusion-driven models such as the Vasicek model [39] or the Cox—Ingersoll—Ross (CIR) model [8]. The popularity of these models stems from their mathematical tractablil- ity. Soon after this, extensions of these simple models were introduced by either extending the fixed parameters to time-dependent parameters such as in the popular Hull-White model [17], or to include multiple factors. Opening the door to multiple factors brought in a whole range of new models. A popular approach is to include a second factor either in the form of stochastic volatility of the short rate or a factor representing the long term rate. However, since the short rate itself is not observable in the market, its value has to be approximated by the interest rate from the market with the shortest possible tenor. A popular choice is the 3-months interest rate, as it is commonly accepted that this should provide a good candidate to drive the rest of the yield curve. Clearly, one factor models only allow for a restricted dynamic behaviour of the yield curve, since the correlation between the short end and the long end of the yield curve is only obtained through the parameters 2 of this stochastic driving force. As a consequence, for such models the yield curve shape does not always correspond to the shape of the curve in the market, introducing an arbitrage between the model and the market. The starting point for the first models, such as Vasicek [39], was to focus completely on the non-arbitrage condition. In fact, such models are set up by first defining the dynamics for one or more of the factors and then using non-arbitrage relationships to price contingent claims. However, the other stream of methods called equilibrium models, start from a descrip- tion of the underlying economy and focus on the fundamental factors that make up the market. Those insights are then used to set up a model for the short rate. The advantage of fundamental models is that both the term structure as well as its dynamics are endogenously defined (see e.g. [25]). In the 1990’s, a big break-through in interest rate modeling came about through the Heath—Jarrow—Morton (HIM) [l6] framework. In this approach, the entire yield curve was modeled by means of the forward rate. In particular, it can be shown that all short-rate models can be obtained within this framework. For about 10 years, implementation of this model was the standard in the industry for modeling the interest rate curves. However, these models were based on multi-factor Brownian components, basically assuming that the interest rates move continuously. To a large extent, the development of interest rate models is driven by the need to price and hedge interest rate derivatives. Therefore, the new wave of models were extended to include stochastic volatility, for example Stochastic-Alpha-Beta—Rho (SABR) model, as a means to incorporate the market price of the risk for the uncertainty encountered while hedging options. Ever since its introduction in 2002, the SABR [15] model has been con- sidered the market reference for pricing derivatives. Throughout the thesis, one shall en- counter many examples, one of which will be the deterministic SABR model known as the Constant Elasticity of Variance (CEV) model. 3 1.1.2 Empirical facts for Bond Prices In a series of papers ([9],[4],[11],[35],[10]), it was shown that the financial returns from eq- uity prices, indices, FX-rates and bond prices, modeled using Lévy processes provide for a much better fit compared to models based on Brownian motion. The rest of this section will illustrate the lack of normality in bond price data and interest rates by studying a historical time series for the 3 months Libor US rate, ranging from December 6, 1984 till May 27, 2010. In Figure 1.1, a QQ plot is shown of both the logretums log(r (t + At) /r(t)) and the absolute changes r(t + At) — r(t). In this analysis the distance At is taken to be one work- ing day. One can observe from these figures that the data does not support the assumption of normality. Any model based on Brownian motion will therefore poorly characterise its dynamics. -3 0.15 Returns vs Standard Normal 6x 10 Changes vs Standard Normal .9 _ l 4 ‘” l “as = .. E005; 52 .-' Dad) 1 '5 e “a o g s I i9 l 130 “-0.05 '5 g l .4 §_2 0-0.1 i. O .- | i -0.15;g '4 i O-4-3-2-101234-64-3-2-101234 Standard Normal Quantiles Standard Normal Quantiles Figure 1.1: The figure contains the QQplot for the logretums (left) as well as the changes (right) versus standard normal numbers. The analysis of the basic statistics in Table 1.1 of this time series shows how big the deviation from the assumption of normality is. For observations coming from a normal 4 distribution, the skew would be zero and kurtosis would equal 3. mean standard deviation skew kurtosis returns -0.0004 0.01 l 1 -1.2630 30.0985 changes -0.0000 0.0006 -0.6186 18.4349 Table 1.1: The statistics for the daily logretums and the absolute changes of the 3 month Libor from December 6, 1984 till May 27, 2010. 1.2 Asymptotics and applications In this thesis, the topic of interest is the asymptotic stability of stochastic interest rate models. In mathematics, there are several notions of stability. However, this study focusses on the concept of exponential 2—stability and ultimate 2—boundedness. Let the short rate process be given as the solution of a stochastic differential equation (SDE). Then, if the process is exponentially 2—stable, its second moment decays exponentially fast to zero, as time goes to infinity, for any initial value. For an ultimately 2—bounded process the second moment of the solution process is asymptotically bounded. In finance, this provides a way of comparing two different interest rate models. Typ- ically, it is seen that interest rates tend to revert to a long term average r*. The view on this long term average may change from time to time and is influenced by the policy of the central bank, long term inflation, economic growth, globalisation, etc. However, it is commonly accepted that interest rates will not blow up or go too negative. Therefore, it is natural to explore the concept of stability as described earlier, for different interest rate models used in the market. 1.3 Overview thesis In this thesis, the Lyapunov technique is presented for determining whether a solution to an SDE is exponentially stable or ultimately bounded. This is an extremely useful method as 5 it provides an way of determining a priori the stability of a model by means of a Lyapunov function, without having to explicitly solve the SDE. It also allows one to derive conditions under which an invariant measure exists. The questions that are addressed in this thesis include how to relate such a Lyapunov function to the different concepts of stability. One of the main challenges is to find the right Lyapunov function to establish a particular result. In some cases, the existence of a Lyapunov function is not only a sufficient, but even a necessary condition for testing for stability. All definitions and concepts required throughout the thesis are summarized in Chapter 2. In Chapter 3, different notions of stability are defined, and their application to inter- est rate models driven by Brownian motion is discussed. Chapter 4 extends the different stability results for interest rate models driven by Lévy noise using Lyapunov functions. stochastic control and dynamic programming for the Lévy driven SDEs is studied in Chap- ter 5. Chapter 6 presents extensions for possible future work. £1... “.3 Preliminaries 2.1 Basic definitions In this chapter, the general definition of a Le’vy process and some of its properties are dis- cussed. In addition, results that are necessary building blocks for the subsequent chapters are also presented here. Notation 2.1. The following notations shall be followed throughout the thesis: 1. The collection of all Borel sets on R is denoted by B (R) . 2. n0=n\{0}. 3. R+={$€R,x20}. 4. The indicator function of a set A is denoted by X A (-) . For intervals, X (a _<_ a: S b) = X[a,b] (1") ' 5. Let C 2(IR) denote be the set of twice continuously differentiable functions on R. 6. Let CEUR) denote the space of bounded, twice continuously differentiable functions on IR. 7. Let C8 (1R) denote the set of all compactly-supported functions in C2 (R) . 8. N :2 {0,1,2,3...}. \O . N0 := {1,2,3...} . Definition 2.1. ([2], pg 39)(Lévy process) Let (Q, .7, {ft}t_>_.0 , P) be a filtered probabil— ity space. An ft-adapted stochastic process {X t }t>0 C IR is a Le’vy process if it satisfies (LI) X0 = 0 ((1.8), (L2) X has independent and stationary increments, (L3) X is stochastically continuous (continuous in probability): lirn P(|Xt ——X3| > a) :0, Va > 0,3 _>_ 0. t—>s Remark 2.1. Every Lévy process has a cadlag modification that is itself a Lévy process [2]. Set Xt— = li1nt1Xtand AXt = Xt — Xt—' 5 An analytic characterization of a Le’vy process is given by the celebrated Lévy-Khintchine formula described below. Theorem 2.1. ([36], Thm l.2.14)(Lévy-Khintchine) Let {Xt, t 2 0} be a Le’vy process. Then, its characteristic fimction is given by E (eiuxt) = 6mm), t 2 0, u E R, where, 7] (u) = ibu — 502112 Jilly _ _ .j ' . , . + /R0 [C 1 my X{0<|y|<1} (31)] l1(dy)- (2.1) In this representation b E R, o > 0 are constants and u is a Levy measure i. e. l/ is a o-finite Borel measure defined on 8 (R0), that is finite outside the neighborhood of zero and satis- fies fRO (yz /\ 1) l/ (dy) < 00. Together; (b, 02, u) are Imown as the Lévy characteristic tn'plet and uniquely define the process X t- 2.2 The jumps of Lévy processes - Poisson random mea- sures The following section introduces concepts related to the jumps of a Le’vy process. Let (52, f, {ft}t>0 , P) be a filtered probability space, (S, B (S )) be any measurable space, and A denote Lebesgue measure on UN”. Definition 2.2. A Poisson random measure on B (R+ x S) relative to {ftItZO is an integer valued random variable N such that the following hold. (2') There exists a o- finite measure u on (S, B (S )) such that, E(N(A))=(/\o¢u)(A), AEB(IR+ x5). (ii) Forevery s E R+ andA E B (R+ x S) such that/1 C (3, 00) x SandE(N (A)) < 00, N (A) is independent of f5. For a Poisson random measure N, the measure [7’ = A0012 is called the intensity measure and is also referred to as the compensator of N ([19], 11.1.21). From Theorem 11.4.8 of [19], one obtains the following result. Theorem 2.2. Let N be a Poisson random measure on B (R+ x S ) with the compen- sator B. Then,forany disjoint sets {Aj E B (1R+ x S) :3 (Aj) < oo;1§j$ k} , the random variables {N (Aj) , 1 S j S k} are independent. For A E B (R+ x S) with B(A) < 00, define q(A) = N (A) -— B(A). Then, q is called the compensated Poisson random measure associated with N and it satisfies Let A 6 [3(8) with u(A) < oo. Define th = q([0,t] x A) ,t 2 0. Then, th is an ft-martingale. Remark 2.2. Let Xt be a Levy process with the triplet (0,0, u) as described in Theorem 2.1. Let C denote the ring of all Borel sets A E 8 (R0) for which V (A) < 00. Fort 2 0, A E C, let N([0,t],A)=#{OSs§t;AX36A}: Z XA(AXS). OSsSt Then, N is a Poisson random measure on B (R+ x R0) with the compensator AV and q is the associated compensated Poisson random measure (see [19], chapter 11). One can represent the Lévy process as follows. Theorem 2.3. ([36], Thm 8.1)(Lévy-It6 decomposition) Let Xt be a Le’vy process. Then, there exist b 6 IR, 0 > 0, a Brownian motion B and an independent Poisson random measure N on B (IR+ x R0) , with associated compensated Poisson measure q, such that, 10 for each t 2 O, Xt = bt + UBt +/ 0<|:c|<1 xq (t, dsc) +/ LEN (t, dx). ll‘lZl Remark 2.3. From the above theorem, it can be shown that every Lévy process is a semi- martingale ([19], 2.6). Therefore, in order to define a stochastic integral with respect to a Lévy process, one can use the well developed theory of semi-martingales (see [34]). However, for completeness we give the needed theory for the case of Lévy process. 2.3 Lévy stochastic integrals Let (l2, .7, {It}t>0 , P) be a filtered probability space and {Bt}t>0 be a Brownian mo- tion. Forp 2 land 0 S T < 00, define MT 2 {f (t,w) : f : [0,T] x O —> R s.t. f (t, -) is Jig-measurable Vt}, T / |f(t,w)|pdt] < 00}. 0 Then, [32] defines the stochastic integral Mt == f6 f (s,w) dB (3) , for any f 6 Mg“ and Mgw={f:f€MTandE t E [0, T]. Further {M1}, t 2 0} is a continuous ft-martingale and it satisfies Ito’s isometry, t 2 t _ 23 8. E(/Of(s,w)dB(s)) —E/Of (,w)d t t Xt—X0=/0 a(s)ds+/0 o(s)dB(s), (2.2) namely, Let, where a E M% and o 6 M%. Then, for any f E 62 (R), f(Xt) satisfies the Ito’s formula: 02 (s) II t I . 2 f (XS)) ds+/0f(Xs)0(5)dB(5)- f(Xt)—f(X0)=/ t [X 0 (f( s)a(3)+ 11 For the corresponding results in the non Gaussian Lévy case, let (0, .F, {ft}t>0 , P) be a filtered probability space and N be a Poisson random measure on B (1R+ x R0) with its compensator B = /\1/. Forp 2 1and0 S T < 00, define: MT =:{f [0, T]XR0xQ—>Rs..t f( (t, u, -) is If -measurable foreachtandu}, MT:{feMTandE(/OT /0|( f(t,u,)0 be a stochastic process given by t =Y0(s)-+-/a )ds+/0/b( s, u)q ((1.9, du) +/Ot/G( s, u) N(,ds du), (2.3) 0 0<|u|<1 lulzl where a 6 MT and E (jg |a(s,w)|ds) < 00, b and G E 191%». Then, {Yt,0 _<_ t g T} is well defined (see [18]) and is called a jump-Lévy type stochastic integral. The next theorem presents the Ito formula for a jump-Lévy type stochastic integral. 12 Theorem 2.4. ([18] Theorem 4.1, page 66): Let Yt be the Lévy type integral given by (2.3) . Then, for any f E C2(IR) the following holds with probability 1 : t f (Yt) — f (Y0) = / f’ (Y5) ads 0 f {f (Ys— + Gm» — f (Y,_)} N(ds,du) ['uIZI / { f (Y8- + b(s, 1.)) — f (Y,_) } q(ds, du) b(s, u)f’ (Y3_) u(du)ds. (2.4) Henceforth, for any stochastic process defined in the form (2.2) or (2.3), we shall as- sume that the integrands belong to the appropriate function space (as discussed above) such that their integral is defined. 2.4 SDEs driven by Lévy processes Let {Yt’ t 2 0} be a jump - Lévy process as in (2.3) . In order to study this process, one can, as a first step, study the modified SDE dYt = a(Yt)dt+ / | b(Yt,u)q(dt,du), Y0 = y. (2.5) [u <1 Then, (2.3) can be recovered from (2.5) by using what is known as the technique of inter- lacing ([2], pgl 1 1). Under the appropriate conditions, the results obtained for the reduced 13 system (2.5) can be extended to the general SDE (2.3) . Consider the following two conditions: a Growth condition: there exists a constant C > 0, such that Vy E R, |a(y)| + | (yin)| V( 'u) S 1+|y| - (A1) [u|<1 e Lipschitz condition: there exists a constant L > 0 such that, Vy1,y2 E R, — 2 b‘ —b 2 d < L — 2 (A2 100/1) a(y2)| + l |<1| (ywl) (312.201 1/( u) _ |y1 yzl - . ) ’11. Theorem 2.5. (([2], pg 304), [26]) If the coefficient functions of (2.5) satisfy (A1) and (A2) , then there exists a unique cadlag solution process {Yb t 2 0} of (2.5). Furthermore, under the following additional condition on C, y ——> C (y, u) is continuous, V In] 2 1, (A3) a unique solution to (2.3) exists and is cadlag. Remark 2.4. The above result is proved even under local Lipschitz condition in [26]. For a time-homogeneous Markov process {Ir/t, t 2 0} define, P(t,y,A) = P9 (Yt e A), A e BUR), t2 0, y 6 R. (2.6) Then, P(t, y, A) denotes the transition probability of Y and can be interpreted as the con- ditional probability that Yt is in A, given Y0 = y. In [2], it is shown that the transition probabilities of a Markov process define a transition semigroup, which in turn uniquely defines the infinitesimal generator. 14 Definition 2.3. A family {Tt, t 2 0} of bounded linear operators on a Banach space B, is called a strongly continuous semigroup, if for all t, s E [0, 00), (a) TtTS = Tt+s’ (b) [33th = f. Vf e B. Definition 2.4. The infinitesimal generator L of a semigroup {Tb t 2 0} is defined by the formula, T _ £f = limJLi. hi0 h Its domain ’Dfi consist of all f E B, for which the above limit exists. Let {Yb t _>_ 0} be the solution process of (2.3) under (A1) — (A3) . For f E Cb(IR), SCI th(l/) = / mam/ex) = E?! (f (Ya), y e R, where E y denotes the expectation given the initial condition Y0 = y. Then, it is shown in [26] that {Tt;t 2 0} constitutes a continuous semigroup and its infinitesimal generator is given in the following theorem. Theorem 2.6. (Infinitesimal Generator): Let Yt be the unique solution of the stochastic differential equation (2.3). Then, for any fixnction f E C2(IR) and y 6 IR, its infinitesimal generator [I is given by £f(y)=f'(y)a(y)+ / {f (y+G(y,u))— f(y)}z/(du) (2.7) Inlzl + / [f (y + My, 20) — f(y) - bun) f’0 be 1-dimensional Brownian motion. The solu- l7 tion of dYt = a(Yt)dt + bO/tldBtt Y0 = y. (3.1) is said to be exponentially p-stable, p > 0, if there exist some positive constants A and a such that‘s/y 6 IR, t 2 O, Eylmp s Alylpe“at- Definition 3.2. ([12]) Let Yty denote the solution to (3.1) with initial condition Y0 = y. Then, the solution of (3.1) is called exponentially stable in quadratic mean (ESQM) if there exist positive constants A and a such that, Vy, y* 6 IR, t Z 0, 11* 2 2 t * _. Emil—Y, | _<_Aly—yl e ‘1. Definition 3.3. ([29]) A stochastic process Yt is said to be exponentially p-ultimately bounded, p > 0, if there exist positive constants C, [3, M such that,\7’ y 6 IR and t Z 0, Ey [Ytlp < Ce—fltlylp + M. Definition 3.4. ([29]) A stochastic process Yt is said to be p-ultimately bounded, p > 0, if there exists a constant M such that,\7’ y 6 IR, limsup Ey [Ytlp S M. t—>oo The earliest development in the short rate models were the one-factor models, given in its generic form as dl‘t = a(t, T't)dt + 0(t, 71)de (3.2) where (Bt )tZO is 1-dimensiona1 Brownian motion. By suitably choosing the drift function a and volatility function b, one can obtain a variety of models from (3.2), for example 18 Vasicek [39] and CIR [8]. All the models discussed in this chapter are given in [37]. The main objective of this chapter is to analyze asymptotic stability of interest rate models. This is important from a portfolio stress testing point of view. Most fixed income derivatives depend on future interest rates, which in turn are estimated using the model for the short rate. Therefore, if a short rate model exhibits exponential stability, then any stress test will fail to yield different results with respect to its long term behavior (future interest rate), because it only tampers with current input values. Thereby, the very purpose of the stress test gets defeated. The concept of ESQM allows one to study the proximity behavior of solutions with different initial conditions. This is particularly useful in the context of interest rate models, where the short rate itself is not observable and hence its initial value needs to be estimated from the market. If the short rate models exhibit ESQM, then small errors on this estimation will not lead to solutions that are fundamentally different asymptotically. From a financial point of view however, it is unrealistic to demand models to asymp- totically converge to a degenerate point as in Definition 3.1 of exponential stability. It is therefore more natural to start from the concept of recurrence of interest rates (see Defini- tion 3.6). This concept is closely related to the other notion of stability, namely ultimate boundedness developed in [29]. This concept of asymptotic behavior allows one to assess whether there exists a finite interval (recurrence set), in which the process is positively recurrent. The subject of stability of financial models has recently gained a lot of attention (see [12], [13], [38]). However, in these papers, the focus has been on explicitly solving indi- vidual models to examine their asymptotic behavior. These results can be obtained more elegantly (see [5]) by applying the work of Khasminskii [20] and Miyahara [29], [30]. The Lyapunov method for studying stability of stochastic systems, driven by Brownian motion, was mainly developed by Khasminskii [20]. The extension to ultimate bound- edness of stochastic differential equations was first analyzed by Miyahara [29]. In this 19 chapter, those techniques shall be exploited in order to study the asymptotic stability and asymptotic boundedness of solutions of different short rate models. In particular, the re- currence behavior of these solutions shall be examined and in some cases the recurrence region will be explicitly computed. This study is useful from an investment point of view, as one can choose to buy or sell bonds, depending on the level of interest rates with respect to the recurrence set. The proofs of the general theorems of [20] and [29] are omitted in this chapter. 3.2 Ultimate boundedness The following results are taken from [29] in order to make the thesis self contained for the reader. It is assumed that (3.1) has a unique (see Theorem 2.5) solution. Recall its infinitesimal generator £ given by (2.8). The following theorem gives a sufficient condition for the solution of (3.1) to be ultimately bounded. Theorem 3.1. ([29], Thm 3.1) Let Yt be the unique solution to (3.1) , and let p be a positive number. Suppose there exists a fitnction A E C2 (IR) satisfying, Va: E IR, (1) -a1+ C1 lxlp S MI), (2) [IA (:r) g —C3A (:r) + (13, where cl, C3 > 0 and a1, 0.3 E IR are constants. Then, the process Yt is p-ultimately bounded. Ifin addition to (1) and (2), V1: E IR, (3) A (:r) 3 a2 + c2 [1:|p,f0r some 02 > 0 and (1.2 E IR, then, the process Yt is exponentially p-ultimately bounded. The function A in the above theorem is called the Lyapunov function of the process {th t 2 0} . The biggest challenge to establish stability of a stochastic system is to be 20 able to find an appropriate choice of a Lyapunov function. There is no straightforward technique to do this in general. However, in certain special cases the following theorem explicitly gives the Lyapunov function. Theorem 3.2. ([29], Thm 4.1) Consider the system (3.1) . In addition to (A1) and (A2), assume the coefi‘icient functions a (-) and b (-) are in the class Cl? (IR). Under these as- sumptions and if Yt is exponentially p-ultimately bounded, for some 19 > 0, then there exists a function A E C2 (IR) satisfying conditions ( 1) — (3) of Theorem 3.]. Remark 3.1. If the coefficients functions of the system (3.1) are linear, i.e a (3:) = as: and b (9:) = bx, then they satisfy conditions of Theorem 3.2. Further, for this case, the Lyapunov function is given by A (:r) = 0:122, for some 0 > 0. In [29], the author studied the relationship between recurrence and ultimate bound- edness of the system (3.1). It was shown that if the solution of (3.1) is exponentially 2-ultimately bounded, then it is weakly positive recurrent. Definition 3.5. A time homogeneous Markov process Yt is said to be weakly recurrent, if there exists a constant K, such that, Py (IYtI S K, forsomet Z 0) = 1, Vy E IR. The set {2: E IR : [2:] s K } is called the recurrence region. Letft = o{Bs, s S t}. Definition 3.6. A time homogeneous Markov process Yt is said to be weakly positively recurrent, if there exists a constant K, such that Yt is weakly recurrent and the expectation of its first hitting time T w.rt. {7,5}, of the recurrence region {3: E IR : [:c] S K} is finite, i.e. Ey('r) < 00, for any y E IR. 21 Note that weak recurrence is a rather mild condition, since even though the process is recurrent, there is no guarantee that it shall return to the recurrence region within finite mean time. This can however be overcome in the case when the process is exponentially ultimately bounded. Theorem 3.3. ([29], Thm 5.2 ) If the process Yt defined by (3.1) is exponentially 2-ultimately bounded, then it is weakly positively recurrent with the recurrence region given by {IEIR:|1‘|oo = O and lim sup III—>00 =0,Vt_>_0. Then, the solution of (3.3) is exponentially 2-u1timately bounded. As a consequence of Example 3.1, many well-known l-dimensional interest rate mod- els can be analyzed. 22 Example 3.2. Let (x, [3, a > 0. The CIR model is given by (17,3 2 —-artdt + aBdl + ofldBU), (3.4) fort > 0. Substituting f(t,:c) = 0213 and b(t,:z:) 2 o a: in Example 3.1, the exponen- tial 2—u1timate boundedness of rt follows. Consequently by Theorem 3.3, it is weakly positively recurrent. In order to obtain the recurrence region, consider ,802 2 —(ii _ —2(.yt (c e )+ 2a . 0 E7 2: __ 1TH r20 + (re—Oil. + ,B(1 _ 6—06))2 (1 _ e_"t)2 S 1.5]7‘I28—at + K, where 4 2 1 0 Ba 2 K=—— — 2 is the constant determining ultimate boundedness of rt. Therefore, using Theorem 3.3 the recurrence region (see Figure 3.1) is given by {z : 0 < z < x/K} . Example 3.3. The geometric Brownian motion is given by the SDE: dYt = [,tI/tdt + O'YtdBt. (3.5) Since the SDE is linear, the Lyapunov function is Mat) 2 3:2. Then, by Theorem 3.2, for 2 u + 92— < 0, one gets that Yt is exponentially 2-ultimately bounded. Example 3.4. Let B > 0, a > 0, a > 0. The Vasicek model is given by: th = -/i'l‘tdt + (Jill. + (Id/3t. (3.6) 23 Taking f (t, 2:) = a and b(t, 3:) = a in Example 3.1, one gets the exponential 2-ultimate boundedness of rt and hence by Theorem 3.3, it is weakly positively recurrent. Consider the following, Ey (rt) 2 ye_'8t+% (1— 6—5t). Vy(rt)= 627—:— (1— €~2fit). Then, 2 Ey Iril2 = i'—(1 — 6‘2“) + (ye‘fl‘ + 9-(1— 6—2))2 213 2 l3 s 213/126—2m +12%)2 + (El—3)]- Therefore, the recurrence region (see Figure 3.1) is given by, zzlz] < \/2(%)2+g—; Example 3.5. The CEV model is defined by drt = (a — brt)dt + oerBt. (3.7) Using Example 3.1 with f (t, 3:) = a and b(t, 2:) = 0:57, one gets that the solution of (3.7 ) is exponentially 2-u1timately bounded for ”y < 1. Further, it is recurrent and its recurrence region is given by {z : [z] < %+1} . 3.3 Stability The following theorem provides a sufficient condition for exponential p-stability for the system (3.1), in terms of the Lyapunov function. These results are from [20] (see also 24 [21]) and are quoted here for the convenience of the reader. Theorem 3.4. ([20], Thm 5. 7.1 )(Su/ficient): The solution of the process (3.1) is expo- nentially p-stable, p > 0, if there exists a fimction A E C2 (IR) and positive constants obi = 1,2,3 such that Va: E IR, (1) c1 lep s A (x) 5 c2 lrvlp. (2) £A(a:) S —C3A (:13) Theorem 3.5. ([20], Thm 5. 7.2 )( Necessary): If the solution of the system (3. 1) is exponen- tially p-stable and the coefficient functions a () and b () are in class Cl? (IR) , then there exists a Lyapunov function A E C 2 (IR) that satisfies condition (1) and (2) of the above theorem. Remark 3.2. (see Lemma 2.1, [21]) In the case when the coefficient functions are linear, 2 the Lyapunov function is given by A (2') 2 ca: , c > 0. Next one shall study the exponential stability of the examples given in Section 3.2. Example 3.6. The stock price, modelled using the Black Scholes model, follows (3.5). The discounted stock price model is given by X t = St/Mta where Mt is the money account. Under a fixed interest rate r, the dynamics for both quantities are given by dA/It = thdt, dXt = Xt [([t - 7‘)dt + OdBt]. From Remark 3.2, it follows that the Lyapunov function of the above models is given by cx2, c > 0. Therefore, the necessary and sufficient condition for exponential 2—stability for the Black Scholes model and the discounted stock price model is given respectively by, u+-Q—<0andtt—r+—2—<0. 25 Example 3.7. The Vasicek model is determined by (3.6) . Let rt and r; be the solutions of (3.6) with initial conditions r0 and r6 respectively. Then, d(rt - r21): “5(7): — r;)dt. Since 5 > 0, then from Definition 3.2, it follows that the solution of (3.6) is ESQM. Example 3.8. Consider the mean reverting stock price model: dSt = q(L — St)dt + oStdBt. (3.8) Let Si" be another solution to (3.8) with initial condition S6 = 3* . Then, as, — 53‘) = as, — spat + 0 (St — 53‘) dBt. (3.9) Since the coefficients of this system are linear functions, then it follows that (2a + 02) < 0 is the necessary and sufficient condition for the solution of (3.9) to be ESQM. 26 Vasicek Model ‘ 0 5 .1015 20 25 '0 5 .1015 20 25 time (years) time (years) Figure 3.1: The recurrence region for the CIR model (left) and the Vasicek model (right) for a particular choice of the parameters. 27 Stability of stochastic interest rate models driven by general Lévy noise 4.1 Introduction In the previous chapter, the recurrence property of term structure models driven by Brow- nian motion was studied. However, in view of the work in [11], interest rate models that are driven by Le’vy processes, provide a more realistic fit to the market. In order to explore the recurrence properties of such models, one first needs to study the exponential ultimate 28 boundedness of the solution of following type of SDE: dYt=a(Yt)dt+/ b(Yt,u) q (dt,du) +/ |u|<1 lul>1 C(Yt,u)N(dt,du), Y0 = y. (4.1) Recall from Chapter 2 that one can construct the solution of (4.1) using the modified SDE: dYt = a(Yt) (1t +/| I b(Yt,u) q (dt,du), Y0 = y. (4.2) u <1 As in Chapter 3, it will be assumed here as well that the coefficient functions of (4.2) satisfy (A1) and (A2). Hence, (4.2) has a unique solution. From Chapter 2, it also follows that the solution of (4.2) is a homogeneous Markov process for which the transition probabilities constitute a Feller semigroup. In a recent paper [3], the stability problem for the solution of (4.2) was studied. How- ever, it only gives a sufficient condition for the stability of the solution, provided a Lya- punov function exists. As discussed previously in Chapter 3, in order to successfully derive stability results of any stochastic model, the main challenge is to appropriately construct such a Lyapunov function. This chapter addresses exactly this issue by explicitly construct- ing such a function for models of the form (4.2) . Lyapunov function is then extended to general models of the type (4.1) through approximation schemes also discussed here. As pointed out in [3], the work [14] also studied a similar stability problem. However, their results were developed only for SDE’s driven by compound Poisson processes. The models discussed here are much more general. In [27], a class of SDEs, driven by semi martingales with jumps was studied for stabil- ity. But the conditions that were imposed there, are not easily applicable to the Lévy noise case of interest here (see [3]). Since exponential 2-stability of (Yt)t>0 leads to an invariant measure that is degen- erate (or trivial), it is not suitable for spot rates models. While exponential 2-ultimate 29 boundedness, which leads to a non-trivial invariant measure and also implies weak positive recurrence, provides a more realistic] and useful feature for the study of spot rate models. The techniques developed in this chapter allows one to directly investigate these properties for different interest models, which is an improvement over [3] and [12]. Further, the def- inition for stability in probability studied here is based on [20] and is different from that used in [3]. In conclusion, the focus of this chapter will be on exponential ultimate boundedness, which leads to recurrence. As an application of the results, it will be shown that the p- exponential stability from [3] readily follows from the techniques introduced in this chapter. In the last part of the chapter, the general theory will be applied to term structure models, including the models in [12] and [38]. 4.2 Exponential p-stability Henceforth, one shall distinguish between (4.1) and (4.2), by referring to them as big jump SDE and small jump SDE, respectively. Recall the infinitesimal generator L of (4.1) from (2.7 ). The following theorem gives a sufficient condition for the exponential 2-stability. We include an elementary proof. Theorem 4.1. The solution Yt of the big jump SDE (4.1) is exponentially p-stable, if there exists a function A E C2(IR) such that, V51: E IR, 6) Cilxlp 3 Abs) .<. C2lxlp. (ii) £(A(:1:)) S -C3A($), for some positive constants c1,c2, C3. 30 Proof Since A e 62(IR), it follows from Ité’s formula [36] and (2.7) t Ely/\(Yl) — My) = / Eyc (A(Ys))ds. 0 Define <1>(t) = EyA(Yt). Differentiating both sides w.r.t. t, I _ (t) = M (Am. Then, from from condition (ii) one obtains, cr'n) g —C3<1>(t). This implies that giec3t<1>(t) = 6031: ((D,(t) + C3<1>(t)) S 0. Hence, it is a decreasing function oft and ec3tCI>(t) _<_ (13(0), fort > 0, where (0) = My). Using this and condition (i) on A, C1Ey|Ytlp S EyMYl) S A(y)e'c3t S 02|ylpe_c3t- Hence, Yt is exponentially p-stable. El Analogous to the previous chapter, we call A E C2 (IR) satisfying conditions of Theo- rem 4.1 a Lyapunov function. Remark 4.1. In the case where the coefficient functions of (4.2) are linear in the state variable, the SDE is of the form dYt = aYtdt + Yt /| | b(u)q(dt, du), Y0 = y. (4.3) . u <1 One shall refer to this as the linear SDE. The next theorem provides a necessary condition for exponential 2-stability of the so- 31 lution of the linear SDE by constructing a Lyapunov function under exponential 2-stability. Theorem 4.2. Let {Yb t 2 0} be the unique solution of the linear SDE (4.3). If it is exponentially 2—stable, then the fixnction, oo , 2 My) = /0 By (Yt) dt, y e R, (4.4) is finite, A E C2(IR) and it satisfies conditions (i) and (ii) of Theorem 4.1 with p = 2. 2 Proof Applying the infinitesimal generator L of (4.3) to f (2:) = :1: , :1: E IR, one obtains 1: (x2) = cx2, (4.5) where c = (20. + f[u|<1 b2 (u) u ((111)) . Then applying Ité’s lemma to f (1:) = :1:2 and subsequently using (4.5) yields, 2 2 t 2 Ey (Vt) — y = /0 Emir.) ds t 2 = 6/0 By (Y3) d3, 3,] E IR. (4.6) By exponential 2-stability of Y, E9 (Yt)2 —> 0 as t —> 00. Therefore, upon taking the limit it ——> 00 on both sides in (4.6), one obtains —1 My) = —-y2. C Since from (4.4), A must be positive, one obtains that c < 0. Let 7 = ’71, then 7 > 0 and A(y) = 73/2. From this it is clear that A E C2(IR) and condition (i) of Theorem 4.1 is satisfied. As done in [21], one can also prove this by using the fact that the solution of (4.3) is linear in y. For the second condition (ii), recall that the unique solution process {Yt’ t 2 0} is a 32 Markov process. Define J-"tY = o{Y5, s g t}. Then, by choice of A, 0° Y 2 EyA(Yt) = Ey/ E t(Y3) d3. 0 Since Yt is Markov, 0" Y 2 Y EyAm) = 13/0 E t[(Y3) if, ]ds. Then, by the uniqueness of the solution, 00 OO EyMYt) Ey[/O [(n+.)2lf2” 1]: /t Etc/3%. 0° . 2 t 2 =/ Ego/S) (ls—[0 Ey(Y3) ds. 0 The first integral on the r.h.s. is A (y) and one obtains t 2 mm.) :— My) — [0 13W.) «is Divide both sides by t and take the limit t —-> 0. Then, using continuity of E31 (Yt)2 in t at t = 0, and using Y0 = y, one obtains — [5 By (Y3)2 ds 2 2 y ’ _ lim E Ant) AW) = lim —y . t——>0 t t—+0 t Since A E C2(IR), then on the l.h.s. one gets LA(y). Therefore, LA(y) = —y2. From the lower bound of condition (i) of Theorem 4.1, one gets LA(y) g —C3A(y), for some C3 > 0. Thus, condition (ii) of Theorem 4.1 is satisfied. C] (1 Let us denote forafunctionb(:c,u) oan x R0, 1);); (23,11) = EEMx, u) and ban; (3,1)) = d2 33 Theorem 4.3. Let Yt be the unique solution of the small jump SDE (4.2). Assume a E C; (IR) andboth integrals flul<1 bx (11:, u) l/ (du) and 'IIUI<1 buy (2:, u) u (du) are bounded for any at E IR. Also, assume 0(0) = 0 and quI<1 b (0,11) u (du) = 0. Then, ifYt is expo- nentially 2-stable, there exists a Lyapunov function A E C 2(IR) that satisfies conditions (i) and (ii) ofTheorem 4.] with p = 2. Proof. Under the assumptions imposed on the coefficient functions and exponential 2- stability of Yt, it follows from Corollary 9.7 in [1] that My) = [6’0 EV (Yt)2 dt E C2 (IR) , y E IR. Next, one shall show that A is indeed a Lyapunov function for (4.2) . From exponential 2-stability of Y], one obtains for all y E IR, 0° , 2 2 0° t 2 A(y) 2/ Ey (Y1) (it 3 Ag f0 6—0 dt = c2 [3]] . (4.7) 0 Using the smoothness conditions imposed of the coefficient functions and the initial condi- tion (1(0) and qu|<1 b(0, u)1/ ((111) = 0 one has, by Lipschitz condition [a(:c)| S clcrl and fl I |b(rc,u)|2u(du) S clxlz, Va: E IR. (4.8) u <1 Applying L to f (:12) = 2:2, :5 E R yields, Lf(2:) :2xa.(zc)+/ (b(x,u))2u(du). |u|<1 Hence, by (4.8) [£(x2lI s K I242. (4.9) for some K > 0. Applying Ito’s lemma to f (2:) = 2:2 leads to t E9 (in? — ,2 = /O Egan/52o. 34 Applying (4.9) on the r.h.s., 2 2 t 2 Ey(Yt) —y 2 —K/ Ey(Y5) ds. (4.10) 0 Take limit 1 —+ 00 on both sides. Then, by the exponential 2-stability of Y, E31 (Yt)2 -—> 0 as t —+ 00. On the r.h.s. f6 Ey (Y3)2 ds —) A(y) as t —-> 00. Therefore, (4.10) reduces to —y2 2 —KA(y) or equivalently My) 2 c1 [y]2 for some c1 > 0. Using this and (4.7) , the condition (i) of Theorem 4.1 is satisfied. For proving condition (ii), one follows the same argument as used in the proof of Theorem 4.2. 1:] Theorem 4.4. Let Yt be the unique solution of the linear SDE (4.3) and assume that it is exponentially 2-stable. Let Yt denote the solution to the small jump SDE (4.2). Further assume that the coefficients satisfy (1(0) and fl b(0, u)l/ ((111) = 0 and u[<1 2 [11:] |a(:1:) — (1:1:| +/I I 1|b(1:,u) — lb (11)] [b(:1:, u) + arb(u)[1/(du) g C4IIIII2, (4.11) 11 < for all a: in a small neighborhood of the origin, for a small constant C4. Then, it is also exponentially 2-stable in that neighborhood of zero. Proof Let L and L0 denote the infinitesimal generator of the small jump SDE (4.2) and linear SDE (4.3) , respectively. Let y be in a neighborhood of zero, such that condition (4.11) holds. By Theorem 4.2, there exists a Lyapunov function of the form A (y) = 7.1/2 for some '7 > 0. Then, A E ’D L and one obtains, £(A(y)) —co(My)) = A’(y)(a(y) —ay) + / lMy + My. u)) — My) — b(y, u)A’(y)Iz/(du) lul<1 — / [My + by2yl s 2, lyl |a(y) -— “""Hl/loll’iy’ u) — yb 0. Hence, ( 11)) _ C3|yl + Will/l - Then, for C4 < E73, one has £(A(y)) s —c51y12. for some positive constant c5. The result then follows from Theorem 4.1. CI 4.3 Stability in probability In this section, the results of [20] on stability in probability are extended to systems driven by Levy-noise. Definition 4.1. The solution of general SDE (4.1) is stable in probability, if for any 6 > 0, lim Py (sup [Y(t)] > e) = 0. [HI—*0 t 36 The following theorem gives sufficient conditions for stability in probability of the so- lution of (4.1), in terms of a Lyapunov function. Theorem 4.5. Suppose there exists a function A with the following properties: (1) A e Camus) and |A(a:)| 3 00(1 + ]:1:]),C'0 > 0, :1: e 111, (ii) A(:c) —> 0 as [r] —> 0, (iii) inf A(1:) 2 Ac > 0, V c > 0, ]:1:]>c (iv) L(A(:c)) g 0, |:1:| < (5,for some (I > 0. Then, the solution of (4.1) is stable in probability. Proof. Observe it is enough to show that Py(sup IYt] > r.) S Aiy), (4.12) t 6 since in this case, the result follows immediately from (ii) and (iii). Let T5 = inf[t > 0 : [Y1] > 6}. Then T5 is a stopping time. From condition (ii), A E ’D L and by Dynkin’s formula, tATg EyAlnAT.)—A 71)]- From (iii), one has that A is non-negative and therefore, EyMYt/WEI Z By IA(YTE)X(t > 76)]- Again, from condition (iii), EyMYt/wé) 2 E111 [A(YT()X(t > 7.)] _>_ AePym < t). Now applying (4.13) to EyAO/t/(TE) EyMYyMC) < M y < . P (TE < t) __ A6 _ /\€ Since the right hand side does not depend on t, A(y) A. ' Py(sup]Yt] > c) _<_ t Then, the result follows from (ii) by letting ]y] —> 0. Cl One may ask the following question. Is there is a direct relation between exponential 2-stability and stability in probability? The answer is yes and can be summarized in the following theorem. Theorem 4.6. If the solution Yt to the linear SDE (4.3) is exponentially 2-stable, then it is stable in probability. Proof. From Theorem 4.2, one can show that the Lyapunov function for (4.3) is A (:1) 72:2, for some '7 > 0. It satisfies LA(:L:) < —C3A(:c), for some positive constant C3. Then, 38 it can be readily seen that A satisfies all of the conditions of Theorem 4.2 and hence the result follows. 1:] Remark 4.2. If the coefficients of the small jump SDE (4.2) satisfy the conditions of Theorem 4.3, then exponential 2-stability of solution of (4.2) implies that it is stable in probability. The next results are analogous to Theorem 4.4 for stability in probability. The proof is omitted here, as it follows from the same argument used in the proof of Theorem 4.4. Theorem 4.7. Let Y}: be the solution of the linear SDE (4.3) and assume that it is expo- nentially 2-stable. Let I7t be the solution to the small jump SDE (4.2) . If the coejficient functions of the two systems satisfy the following inequality, 2 [:13] ]a(:l:) — 11.x] +/I‘ I ]b(:1:, u) — 23b (11)] |b(:1:,u) + :cb(u)] S C4]CL‘]2, u <1 for all :r in a small neighborhood of the origin, for a small constant C4, then the solution of the non linear system (4.2) is also stable in probability. 4.4 Exponentially ultimately bounded In this section, the analogous results to Chapter 3 for stochastic systems driven by Lévy noise are presented. The following theorem gives a sufficient condition under which the exponential ultimate boundedness of the system (4.1) is guaranteed. Theorem 4.8. The unique solution of the system (4. 1) is exponentially p-ultimately bounded if there exists afitnction A E C2(IR) such that, V51: E IR, (1) c1|;1:]p — 121 g A(:1:) S 62 ];1:]p +1112, (ii) £ (A($)) S -631\(17) + (C3) 39 for some positive constants C1 . (:2, C3 and real constants k1, kg, 193. Proof. Applying the It?) formula to A and taking expectation on both sides, leads to t EyA(Yt) — My) 2 /EyL (A(Y3))ds,Vy E IR. 0 Let (t) -—- EyA(Yt). Differentiating both sides w.r.t. 1, yields (1 7,341) = Eyc (Mm). ( Using condition (ii) on the right hand side, (1 Then, Therefore, where (0) = A( y). Or equivelently, )~ ' k Mt) s —3 + (My) — —3) e—C3t. ’3 C3 Then, from (i), Ey m]? g Cc_fit [ylp + M. C] The following theorem gives a necessary condition for exponentially 2-ultimate bound- edness for the linear system (4.3). 40 Theorem 4.9. Let Yt be the unique solution of the linear SDE (4.3). If it is exponentially 2-ultimately bounded, then there exists a function A E C 2 (IR) satisfying the conditions of Theorem 4.8. Proof. Consider, T . My) = [O Ey(YtI2d1‘-, where T is a certain positive constant to be determined later. Since Yt is exponentially 2-ultimately bounded, T C My): [0 Eymfdts -3(1—e‘BT)lyl2+MT = C2 |y]2 + 12. (4.14) Applying L to f (2:) = 22, 2: E IR, 172 20.117317 :13 111132—232— 'U) 331/ U. £( ) 2 +/IIII<11< +b< )) b( )21 (d ), and following the same technique as in Theorem 4.2, one obtains [(2:2) _<_ 2a2:2 + 212/ (b(u))2 l/(du), ]u]<1 or ]L(2:2)l 3 K22, (4.15) 2 for some K > 0. Applying Ito’s formula to f (2:) = 2: and taking expectation leads to 2 2 T 2 Ey (YT) — y = /0 EyL (Y3) ds. Using the fact Yt is exponentially 2-ultimately bounded, T Ce—fiTy2 + M — y2 2 [0 EM (Y3)2 (is. 41 From the inequality L (2:2) 2 —K2:2, one obtains Clo-5713,12 + M — ()2 2 —K /0T lat/(1(3)2 do. Then, — (1 — C€_’8T) y2 + M _>_ —KA(y). Hence, for T such that 1— (Jo—3T > 0, (4.16) one gets 01y? — 11 3 My) (4.17) where c1 = (1 — C€_’BT) /K and k1 : M/K. Equations (4.14) and (4.17) together. imply condition (i). For the second condition, recall that the unique solution Yt is a Markov process. Define FtY 2* o{YT, r _<_ t}.Then, T T EyA(Yt) = E31 /0 EYt(YS)2ds = [0 E9 [EYt[(Y5)2 my I] do. Since the solution is unique, T 2 EyAO/t) = f 1:31 (Yt-l—s) d3. 0 Then, By choice of A, one has, t , 2 t 2 EyA(Yt) = My) —-/0 EV (Y5) ds +./() By (YT—+3) ds. Therefore, 4 t 2 t 2 EyA(Y (t)) — A(y) 2/0 Ey (YT+5) d5 -—/0 By (Y5) d5. Divide both sides by t and the take limits t ——> 0 . EyMYl) —My) . 1 ft , 2 t 2 11m = 11111 - Ey Y . ds—/ Ey Y (15 . t—+0 t t—>0t 0 ( T”) 0 ( 8) Since A E C2(IR), the l.h.s. reduces to LA(y) and one obtains LA(y) = Isl/(11p)2 — 1}. On the r.h.s., by using exponential 2-ultimate boundedness of Yt, one gets LA(y) S CC_BTy2 + M — yz, 01' (511(1)) 3 — (1 -— Con/371)];2 + M. (4.18) From condition (1) of the sufficient theorem, one can observe that 2 1 - <-—A- k. y _ 02 (10+ 2 43 Therefore, ch) _<_ — (1— Ce‘fiT);1-A(y)+ 12 + M, 2 = “0311(9) + ]‘83, for some constants C3 > 0 and k3 E IR. This proves (ii). [:1 The following theorem is analogous to Theorem 4.3 for ultimate boundedness. The proof is omitted and can be obtained from the proof of the corresponding Theorem 4.3. Theorem 4.10. Let Yt denote the unique solution of the small jump SDE (4.2) . Assume that the coeflicient functions satisfy the conditions of Theorem 4.3. If the solution of (4.2) is exponentially 2-ultimately bounded, then there exists a Lyapunov function A E C2 (IR) satisfying all the conditions of Theorem 4. 8. Theorem 4.11. Let Yt be the solution for linear SDE (4.3) and assume that it is exponen- tially 2-ultimately bounded. Let I’t be the solution to the small jump SDE (4.2). F urther; if the coefficient functions satisfy, V2: E IR, 2 I23] ]a(2:) — 112:] +/ I We, 11) — 2'b(u)] ]b(2:, u) + 2:b (11)] 1/ (do) S C4|2:]2 + k4, [1). <1 (4.19) with constants C4 and k4, such that 1 — "53 C4 S max —£§——— , (4.20) 8) In C 73— 73' where the constants c and ,8 are taken from Definition 3.3, then the solution IQ is also exponentially 2-ultimately bounded. Proof. From Theorem 4.9 we know that the Lyapunov function for (4.3) is T 2 My): [0 E9111) 41, yea. (T such that (4.16) holds) and it satisfies all the conditions of Theorem 4.8. Further, using a similar argument as in the proof of Theorem 4.9, it can be shown that My) is of the form Ky2 for some K > 0. Then from (4.14), Ky2 < %(1 — e‘fiTM/Z + MT. Let L and .A denote the infinitesimal generators of (4.2) and (4.3) respectively. Then, A E D 5 and one can show the following £A(y) - 44(3) = 2K1: (0 (y) — ay) + [(11le { 0 and k5. Using A(y) = Ky2, the inequality LA(y) S —c6A(y) + k5 holds and the result follows from Theorem 4.8. E] 4.5 Recurrence In [29], the recurrence property of stochastic differential equation (driven by Brownian motion) is studied in terms of the Lyapunov function. In particular, [29] showed that if the solution process is exponentially 2-ultimately bounded then it is weakly positively re- current. In this section, the same type of results are obtained for SDE’s driven by Lévy noise. Recall Definition 3.5 and Definition 3.6 for the concept of recurrence. The following results for general Feller Markov processes are due to [29] (pg 140). Lemma 4.1. ( [29], Lemma 5.1) Let Yt be a Markov process. If there exits a positive Borel measurable function p, defined on IR, and two positive constants K and a such that Vy E IR, Py(lY(t+/)(y))l S 102 a > 0. then the process Yt is weakly recurrent and { :1: : |2:] S K} is the recurrence region. Theorem 4.12. If the process Yt defined by (4.2) is 2-ultimately bounded then it is weakly recurrent. 46 Proof. Recall that the solution of (4.2) is a Markov process. Since Yt is ultimately 2- bounded, lim sup 139m]? 3 M, for any 1] e 111. t—yoo . Also note that Ey|Yt|2 is continuous in y for a fixed t by feller property and continuous in t for a fixed y. Therefore, from [23], one can define a Borel function ,0 such that Ey Y 2 < M ] t+p(y)] — ' Take K = x/M +6, 1: > 0. Then, P9 Y K y 2 K2 (] t+p(y)] > ):“P y] t+1411) )] > E- [Y 12 S t+p(y) S_ M _(1 _a) < 1 K2 K2 Then, from Lemma 4.1, Yt is weakly recurrent and the recurrence region is specified by {2:;]:1:]<\/M+c} foranyc>0. El Recurrence of Yt by itself does not ensure that the process gets to the recurrent set in finite mean time. However, this is the case if Yt is weakly positively recurrent. 1n the following theorem, it is shown that if Yt is exponentially 2-ultimately bounded, then it is weakly positively recurrent. Lemma 4.2. ([29], Lemma 5.2) Let Yt be a Markov process. If there exists a positive non- decreasing function W() defined on [0, 00) and two positive constants K and p such that Vy E IR, Ey IYoIP s KP for s 2 W(Iyl), (4.21) and 00 Zlipl l)A’) < 00, for any N > 0. (4.22) 47 Then, Yt is weakly positive recurrent. Theorem 4.13. If the solution Yt of (4.2) is exponentially 2-ultimately bounded then it is weakly positively recurrent. Proof. Let M be the constant determining the ultimate boundedness of Yt in the Definition 3.4. From Chapter 2, it is known that Yt is a Feller Markov process. Let W(r) = dllog(1+ r) + d2; where d1, (12 are constants and depend on the exponential 2-ultimate boundedness of Y and r > 0. Then for this choice of W (-), both the conditions of Lemma 4.2 are satisfied for K = x/M— and p = 2. Hence, Yt is weakly positively recurrent. C] 4.6 Invariant measure In [30], the author showed that every Feller process that is ultimately bounded, has a finite invariant measure (sec (4.24) below). The results are recalled here for the convenience of the reader. The following theorem gives a sufficient conditions for ultimate boundedness for the solution of the system (4.2) . The proof is omitted here and follows directly from the proof of Theorem 4.8. Theorem 4.14. The solution of the system (4.2) is p-ultimately bounded if there exists a function A E C2(IR) such that,V.r E IR, (i) Cll'p — k1 S A(2:), (ii) L(A(.’c)) S —C3A(2:) + 1:3, for certain positive constants (:1, C3 and real constants k1, k3. Next, the definition of an invariant measure is given and its dependence on ultimate boundedness is established through Theorem 4.15. Recall from (2.6) the transition func- tions for a Feller Markov process. 48 Let 73 (IR) denote the collection of probability measures on l-dimensional Euclidean space. For y E IR, T > 0 and A E 8 (IR) , define y 1 T y (A) = f/O P (Y3 E A)ds. (4.23) Then, [ta/4H E ’13 (IR) and consequently M := {u%(-),y E RT 2 0} is collection of probability measures in 1’ (IR). In Theorem 4.15, one shall show that the family M is tight. Then, by Prokhorov’s theorem (Theorem 6.1 [6]), there exists a subsequence in M and a probability measure 11 E ”P (IR) to which the subsequence converges weakly. Further, )1 satisfies: MA) = 111 Py(Ys e A)y(dy), v A e 8 (IR) (424) i.e. 11 is an invariant measure (see [24]). Theorem 4.15. If the solution Yt to (4.1) is p-ultimately bounded, p > 0, then its invariant measure exists. Proof. Let M denote the constant that determines the ultimate boundedness of Yt (see Definition.3.4). Then, for any y E IR, one can get a T0 = T0(y) such that Eletlp S M, t 2 To. From this, one can show the following, 1 T 1 T E?! Y P lim sup —/ Py(|Yt] > K)dt S lim sup — —l——t|—-dt K—mo T T O K—-)oo T T 0 KP — lim isupl/TEyIYlpdt K—>oo KP T T 0 t I . 1 11m —sup— < —K—)oo KP T T TO T / Elet|pdt +/ 131/12,]de 0 T0 =0. This proves that M is tight. Then, the result follows from Prokhorov’s theorem. CI 49 4.7 Examples Example 4.1. Adding a little pertubation to an already exponential stable system does not affect the stability of the system. Consider the exponentially stable system i‘t = ——a2:t. where a > 0, 2' (0) = 2:0 E IR. Consider the perturbed system: dCL‘t = —a.;1‘tdt + brtdYt, (4.25) where Yt is a pure jump Lévy process of the form dYt = / uq (dt,(1u) , 0<]u| 0. Using the infinitesimal generator of the process (4.25) on A (2:), one is led to LA (2:) = )3[—2a, + b2/ 1121/ (du)]2:2. 0<]‘u]_C Note that the system is linear. From Theorem 4.2, it then follows that the Lyapunov function is of the form A (21) = 522, for some ,6 > 0. This means that LA (2:) = d[—-2o + 12 / 0<|11|Cu2u(du)+2b/II 111/(d11)]2:2. 112C (4.27) If the system (4.26) is exponentially 2-stable, then its follows that f] 1121/ (du) < oo. u]2c So (4.27 ) can be rewritten as LA (2:) = d [-20. + E (1 + 11(1)2 —- 1] $2, for {—211 + E (1 + bY1)2 — 1] < 0. From this, it can be concluded that the system is still exponentially 2-stable. Example 4.3. (Stock price model with jumps) Let St be the stock price, modelled by (151‘ = ,uSttlt + O’StdBt + VStst, (4.28) where (Bt,t _>_ 0) denotes the Wiener process and st = / 11q (dt, (111) + / uN (dt,d11) , 0<]11]_’C . ]11]2c In case the jumps in the process can be represented as a compound Poisson process, N(t) L1 = 2: U1) 1:1 where N (t) is a Poisson process with intensity )1 and U,- are i.i.d. random jumps with common jump size distribution as of random variable U, then (4.29) is reduced to LA (2:) = [32:2 [211 + (72 + A {E (1 + ’yU)2 — 1} + A2E(7U)2]. It follows that for [211 + 02 + /\ {E (1 + c/U)2 — 1} + AQE (7U)2] < 0the system (4.28) is exponentially-2 stable. This gives an alternate proof for Proposition 6.1 in [12]. Example 4.4. (Vasicek model with jumps) Let rt satisfy the following SDE, drt = ((1 — fir‘t) (ll + (711315 + t‘tst, 52 with starting point r0, where st = / 11q ((11,1111) + / uN (dt,d11), 0<|11| 0, we get LA(2:) C[ 25+/0<]u]oo 2B and by Theorem 4.13, it follows that the recurrent set for the process (4.31) is {$1]$]<\/02Ku/26}- In conclusion, the recurrence set for the Vasicek model is given by {2:1 |2:] < «(fa/(3)2 + 02KV/213}. 54 Example 4.6. (CIR model) The mean-reverting interest rate model, extended to include jumps is given by drt : a (B —- rt) dt + omst (4.33) where st = dBt+ f0oo Ill and It 11m 191 #3)] : ]2:]—>oo I13] Therefore, from the linear approximation Theorem 4.1 1, the system (4.33) is exponentially 2-ultimately bounded. To get the bounds, note that t t Ert — r0 2/0 (QB — aErs) (18 + (IE/0 \/r—tst. But since Lt = Bt +/ 11q (t, 6121) 0<|11|<1 is a martingale, and since the process is exponentially 2-ultimately bounded, t E/ (#7st = 0. 0 So from, t Ert — r0 = /0 (QB — aErS) d5, 55 it follows that d—ggl = (1,6 — aErt. So, Ert = ,8 + (:r — ,6) 6"”. Now applying Ito’s formula to f (t, :13) = e2atx2, leads to d (82(11'7‘?) = €2a£ [20,8 + 02 (Ky + 1)] rt, (4.34) where Ky = f0<|u|<1 U21! (du) . Upon solving (4.34) and setting 7 = [2a,3 + 02 (Ky + 1)] , E 7‘2 g 2T26—2at + 73 + ’72, t 0 20 and consequently, the recurrence region is given by {:13 : 0 < a: < \/ 72% + '72} . 56 Stochastic control and dynamic pro grammin g 5.1 Introduction In this chapter, the application of stochastic control is studied in order to stabilize asymp- totic behavior of a Lévy diffusion process. The two notions that shall be addressed are exponentially 2-stable and exponentially 2-ultimately bounded. The reader is referred to [33] for a detailed discussion of stochastic control of jump diffusion processes. 57 In Chapter 4, one factor jump diffusion models for interest rates were introduced. In this chapter, the focus will be on designing appropriate controls so as to make the interest rates exponentially 2-ultimately bounded and as a consequence of Theorem 4.13, positively recurrent as well. The stochastic control problem to make the interest rates exponentially 2-stable, is investigated as well. The stochastic process at = u (t,w) : IR+ x (2 —) IR is called the control process and the controlled SDE corresponding to the small jump stochastic differential equation (4.2) is given by: dYt = a(Yt, u-t)dt +/ 1b(Yt, ut, z)q (dt, dz). (5.1) 7I 1v The main idea of the control is to steer the stochastic system (5.1) in a desired way. This is done by modifying the coefficients through appropriately choosing the control process at at any instant t. Since the decision at time t must be based on what happened up till time t, it is natural to assume that the stochastic process ”t must adapted to the filtration {1'}. t2 0}- As a first step in the stochastic control problem, an objective is defined. Usually the objective is formulated in terms of a cost function associated with each control u. In gen- eral, this objective or cost function depends on the problem one wants to address. The idea is to penalize undesirable behavior by giving it a large cost, while desirable behavior is encouraged by attaching a low cost to it. The goal of a stochastic control problem is then to find, if possible, an optimal strategy u which minimizes the total cost. The purpose of this chapter is to study the use of stochastic control techniques to make (5.1) firstly exponentially 2-stable and secondly, exponentially 2-ultimately bounded. The main technique used to find an optimal control is the Hamilton—Jacobi—Bellman (HJB) equation. 58 5.2 HJ B Equation Let Yt be the solution of the controlled process (5.1) with the initial condition Y0 = y. Let the cost of using the strategy u be J u (~) and defined as . oc- Ju (y) 2/0 Eylt'(Yt,ut)dt, y E R, where K is a given cost function, continuous in both its arguments. We define a control strategy it as admissible if (5.1) has a unique solution for that choice of u and J u (y) is finite. Then, the optimal control problem constitutes of finding the optimal control 21* (if it exists) and the value function V (y) defined as ., * V(y) = inf J“ (y) = J“ (y), y E R, (52) 116A where A denotes the set of all admissible controls. Note that it could be the case that such u* for which the lower bound is attained, may not exist. In that case one works with E-optimal strategies. However, for the rest of the chapter, it shall be assumed that u* exists. Another desirable feature of at is that at time t, it depends only on the state of the system at the time i.e. at = u (Yt). Such a u is called a Markov control since the corresponding controlled process (5.1) becomes a Markov process and its infinitesimal generator [1” is given by, cud) (y) = a(y,u) as, (y) (5.3) +/|z|<1{¢(y + My, u, z)) _ (My) — 05’ (y) Myra, 2)} u (dz) , for¢EC2(R),yElR. Further, it can be shown that, under mild conditions (see Theorem 11.2.3 in [32]), it is sufficient to consider Markov controls. Therefore, for the rest of the work, only Markov 59 controls shall be considered. Theorem 5.1. ([33], Theorem 3.1) (HJB for optimal control of jump dtfiusion) Given a cost function K E C (1R2) . a) Suppose for all y 6 IR and all admissible controls 11. E A, there exists a smooth function (15 E C2 (IR) satisfying: (i) 12% (y) + K (y. u) 2 0, (ii) Eyqb (Ytu) —-> 0 as t -—> 00. then d) (y) g V (y)for all y 6 IR. b) Furthermore, if for each y E IR there exists u* E A such that (W) L“*(y)¢('y) + K (y,U* (31)) = 0, then u* is an optimal control and, (I) (y) = V (y) = Ju* (y) (y) for all y 6 IR, where V (y) is defined in (5.2) . 5.3 Exponential 2-stabilization The objective of this section is to find an optimal control which makes the system (5.1) exponentially 2-stable. The controls are assumed to be Markovian. Definition 5.1. A control it = u (1') is said to be admissible (denote it E A), if it satisfies the following conditions: (BI) For a given u, the coefiicient function a (2:, u (15)) is twice continuously difi‘erentiable with respect to :17, with bounded derivatives and for any x 6 IR, both the integrals fIZI<1 b3; (:5, u (:r) , z) :1 (dz) and fI3I<1 ban; (3:, u (:13) , 2) [1 (dz) are bounded. (B2) 11(0) 2 0. 60 (BB) The system (5.1) admits a unique solution. The following lemma is by [20] and is stated here as it shall be required later on. Lemma 5.1. ([20], Lemma 5. 7.1) Let Y,“ denote the unique solution of the controlled sys- tem (5.1) , u E A and (BI) holds, then V y E IR, 00 / Eletu|2dtoo In the next theorem, a link between the Lyapunov function and the optimal control that makes the system exponentially 2-stable, is established. Theorem 5.2. Let K E C (IR?) be a cost function. Consider the stochastic system (5.1). If there exists a non negative function A E C 2 (IR) and an admissible control u* E A, such that for some positive constants 19,-, z‘ = 1, 2, 3 the following holds, V11: E IR, 1. A (x) s k. lez. 2. £u*A (:13) + K (1521*) = 0, 3. LuA(:I;)+K(:1:,u) Z 0,u E A, 4. K(:L‘,u) 2 k2 I112. Then, the function u* is the optimal control for the system (5.1) in the sense that it minimizes the cost a: J” (y)— — min Ju(y — min AEy/OOO K(Y s u,u3) d.s uEA It also makes the system (5.1) exponentially 2-stable. Conditions 2 and 3 together are the HJB equation. 61 Proof. Let u be any admissible control. Applying Ito’s lemma to A E C2 (IR) and taking expectation yields, Vy E IR, t 1531A (Ytu)—A(y)= f0 EyLuA(Y3u)ds. (5.6) Setting u = u* and using condition 2 one obtains , * t * * EyA (Ytu ) -— A (y) = /0 —EyK (31,“ ,us) (is, 01', t * * f0 EyK (YSU ,u*) d3 = A (y) — EyA (Ytu ) . (5.7) Since A (-) is non negative, it follows that t , a: / ELI/K (1'3“ ,ug) ds 3 A (y). 0 , * Therefore, taking the limit it —) 00 one has J“* (y) = f6” EyK (Ysu ,ug‘) d3 < oo. * 2 Y,“ I —+0ast—>oo. Hence, Then, using condition 4. and Lemma 5.1, E9 =1: , a: 2 0 g EyA (Ytu ) g A-lEy Iv,“ | —+ 0, as t —+ 00. (5.8) Hence, from (5.7) , upon taking t —> 00, one obtains Ju* (y) = A (y) . Similarly, for any other it E A with J u (y) < 00, it follows from the same argument that EyA (Ytu) —) 0 as t —> 00. Then, (5.6) and condition 3 yield, 52‘) J“ (y) 2 A (y). (5.9) with equality holding for u = (1*. It remains to prove that the system is exponentially 2-stable when u = u*. From The- 62 orem 4.3, it is sufficient to prove that A (y) 2 k3 lyli2 ,Vy E IR for some positive constant k3. Using (5.9), it follows that Then, from condition 4, 00 a: 2 A(y)2k2/ EyIYSU | ds. 0 2 2 )3?) < IEQI—mhen, 00 “*2 A(y)_>_l.:2/0 13le3 I ds From (5.8), one can find T = T (y) such that E9 T * 2 21(2/ EleS“ l ds. 0 Using the inequality £u* (m2) 2 —k4 |:z:|2 ,Va: E IR and subsequently Ito’s lemma, one obtains A (y) 2 —k5 AT EM” ( will?) ds = 1:5 (Iylp — E’y IYF 2) 2 E-IZEE = k3 lylz. This completes the proof. [:1 5.4 Exponential 2-ultimate boundedness and control In this section, one shall develop a control strategy that makes the system (5.1) expo- nentially 2-ultimately bounded. Since exponential ultimate boundedness implies ultimate boundedness, it is therefore enough, to search for an optimal control amongst the set of 63 controls that make the system ultimately bounded. Definition 5.2. A control it = u (1:) is said to be admissible and denoted by u E M, if it satisfies the following conditions: (M 1) For a given 11., the coefficient function a. (1:, u (1:)) is twice continuously differentiable with respect to 1: with bounded derivatives and for any 1: E IR, both the integrals fIZI<1 bx (17, U (513) , Z) V (dz) and fIZI<1 b“; (1:, u (1:) , z) u (dz) are bounded. (M2) The solution of the controlled process (5.1) is unique and is 2-ultimately bounded. The following theorem gives a method for finding the optimal control (that makes the system exponentially 2-ultimately bounded) using the HJB equation. Theorem 5.3. Let K E C (1R2) be a given cost function. Let Y,“ denote the solution of the controlled process (5.1) under control it. Suppose there exists a non negative function A E C2 (IR), 11* satisfying (M 1) and positive constants k1, k2, c1, c2)» such that, V1: E IR, u)ogA(ngkfln2—q, (2) -AA (1:) + £“*A (I) + K(1:,u*) = 0, (3) —/\A (1:) + [WA (1‘) + K (13,11) 2 0, V a E III, (4) K(1:,u) 2 k2 [3:12 — c2. Then, 11* is the optimal control in the sense that it minimizes the cost, * . 0° . J” (y) = min Ju(y) = min Ey/ e_’\'SK(YSu,u3)ds, uErU 1161)! 0 * amongst all admissible controls in M. Further, Y,“ is exponentially 2-stable. The condi- tion (2) and (3) together are the HJB equation. 64 Proof. Let u E M. The function f (t, 1:) : e’MA (y) E 01,2 (R2) . Then, Ito’s lemma is applicable and one obtains, t e-MEy (A (1311)) — A(y) = [0 Eye-*8 (—/\A (1’5“) +1111 (1%)) ds. From condition 3, one finds t e_’\”Ey (A (Ytu)) - A (y) _>_ —/O e_’\sEyK (Ysu,u5) ds. (5.10) For 11* (using condition 2), 21: t , >1: (Mist/(1103“ )) — A (y) = [0 e—ASEyK (Ysu ,ujg) (is. (5.11) —/\t - 11* - - - - - Let (b (t) = e E9 (A (Y, )) . Differentiating w.r.t. to t on both Sides gives, d __ :1: EN) = —c ’VEyK (13," ,ug‘). Then, from condition (4) and (1) (1, u) )> 22M )+ Cl) — 02 and using this on the r.h.s. returns = 415(1) + eta—At. (5.12) for some constant I > 0 and c > 0. Then, from Gronwall’s lemma one obtains, (15(1) ——> 0 as t ——> 00. Therefore, using this :1: condition in (5.11) , one can ascertain J“ (y) = A (y) . Now, for any other control 11 E M, one gets from (M 2) that e_’\tEy|Yt“|2 ——) 0 as t —> 00. Then using condition 1, it follows that e—AtEy (A (Yt“)) ——> 0. Therefore, from 65 (5.10) one observes 21/ _ 21* , , J 111) 2 A(1/)— J (y). W E M- (5.13) From this it follows that 21* is the optimal control. Now in order to show that for 21* the 2|: process Y,“ is exponentially 2-ultimately bounded, it follows from Theorem 4.10, that it is sufficient to show k3lyl2—C3 311(1). for some positive constant A13 and real C3_ From (5.13), one obtains A (y) = J?” (y) = By [000 12—)‘3K (Ys“‘*,u* (Ys“*)) 113. Using condition (4), A (y) 2 A'QEy /OOO e—AS |Y3u*)2ds — C3. But since ¢(t) = e’AtEy lYt’uIQ, starts at (13(0) = 2,12 and (Mt) —-> 0 as t —> 00, a 2 T = T (y) can be found such that (p (T) r: %—. Then, * 2 , T ,. A(y) Zh‘QEy/ (3— '5 0 * Note that for any 1 E IR, Ewe—“V |.2:|2 = (—A [1|2 + £21 |1:|2) 6"”, but from (M1) it 5|: is known that IL" I:1:|2 g k lzvl2 for some k: > 0. Hence, one gets , * III” e—M |1|2l S kyle—At |1:|2, V1: E IR, for some 124 > 0. Then, .* 2 k Y8” ' ds—C3, where A15: ——2— y 21* As ’ > —l.: E — . A(y) _ 5 /O I; 6 k4 66 Now using Ito’s lemma, one finds _ _ , * My) 212 (1112 — e ”By lYi‘ By the choice of T, one has 1’71!k = y2, therefore, A > k 2 — (y) _ 3 l1! 63. which completes the proof. 67 2 )—C3. (5.14) . 1' 1’- .-' 1,. ‘11:. ‘55 ' 1”..."- ‘ ’1‘ ‘ ; i k '1 i , J.“ 7‘9 ‘. . .a ., '1' 1,13‘ ‘ 1'; ~. 3 ‘2, , ' ,3 ax?" Future work In this thesis, the recurrence behaviour of interest rates is studied and it was shown how to control a system to make this behavior possible. Although the results were presented for one dimensional interest rate models, the whole theory can be generalized for higher dimensions in view of Miyahara’s work. The first problem to address is the generaliza- tion with systems driven by two-dimensional Lévy processes, where the components are independent. Suppose that the first component represents the spot rate r? of country A and the second component represents the spot rate 1*? for country B. Then, one can define 68 St = log 6t where et is the exchange rate as a function of the two spot rates, namely at = f (TiAJ‘tB) =¢(TZ*—rtB). Using Ito’s formula, one can study the exchange rate dynamics and also examine the recur- rence behaviour of {st }t>0 . This involves the study of the following questions. 1. 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