X
0<s≤t
f(∆Xs)
=t Z ∞
−∞
f(x)ν(dx). (3.15)
The sum on the left hand side is taken over every time swhen a jump occurs. Taking f(x) =1S(x) where S is some set in Rd the relation tells us that the expected sum of jumps ∆Xs∈S is the integral overS with the L´evy measure.
3.2 Stochastic Differential Equations Driven by L´evy Processes
This section is devoted to develop stochastic calculus for L´evy processes. We follow the lines of Protter (2004), and develop stochastic calculus for a class of stochastic processes called semimartingales. The connection to L´evy processes will become apparent later on. Due to time and space constraints on this work, many details are skipped.
3.2.1 Semimartingales and the Stochastic Integral
We start off with the somewhat involved definition of semimartingales. Semimartingales are roughly speaking the class of stochastic processes where the we can define a stochastic integral in the same manner as the Itˆo integral. Let us start with the integrand:
Definition 3.3: A process H is said to be simple predictable if H has a representation Ht=H01{0}(t) +
Xn i=1
Hi1(Ti,Ti+1](t) (3.16) where 0 = T1 ≤ · · · ≤ Tn+1 < ∞ is a finite collection of stopping times with respect to Ft, Hi is adapted to FTi and |Hi| < ∞ a.s. for 0 ≤ i ≤ n. The family of simple predictable processes is denoted S.
We denoteS with uniform convergence in (t, ω) as topologySu. Moreover letL0 be the space of finite valued random variables topologized by convergence in probability. Be-tween these two spaces we define the map which will be our integral for simple predictable
3.2. STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY L´EVY PROCESSES 17 processes given a reasonable processX. More precisely we define IX(H) :S→L0 as
IX(H) =H0X0+ Xn
i=1
Hi(XTi+1−XTi). (3.17)
Definition 3.4: A process Xt is a total semimartingale if X is cadlag, adapted and IX(H) :S→L0 is continuous.
Let Xt be a process andT a stopping time, then the notation XtT denotes the process XtT ={Xt∧T}t≥0.
Definition 3.5: A process is a semimartingale if, for each t ∈ [0,∞), Xt is a total semimartingale.
Hence a semimartingale is a process that gives meaning to the mapIX defined onSu as an integral for arbitrary finite integration limits.
Following the usual routine of extending integrals of simple functions to more general spaces, Protter (2004) then shows that S dense in the space of cadlag processes. More precisely:
Definition 3.6: Let D andL denote two spaces of adapted processes with cadlag paths.
On these spaces we define a topology:
Definition 3.7: A sequence of processes {Hn}n≥1 converges to H uniformly on com-pacts in probability (UCP) if for eacht >0,sup0≤s≤t|Hsn−Hs|converges in probability.
Further Protter (2004) shows that
Theorem 3.2: The space Sis dense in L under UCP topology.
Finally we define the stochastic integral for a process inL:
Definition 3.8: Let X be a semimartingale. Then the continuous linear mappingIXt : LU CP → DU CP obtained as the extension of IXt(H) : S → D is called the stochastic integral.
Remark 3.1. From (3.17) it is clear that the Itˆo integral is one instance of the stochastic integral, since the integrand is evaluated in the left endpoint, and obviously Xt is a Brownian motion.
Now that we have the stochastic integral we are ready to develop the notion of stochastic differential equation driven by semimartingales.
3.2.2 Stochastic Differential Equations Driven by Semimartingales
Given the results in the previous subsection, we are ready to give meaning to the equation
dXt=f(Xt−)dZt (3.18)
whereZt is a m-vector semimartingale and Z0 = 0. The notation is just the shorthand notation of the integral equation:
Xti =xi+ Xm α=1
Z t
0
fαi(Xs−)dZsα (3.19) where i = 1, . . . , d, Xti denotes the i-th component of the vector process X and Zsα denotes theα-th component of the process Z at times. The coefficient functions fαi : Rd→Rare given and we denote f(x) the d×m matrix function (fαi(x)) admitting the notation
Xt=x+ Z t
0
f(Xs−)dZ (3.20)
which is equivalent to (3.18).
As for ordinary differential equations, we have a theorem that ensures existence and uniqueness of solutions of (3.18). First we need the notion of locally Lipschitz functions:
Definition 3.9: A function f : Rd → R is said to be locally Lipschitz if there exist constants CK only dependent onK such that
|x−y|< K ⇒ |f(x)−f(y)|< CK (3.21) for everyx, y∈Rd and | · | being the Euclidean norm.
It is rather obvious that functions of classC1, that is continuous functions with contin-uous partial derivatives, are locally Lipschitz. We are finally ready to state the main theorem (Theorem V.38 in Protter (2004)):
Theorem 3.3: Let Z and f be as above with f locally Lipschitz. Then there exists a functionζ(x, ω) :Rd×Ω→[0,∞]such that eachx,ζ(x,·) is a stopping time, and there exists a unique solution of
Xt=x+ Z t
0
f(Xs−)dZs (3.22)
up toζ(x,·) withlim supt→ζ(x,·)kXtk=∞ a.s. on {ζ <∞}.
Hence stochastic differential equations with locally Lipschitz coefficient functions have unique solutions up to explosion timesT(ω) =ζ(x, ω).
It is also worth noticing that if the coefficient functions are taken to be globally Lipschitz, the solutions exists and are unique for all timest ∈[0,∞). This is in accordance with the classical result for Itˆo stochastic differential equations given in e.g. Øksendal (2003).
3.2. STOCHASTIC DIFFERENTIAL EQUATIONS DRIVEN BY L´EVY PROCESSES 19 3.2.3 Stochastic Differential Equations Driven by L´evy Processes
Finally we are ready to address stochastic differential equations driven by L´evy processes.
The following results can be found in Protter (2004):
Theorem 3.4: A L´evy process is a semimartingale.
The deterministic functiong(t) =tis a semimartingale since it is the L´evy process with L´evy triplet (0,1,0). Hence any stochastic differential equation on the form
dXt=f1(Xt−)dt+f2(Xt−)dLt (3.23) is well-defined for a (m-vector) L´evy process Lt, and we can apply the results above concerning stochastic differential equations driven by semimartingales.
One important property of (3.23) is that the solution is a strong Markov process. More-over it is in fact true that if the solution of a general stochastic differential equation (3.18) is a strong Markov process, the driving noiseZ is a L´evy process (Protter 2004).
3.2.4 The Infinitesimal Generator
As noted in Protter & Talay (1997), a motivation for the analysis of equations on the form (3.22) is that it can be used to solve the Kolmogorov Backward equation. A nice expression for the generatorA of the process Xt solving (3.22) is at hand:
Theorem 3.5(Protter (2004) Exercise V-8): LetLtbe a L´evy process with L´evy measure ν. Moreover let Lt have the decomposition Lt=bt+cBt+Mt where Mt is a pure jump martingale. Then the generator of the process Xt solving (3.22) is given as
Ag(x) =∇g(x)f(x)b+1 2
Xd i,j=1
µ ∂2g
∂xi∂xj(x)
¶
(f(x)cf(x)>)ij +
Z
ν(dy)(g(x+f(x)y)−g(x)− ∇g(x)f(x)) (3.24) where g∈Cc∞ and ∇g(x) is a row vector.
No general adjoint operator is to our knowledge given in closed form. However we shall see later that it is often easy to find when a specific equation is given.
3.2.5 A Solvable Stochastic Differential Equation
Example 3.4: An important class of stochastic differential equations with applications in mathematical finance are on form (Cont & Tankov 2004):
Xt= 1 + Z t
0
XsdLs. (3.25)
Here Z is a 1-dimensional L´evy-process with L´evy triple(σ2, γ, ν). Its solution is given by the so-called stochastic exponential for rather obvious reasons:
Xt=E(Lt) = exp[Lt−σ2t/2] Y
0<s≤t
{(1−∆Ls) exp[−∆Ls]} (3.26) Notice that for a continuous process, that is with ν = 0, this equation reduces to the ordinary solution of the Black Scholes Equation. In mathematical finance, is often useful to find the stochastic differential equation with solutionYt= exp(Lt)whereLtis a given L´evy process. In the one dimensional case, this can be done following Proposition 8.22.2 in (Cont
& Tankov 2004):
Proposition 3.1: LetY be given as Yt= exp(Lt) where Lt is a L´evy process with L´evy triplet (σ2, γ, ν), then there exists a L´evy process X such that Yt = E(Xt). The L´evy triplet of X,(σX2, γX, νX) is given as:
σX =σ (3.27)
γX =γ+σ2 2 +
Z
R
{(ex−1)1[−1,1](ex−1)−x1[−1,1](x)}ν(dx) (3.28) νX(S) =
Z
R
1S(1−ex)ν(dx) for S ⊂R (3.29)
We see that L´evy stochastic calculus is somewhat more complicated than the Itˆo sto-chastic calculus. Generally we do not get nice expressions involving the driving noise only as the strong solution.