In the paper we establish strong uniqueness of solution of a system of stochastic differential equations with random non-Lipschitz coefficients that involve both the square integrable continuous vector martingales and centered and non-centered Poisson measures.
Stochastic differential equationsnon-Lipschitz coefficientsPoisson measureuniqueness of a solution60H1060G44Introduction
Let (Ω,ℑ,ℑt,t≥0,P) be a complete probability space with filtration. Let {ζk(t),k=1,m‾} be continuous square integrable martingales defined on the stochastic basis (Ω,ℑ,ℑt,t≥0,P) with square characteristics ⟨ζk⟩ satisfying the following relations of orthogonality:
E([ζk(t)−ζk(s)][ζi(t)−ζi(s)]/ℑs)=E([⟨ζk⟩(t)−⟨ζk⟩(s)]/ℑs),k=i,0,k≠i
with probability 1 for any s<t. Let (Θ,BΘ) be a measurable space admitting some expansion Θ=Θ1∪Θ2 with Θ1∩Θ2=∅. Also, let ν([0,t],A) be a Poisson measure with the parameter tΠ(A), A∈BΘ and let ν˜([0,t],A)=ν([0,t],A)−tΠ(A). Suppose that Π(Θ1)=∞, ∫Θ1|θ|2Π(dθ)<∞, Π(Θ2)<∞ and ν([0,t],A) are ℑt-measurable. Also suppose that the family of random variables ν([t,t+h],A),h>00$]]> is independent on ℑt and that the processes {ζk(t),k=1,m‾} are jointly independent with the measure ν([0,t],A). Consider random vector-valued functions a(t,x), bk(t,x), fi(t,x,θ) with the values in Rm, x∈Rm, θ∈Θ and suppose that they are ℑt-measurable and continuous with probability 1 on the aggregate variables t, x, θ. At last, suppose that ξ(0) is a given random variable that is ℑ0-measurable.
In the present paper we generalize the result of Kulinich [4] on a strong uniqueness of the solution ξ for the following system of stochastic differential equations
dξ(t)=a(t,ξ(t))dt+∑k=1mbk(t,ξ(t))dζk(t)+∫Θ1f1(t,ξ(t−),θ)ν˜(dt,dθ)+∫Θ2f2(t,ξ(t−),θ)ν(dt,dθ),t>0.0.\end{array}\]]]>
A strong solution of equation (1) is ℑt-progressively measurable stochastic process ξ(t)=(ξi(t),i=1,m‾), which is continuous on the right and without discontinuities of the second kind and whose stochastic differential takes form (1) (see [2]).
In the paper [4] the system of stochastic differential equations (1) was considered only with f2(t,x,θ)≡0 and nonrandom coefficients under additional condition of boundedness of the functions a(t,x), bk(t,x), ∫Θ1|f1(t,x,θ)|lΠ(dθ), l=1,2. In the present paper these restrictions are not required.
The problem of a strong uniqueness of a solution of the equation (1) with Lipschitz coefficients was considered in a number of works, among which we mention only monograph [2] that contains both an extended bibliography and analysis of the state of art. The problem of a strong uniqueness of the solution in the case of non-Lipschitz coefficients was examined in a small number of papers and only for one-dimensional equations. We point out the papers [5, 6, 8], in which the new original methods of investigation of a strong uniqueness of solution for one-dimensional equations of form (1) without jumps and with ζk(t)=W(t) are proposed. Then in [3, 7, 9] these methods were extended to wider classes of one-dimensional equations of form (1) but also with ζk(t)=W(t).
The most complete study of the problem under consideration is presented in [2], but there the conditions on the smoothness of the coefficients bk(t,x) in the multidimensional case are stronger than in our paper. However, it is important to note that we can significantly weaken the conditions on the coefficients bk(t,x) of equation (1) only for a special class of equations, in particular, for the equations containing Hölder’s coefficients with the index α⩾12. For example, the coefficients b1(t,x)=(t|x1|,sint), b2(t,x)=(cost,|x2|) satisfy conditions of our theorem but were inappropriate for any previous theorems. We apply the method proposed in [8] to prove the main result.
Strong uniqueness
Now we state and prove the main result.
Let for allN>00$]]>functionsqN(r),ρN(r),r⩾0are nondecreasing and continuous and additionallyqN(r)is concave. Suppose that for allt⩽N,|x|⩽N,|y|⩽Nthe following inequalities hold true with probability 1
|a(t,x)−a(t,y)|+∫Θ1|f1(t,x,θ)−f1(t,y,θ)|Π(dθ)⩽qN(|x−y|),|bik(t,x)−bik(t,y)|⩽ρN(|xi−yi|),i=1,m‾,k=1,m‾.Also let∫0+drqN(r)=∫0+drρN2(r)=∞.Then the strong solution ξ of equation (1) is strongly unique.
It is known (see [2]) that in order to prove strong uniqueness of the solution of equation (1), it is enough to establish strong uniqueness of its solution in every random interval [τl,τl+1], where 0=τ0<τ1<τ2<⋯ are subsequent jump times of Poisson process ν([0,t],Θ2), assuming additionally that ξ(τl) is given.
Now, let ξ(t) and ξ˜(t) be two solutions of equation (1) with the same initial condition. Then for τl⩽t<τl+1 we have that ν((τl,τl+1),Θ2)=0, therefore
ξi(t)=ξi(τl)+∫τltai(s,ξ(s))ds+∑k=1m∫τltbik(s,ξ(s))dζk(s)+∫τlt∫Θ1fi1(s,ξ(s−),θ)ν˜(ds,dθ),i=1,m‾. A similar equation we have for the process ξ˜(t).
So,
Δξi(t)=∫τltΔai(s)ds+∑k=1m∫τltΔbik(s)dζk(s)+∫τlt∫Θ1Δfi1(s,θ)ν˜(ds,dθ),i=1,m‾, where
Δξi(t)=ξi(t)−ξ˜i(t),Δai(t)=ai(t,ξ(t))−ai(t,ξ˜(t)),Δbik(t)=bik(t,ξ(t))−bik(t,ξ˜(t)),Δfi1(t,θ)=fi1(t,ξ(t−),θ)−fi1(t,ξ˜(t−),θ).
Then we use the method that was firstly proposed in [8]. According to conditions (2) on the function ρN(r), we can choose a monotone sequence
1=a0>a1>⋯>an→0asn→∞a_{1}>\cdots >a_{n}\to 0\hspace{1em}\text{as}\hspace{2.5pt}n\to \infty \]]]>
such that
∫anan−1ρN−2(r)dr=n,n=1,2,…
Let us construct a sequence of twice continuously differentiable functions φn(r), r∈[0,∞) such that φn(0)=0; φn′(r)=0 for 0⩽r⩽an, the values φn′(r) lie between 0 and 1 for an<r<an−1, φn′(r)=1 for r⩾an−1; φn″(r)=0 for 0⩽r⩽an, the values φn″(r) lie between 0 and 2nρN−2(r) for an<r<an−1, φn″(r)=0 for r⩾an−1. Continuing φn(r) in a symmetric way to (−∞,0), we obtain the sequence of twice continuously differentiable functions gn(z)=φn(|z|), moreover gn(z)↑|z| as n→∞.
From equality (3) and Itô’s formula (see [2]) for τl⩽t<τl+1 we get
gn(Δξi(t))=∫τltgn′(Δξi(s))Δai(s)ds+12∑k=1m∫τltgn″(Δξi(s))[Δbik(s)]2d⟨ζk⟩(s)+∫τlt∫Θ1[gn(Δξi(s)+Δfi1(s,θ))−gn(Δξi(s))−gn′(Δξi(s))Δfi1(s,θ)]Π(dθ)ds+∑k=1m∫τltgn′(Δξi(s))Δbik(s)dζk(s)+∫τlt∫Θ1[gn(Δξi(s−)+Δfi1(s,θ))−gn(Δξi(s−))]ν˜(ds,dθ)=I1(t)+⋯+I5(t),i=1,m‾.
Let us consider the stopping times ζN=inf{t>τl:|ξ(t)|+|ξ˜(t)|>N}∧τl+1∧N\tau _{l}:\hspace{0.1667em}|\xi (t)|+|\tilde{\xi }(t)|>N\}\wedge \tau _{l+1}\wedge N$]]>. Taking into account (4) and the obvious relation gn(z)⩾0 we get the inequalities
gn(Δξi(t))χ{t<ζN}=∑k=15Ik(t)χ{t<ζN}=(∑k=15Iˆk(t))χ{t<ζN}⩽∑k=15Iˆk(t)a.s.,
where
Iˆ1(t)=∫τltgn′(Δξi(s))Δai(s)χ{s<ζN}ds,Iˆ2(t)=12∑k=1m∫τltgn″(Δξi(s))[Δbik(s)]2χ{s<ζN}d⟨ζk⟩(s),Iˆ3(t)=∫τlt∫Θ1[gn(Δξi(s)+Δfi1(s,θ))−gn(Δξi(s))−gn′(Δξi(s))Δfi1(s,θ)]χ{s<ζN}Π(dθ)ds,Iˆ4(t)=∑k=1m∫τltgn′(Δξi(s))Δbik(s)χ{s<ζN}dζk(s),Iˆ5(t)=∫τlt∫Θ1[gn(Δξi(s−)+Δfi1(s,θ))−gn(Δξi(s−))]χ{s<ζN}ν˜(ds,dθ).
Continuity of the functions qN(r), ρN(r) and standard properties of stochastic integrals lead to equalities
E(Iˆ4(t)/ℑτl)=0,E(Iˆ5(t)/ℑτl)=0
with probability 1.
According to the assumptions of the theorem, we have
|E(Iˆ1(t)/ℑτl)|⩽E(∫τltgn′(Δξi(s))|Δai(s)|χ{s<ζN}ds/ℑτl)⩽E(∫τltqN(|Δξ(s)|)χ{s<ζN}ds/ℑτl);|E(Iˆ2(t)/ℑτl)|⩽12∑k=1mE(∫τlt2nρN2(|Δξi(s)|)×χ{an<|Δξi(s)|<an−1}(Δbik(s))2χ{s<ζN}d⟨ζk⟩(s)/ℑτl)⩽1n∑k=1mE(∫τlt1ρN2(|Δξi(s)|)×χ{an<|Δξi(s)|<an−1}ρN2(|Δξi(s)|)χ{s<ζN}d⟨ζk⟩(s)/ℑτl)⩽1n∑k=1mE(∫τltχ{s<ζN}d⟨ζk⟩(s)/ℑτl)=1n∑k=1mE([⟨ζk⟩(t∧ζN)−⟨ζk⟩(τl∧ζN)]/ℑτl)→0asn→∞;|E(Iˆ3(t)/ℑτl)|⩽E(∫τlt∫Θ1[|gn′(Δξi(s)+θ˜Δfi1(s,θ))||Δfi1(s,θ)|+|gn′(Δξi(s))||Δfi1(s,θ)|]Π(dθ)χ{s<ζN}ds/ℑτl)⩽2E(∫τltqN(|Δξ(s)|)χ{s<ζN}ds/ℑτl).
Therefore
E(gn(Δξi(t))χ{t<ζN}/ℑτl)⩽3E(∫τltqN(|Δξ(s)|)χ{s<ζN}ds/ℑτl)+oP(1),
where oP(1)→0 as n→∞ in probability.
From the convergence gn(z)↑|z| as n→∞ and the previous inequality we obtain
E(|Δξi(t)|χ{t<ζN}/ℑτl)⩽3E(∫τltqN(|Δξ(s)|)χ{s<ζN}ds/ℑτl)
for all i=1,m‾. Therefore
E(|Δξ(t)|χ{t<ζN}/ℑτl)⩽CE(∫τltqN(|Δξ(s)|)χ{s<ζN}ds/ℑτl)
and
E(|Δξ(t)|χ{t<ζN}/ℑτl)⩽CE(∫τltqN(|Δξ(s)|χ{s<ζN})ds/ℑτl).
This inequality holds true if t is replaced by the stopping time ηu=u+τl. Therefore
E(|Δξ(ηu)|χ{ηu<ζN}/ℑτl)⩽CE(∫τlηuqN(|Δξ(s)|χ{s<ζN})ds/ℑτl)=CE(∫0uqN(|Δξ(ηv)|χ{ηv<ζN})dv/ℑτl).
Further we apply Jensen’s inequality for concave functions and get
E(|Δξ(ηu)|χ{ηu<ζN})⩽C∫0uEqN(|Δξ(ηv)|χ{ηv<ζN})dv⩽CqN(∫0uE(|Δξ(ηv)|χ{ηv<ζN})dv).
Thus, we have inequality
∫0uα(v)qN−1(∫0vα(z)dz)dv⩽Cu,
where α(u)=E(|Δξ(ηu)|χ{ηu<ζN}). The last inequality together with condition (2) on the function qN(u) implies the equality α(u)=0 for all u⩾0. Therefore ξ(ηu)−ξ˜(ηu)=0 for τl⩽ηu<τl+1. Since ξ(ηu)−ξ˜(ηu) has no discontinuity points of the second kind, we deduce that
supτl⩽ηu<τl+1|ξ(ηu)−ξ˜(ηu)|=supτl⩽s<τl+1|ξ(s)−ξ˜(s)|=0.
Thus, ξ(t)=ξ˜(t) for all τl⩽t<τl+1 and ξ(τl+1−)=ξ˜(τl+1−). The proof will be complete if we show that ξ(τl+1)=ξ˜(τl+1). But it is indeed the case because it follows from the construction of the solution of equation (1) that
ξ(τl+1)=ξ(τl+1−)+f2(τl+1,ξ(τl+1−),θˆl+1)=ξ˜(τl+1−)+f2(τl+1,ξ˜(τl+1−),θˆl+1)=ξ˜(τl+1),
where θˆl+1∈Θ2 is such that ν({τl+1},{θˆl+1})=1 (see also [2]: Theorem 1, §1, Chap. 4). □
If additionally to the assumptions of the theorem the following conditions hold true with probability 1:
there exists a constant C>00$]]> such that
|a(t,x)|2+∑k=1m|bk(t,x)|2+∫Θ1|f1(t,x,θ)|2Π(dθ)≤C[1+|x|2];
for arbitrary t1>00$]]>, x1∈Rmlimt→t1,x→x1∫Θ1|f1(t,x,θ)−f1(t1,x1,θ)|2Π(dθ)=0,
then equation (1) has a weak solution (see [2]: Theorem 3, §2, Chap. 4).
If the assumptions of the theorem and the conditions of the remark are satisfied, then equation (1) has a unique strong solution (see [2]: Theorem 9, §3, Chap. 6).
Conclusion
In this work we investigate the problem of strong uniqueness of a solution for a system of stochastic differential equations with random coefficients and with differentials with respect to martingales, which may be continuous with probability 1 or discontinuous with jumps of Poisson type. Substantial generalizations of previously known results are got for the systems of the special type. For example, coefficients corresponding to the differentials of continuous martingales can be Hölder with index α⩾12. Such systems are of particular interest, for instance, in connection with applications in finance for modeling instantaneous interest rate [1], in investigation of limiting behavior of unstable components of solutions of stochastic differential equations [3].
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