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MSC 60H10, 60H30, 60K30 DOI: 10.14529/mmp150408
ON EXISTENCE OF SOLUTIONS TO STOCHASTIC DIFFERENTIAL EQUATIONS WITH CURRENT VELOCITIES
S. V. Azaria, Kuban State University, Krasnodar, Russian Federation, azarinas@mail.ru, Yu.E. Gliklikh, Voronezh State University, Voronezh, Russian Federation, yeg@math.vsu.ru
The notion of mean derivatives was introduced by E. Nelson in 60-th years of XX century and at the moment there are a lot of mathematical models of physical processes constructed in terms of those derivatives. The paper is devoted to investigation of stochastic differential equations with current velocities, i.e., with Nelson's symmetric mean derivatives. Since the current velocities of stochastic processes are natural analogues of ordinary physical velocities of deterministic processes, such a research is important for investigation of models of physical processes that take into account stochastic properties. An existence of solution theorem for those equations is obtained.
Keywords: mean derivatives; equations with current velocities; existence and uniqueness of solutions
Introduction. The notion of mean derivatives was introduced by E. Nelson [1-3] for the needs of the so-called Nelson's stochastic mechanics (a version of quantum mechanics). Later a lot of applications of mean derivatives to some other branches of science were found. It should be pointed out that among Nelson's mean derivatives (forward, backward, symmetric and antisymmetric, etc.) the symmetric derivatives called current velocities, play the role of natural analogue of physical velocity of deterministic processes. That is why inestigation of equations with with current velocities is very important for stochastic models for many physical processes.
In this paper we investigate those equations and obtain an existence and uniqueness theorem for their solutions.
Some remarks on notations. In this paper we deal with equations and inclusions in the linear space Rn, for which we always use coordinate presentation of vectors and linear operators. Vectors in Rn are considered as columns. If X is such a vector, the transposed row vector is denoted by X*. Linear operators from Rn to Rn are represented as n x n matrices, the symbol * means transposition of a matrix (pass to the matrix of conjugate operator). The space of n x n matrices is denoted by L(Rn, Rn).
By S(n) we denote the linear space of symmetric n x n matrices that is a subspace in L(Rn, Rn). The symbol S+(n) denotes the set of positive definite symmetric n x n matrices that is a convex open set in S(n). Its closure, i.e., the set of positive semi-definite symmetric n x n matrices, is denoted by S+(n).
Everywhere below for a set B in Rn or in L(Rn, Rn) the notation ||B|| means sup ||y||.
v&b
For the sake of simplicity we consider equations, their solutions and other objects on a finite time interval t E [0,T],
We use Einstein's summation convention with respect to shared upper and lower indices.
1. Preliminaries on the Mean Derivatives. Consider a stochastic process £(t) in Rn,
t E [0, l], given on a certain probability space (Q, F, P) and such that £(t) is Li-random t
Every stochastic process £(t) in Rn, t E [0, l], determines three families of a-subalgebras of a-algebra F:
(i) the "past" V generated % pre-images of Borel sets in Rn by all mappings £(s) : Q ^ Rn for 0 < s < t;
(ii) the "future" Ft generated by pre-images of Borel sets in Rn by all mappings £(s) :Q ^ Rn for t < s < Z;
(iii) the "present" ("now") Ntt generated by pre-images of Borel sets in Rn by the mapping £(t).
0
For convenience we denote the conditional expectation of £(t) with respect to N by
Et ()
Ordinary ("unconditional") expectation is denoted by E.
Strictly speaking, almost surely (a.s.) the sample paths of £(t) are not differentiate for t
To avoid using the generalized functions, following Nelson (see, e.g., [1-3]) we give
Definition 1. (i) Forward mean derivative D£(t) of £(t) at time t is an Li-random, variable of the form
j £(t + At) — £(t), AtT+o^t ( At
where the limit is supposed to exists in Li(Q, F, P) and At ^ +0 means that At tends to 0 and At > 0.
(ii) Backward mean derivative DO£(t) of £(t) at t is an Li-random variable
«) = АЙ+, * () «
where the conditions and the notation are the same as in (i).
Note that mainly D£(t) = DO£(t), but if, say, £(t) a.s. has smooth sample paths, these derivatives evidently coinside.
D£(t)
DO£(t) can be represented as compositions of £(t) and Borel measurable vector fields (regressions)
Y0(t,x)= Ita E(£(t + At) - £(t) \£(t)= x)
K ' У AtT+0 At
YO(t,x) = Aim E(£(t - A - At) \£(t) = x) (3)
AtT+O At
on Rn. This means that D£(t) = YO(t,£(t)) and DO£(t) = Y°(t,£(t)).
Definition 2. The derivative DS = 2(D + DO) is called symmetric mean derivative. The derivative DA = 1 (D — DO) is called anti-symmetric mean derivative .
Consider the vector fields v^(t,x) = 1 (Y°(t,x) + YO0(i,x)^d U(t,x) = 1 (Y°(t,x) — YO(t,x)).
Definition 3. vf(t) = vf(t,£(t)) = DS£(t) is called current velocity of £(t); uf(t) = uf(t,£(t)) = Da£(t) is called osmotic velocity of £(t).
For stochastic processes the current velocity is a direct analogue of ordinary physical velocity of deterministic processes (see, e.g., [1-3,8]). The osmotic velocity measures how fast the "randomness" grows up.
Recall that Ito process is a process £(t) of the form
£(t) = £0 + J a(s)ds + J A(s)dw(s), 0 0
where a(t) is a process in Rn whose sample paths a.s. have bounded variation; A(t) is a process in L(Rn, Rn) such that for any element Aj (t) of matrix A(t) the condition P(u\ /J (Aj)2dt < m) = 1 holds; w(t) is a Wiener process in Rn; the first integral is the Lebesgue integral, the second one is Ito integral and all integrals are well-posed.
Definition 4. An Ito process £(t) is called a process of diffusion type if a(t) and A(t) are not anticipating with respect to Vf and the Wiener process w(t) is adapted to Vf. If a(t) = a(t,£(t)) and A(t) = A(t,£(t)), where a(t,x) and A(t,x) are Borel measurable mappings from [0,T] x Rn to Rn and to L(Rn, Rn); respectively, the ltd process is called a diffusion process.
In the latter case with Borel measurable a(t,x) and A(t,x) process £(t) is supposed to be a weak solution of the above equation.
Below we are dealing with smooth fields of non-degenerate linear operators A(x) : Rn ^ Rn x E Rn (i.e., (1, l)-tensor field on Rn). Let £(t) be a diffusion process in which the integrand under Ito integral is of the form A(£(t)). Then its diffusion coefficient A(x)A*(x) is a smooth field of symmetric positive definite matrices a(x) = (aij(x)) ((2, 0)-
Rn
the smooth field of converse symmetric and positive definite matrices (a j). Hence this field can be used as a new Riemannian a(■) = a jdxl ® dx^n Rn. The volume form of this metric is Aa = ^det(aij)dx1 A dx2 A ■ ■ ■ A dxn.
Denote by (t, x) the probability density of random element £(t) with respect to the volume form dt A Aa = ^Jdet(ai^j)dt A dx1 A dx2 A ■ ■ ■ A dxn on [0, T] x Rn, i.e., for every continuous bounded function f : [0,T] x Rn ^ R the relation
t t / \
J E(f (t, £(t)))dt = j I J f (t, £(t))dP J dt = f (t,x)p*(t,x)Aa\ dt
0 0 \ n /
holds.
Lemma 1. [9,10] Let £(t) satisfy the Ito equation
£(t) = £0 + J a(s, £(s))ds + J A(s,£(s))dw(s). 00
Tken C(t ) 1 (aij(t,x)) d
_U(t,x)=2 pf (t,x) dx,_^
t
t
t
t
where (a%j) is the matrix of operator AA* under the assumption that pt (t,x) is smooth and nowhere equal to zero.
Remark 1. Denote by S(x) the vector field whose coordinate presentation is djr -¿ji- One can easily derive from (4) that ut(t,x) = 1 Gradlogpt(t,x) + 2S(x) where Grad denotes
the gradient with respect to metric a(, ■). Indeed, 9x3 ^fj) ' ^ -j = 1a%jdj + 1djjdj
where aij ^ d = Grad log pP and djj d =
ox1 0 ' ox3 ox7-
Lemma 2. [3,8] For vt(t,x) and pP' (t,x) the following interrelation
dpt(t,x) Div(vt(t x)pt
dt
—DivV (t,x)p5 (t,x)), (5)
(known as the equation of continuity) takes place where Div denotes the divergence with respect to Riemannian metric •).
Following [7, 8] we introduce the differential operator D2 that differentiates an L2 random process £(t), t E [0,T] according to the rule
D£(t) = 1m Щ(£t + At) — £(У + At) —£(t))'). (6)
2S w At^+o tK At
с
where (£(t + At) — £(t)) is considered as a column vector (vector in Rra), (£(t + At) — £(t))* is a row vector (transposed, or conjugate vector) and the limit is supposed to exists in Li(Q, F, P). We emphasize that the matrix product of a column on the left and a row on the right is a matrix so that D2£(t) is a symmetric positive semi-definite matrix function on [0,T] x We call D2 the quadratic mean derivative.
Theorem 1. [7,8] For an ltd diffusion type process £(t) the forward mean derivative D£(t) exists and equals E5(a(t)). In particular, if £(t) a diffusion process, D£(t) = a(t, £(t)).
(t) Then D2£(t) = El [a(t)] where
a(t) = AA *. In particular, if £(t) is a diffusion process, D2£ (t) = a(t, £(t)) where a = AA * is the diffusion coefficient.
Lemma 3. [7,8] Let a(t,x) be a jointly continuous (measurable, smooth) mapping from [0,T] x Rn to S+ (n). Then there exists a jointly continuous (measurable, smooth, respectively) mapping A(t, x) from [0, T] x Rn to L(Rn, Rn) such that for allt E R, x E Rn the equality A(t,x)A*(t,x) = a(t,x) holds.
2. Main Results. As it is mentioned in Section 1, the meaning of current velocities is analogous to that of ordinary velocity for a non-random process. Thus the case of equations with current velocities is probably the most natural from the physical point of view. The system of the form
{ Ds£(t) = v(t, £(t))
X D2£(t) = a(t, £(t)) { i)
is called a first order differential equation with current velocities.
Definition 5. We say that (7) has a solution on the interval [0,T] if there exists a probability space (Q,F, P) and a process £(t) given on (Q, F, P) for t E [0,T], that satisfies (7).
Theorem 3. Let v : [0,T] x Rn ^ Rn be smooth and a : Rn ^ S+(n) be smooth and autonomous (so, it determines the Riemannian metric a(■, ■) on Rn; introduced in Section 1). Let them also satisfy the estimates
\\v(t,x)\\ < K(1 + \\x\\), (8)
tr a(x) <K(1 + \\x\\2) (9)
and for all indices ij let the elements of matrix a(x) satisfy the inequality
daij
\d^j(x)\ < K(1 + \\x\\) (10)
for some K > 0. Let £0 be a random element with values in Rn whose probability density p0 with respect to the volume form Aa of a(■, ■) on Rn (see Section 1), is smooth and nowhere
£(0) = £0
well posed on the entire interval t E [0,T] and unique as a diffusion process.
Proof. Since v(t, x) is smooth and estimate (8) is fulfilled, its flow gt is well posed on the entire interval t E [0,T], By gt(x) we denote the orbit of the flow (i.e., the solution of equation x'(t) = v(t, x)) with the initial condition g0(x) = x. Since v(t,x) is smooth, its flow is also smooth.
Continuity equation (5) obviously can be transformed into the form
dp = -a(v, Grad p) — p Div v. (11)
Suppose that p(t,x) nowhere in [0,T] x Rn equals zero. Then we can divide (11) by p so that it is transformed into the equation
dp
— = —a(v, Grad p) — Div v (12)
where p = log p. Introduce p0 = log p0. Show that the solution of (12) with initial condition p(0) = p0 is described by the formula p(t,x) = p0(g-t (x)) — J'0(Div v)(s ,gs(g-t(x)) ds.
[0 , T ] x R n p0
surface (0,Rn). Consider the vector field (1,v(t,x)) on [0,T] x Rn. The orbits of its flow gtl starting at the points of (0,Rn), have the form gt(0,x) = (t,gt(x)) and the flow is smooth as well as gt. Also introduce on [0,T] x Rn the Riemannian metric &,(■, ■) by the formula a((X1,Y1), (X2,Y2)) = X1X2 + a(Y1,Y2). Notice that for any (t,x) the point g-t(t, x) belongs to (0, Rn) where the function p0 is given. Thus on the
(1, v)p(t, x) p(t, x) (1, v)
equals —Div v(t, x). And on the other hand one can easily calculate that (1,v)p(t,x) = |p(t,x) + a(v(t,x),Grad p(t,x)). Thus (12) is satisfied.
Notice that p = ep is indeed nowhere zero and so our arguments are well-posed. From the construction it follows that for a given field a and initial density p0 satisfying the hypothesis, the densities of constructed type and the smooth vector fields having
p(t, x)
for the solution of (7), we can find also the osmotic velocity U (t,x) by formula (4), i.e., u = 2Grad p + |S = Gradlog^fp + |S. Note that u is uniquely determined by p and a and so the forward mean derivative of the solution is also uniquely determined by the formula a(t,x) = v(t,x) + |Grad p + iS = Gradlog^fp + iS. From Lemma 3 and from the hypothesis of Theorem it follows that there exists smooth A(x) such that A(x)A*(x) = a(x) and the relation ||A(t,x)|| < K(1 + ||x||) holds. Then from the general theory of equations with forward mean derivatives it follows that C(t) having the density p(t, x)
t t C(t) = C0 + j a(s, C(s))ds + J A(s, C(s))dw(s). (13)
0 0
From the hypothesis and from results of [4] it follows that (13) has has a unique strong solution C(t) with initial density p0 well-posed for t E [0,T], Thus, by Theorem 2 D2C(t) = a(C(t)). The fact that DSC(t) = v(t, C(t)) follows from the construction.
□
Lemma 4. Let a(x), p(t,x) and Ла be the same as in Theorem 3. Let also the vector field v from Theorem 3 be autonomous. Then the flow gt of vector field (1,v(x)) on [0, T] x Rn preserves the volume form p(t, x)dt Л Ла (i.e., g*(p(t, x)dt Л Aa) = p0(x)dt Л Ла where g* is the pull back) and so for any measurable set Q С Rn and for any t E [0,T]
j p0(x)Aa = j p(t,x)Aa.
Q gt(Q)
Proof. It is enough to show that L(i,v)(p(t,x)dt Л Aa) = 0 where L(i,v) is the Lie derivative (1, v)
L(i,v)(p(t, x)dt Л Aa) = (L(i,v)p(t, x))dt Л Aa + p(t, x)(L(i,v)dt Л Aa).
For a function the Lie derivative coincides with the derivative in direction of vector field, hence L(i,v)p(t,x) = 'dp+a(v, Grad p) (see the proof of Theorem 3) and so (L(i,v)p(t,x))dtЛ Aa = (dp + a(v, Gradp))dtЛAa. Since neither the form Aa nor the vector filed v(x) depend on t, L(i,v)dt Л Aa = dt Л (LvAa) = Divv ( dt Л Aa) as the Lie derivative along v of the volume form Aa equals (Divv)Aa (see, e.g., [6]). Taking into account (11), we obtain L(iv) (p(t,x)dt Л Aa) = 0. □
References
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This research is supported in part by Russian Scientific Foundation (RSCF) Grant 14-21-00066, being carried out by Voronezh State University.
Received May 21, 2015
УДК 517.9+519.216.2 DOI: 10.14529/mmpl50408
О СУЩЕСТВОВАНИИ РЕШЕНИЙ СТОХАСТИЧЕСКИХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ТЕКУЩИМИ СКОРОСТЯМИ
C.B. Азарина, Ю.Е. Гликлих
Понятия производных в среднем были введены Э. Нельсоном в 60 —X годдх XX века, и в настоящий момент имеется много математических моделей физических процессов, построенных в терминах этих производных. Статья посвящена исследованию стохастических дифференциальных уравнений с текущими скоростями, т.е., с нельсоновскими симметрическими производными в среднем. Поскольку текущие скорости случайных процессов являются естественными аналогами обычных физических скоростей детерминированных процессов, изучение таких уравнений важно для исследования моделей физических процессов, которые учитывают стохастические свойства. Получена теорема существования решения для указанного типа уравнений.
Ключевые слова: производные в среднем; уравнения с текущими скоростями; существование и единственность решений.
Светлана Владимировна Азарина, КсШДИДсХТ физико-математических наук, доцент, Кубанский государственный университет (г. Краснодар, Российская Федерация), azarinas@mail.ru.
Юрий Е ВГбНЬбВИЧ Гликлих, доктор физико-математических наук, профессор, кафедра «Алгебра и топологические методы анализа:», Воронежский государственный университет (г. Воронеж, Российская Федерация), yeg@math.vsu.ru.
Поступила в редакцию 21 мая 2015 г.
