N. V. Krylov. Controlled diffusion processes. Springer-Verlag, New York, NY, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....trajectories is not necessary. The specification of K is not required by our algorithms due to stationarity, which will be discussed in Section 2.3.4. The formalism could also be defined in sufficient generality without discretizing time, and consequently defining controlled diffusions [104]; the definitions that would follow require the use of stochastic differential equations and measure theoretic concepts. For our context, a discretized representation of time facilitates the development of the numerical computation approach. Furthermore, actual robot systems will be limited to ....

N. V. Krylov. Controlled diffusion processes. Springer-Verlag, Berlin, 1980.

....proof we use the following results. Proposition 24 For all u 2 A t; m 0, we have E t;x [kX k m 1 ] Bm (1 jxj m ) where the norm k k is the sup norm in C( t; T ] and the constant Bm depends only on T and the constant C in assumptions (16) Proof. See [14] pagg. 398 399 and [19] pag. 85. 2 From this result, we get that for M 0, PfkX k 1 Mg 1 M E [kX k 1 ] B 1 M (1 jxj) Besides, by de nition of V , if L and have polynomial growth, then jV (t; x)j M(T t) 1 jxj k ) where the constant M depends only on C 1 , C 2 and C 3 in assumptions (19) and ....

N. V. Krylov, Controlled diffusion processes, Springer, 1980

.... matrices u 2 K cd such that 1=k det(uu T ) and let I be the set of all stochastic integrals I = f Z t 0 y( s)dw( s) y 2 A( Omega ; J)g: The preceding corollary shows that I is contained in a neocompact subset of M( Omega ; R c ) We apply the following inequality of Krylov [16]. 53 Lemma 11.4 There is a constant b depending only on k; c, and d such that for each x 2 I and Borel function h : 0; 1] Theta R c R, E[ Z 1 0 jh(t; x( t) jdt] bkhk c 1 :2 (8) Definition 11.5 Let H be the set of all adapted processes x 2 L 0( Omega ; C( 0; 1] R c ) such that ....

....of our method, we give a short proof of an existence theorem from [13] see also [1] and [5] for differential equations where the coefficient matrix g(s; x) does not depend on but is only measurable rather than continuous in x. The analogous weak existence theorem was proved earlier by Krylov [16] using the same inequality which we used in Theorem 11.8. We assume that g(s; x) is nondegenerate, that is, g maps [0; 1] Theta R c into the set J = fy 2 K cd : det(yy T ) 1=kg: Let w be a d dimensional Brownian motion on Omega Gamma and recall that I = f Z t 0 y( s)dw( s) y 2 ....

N. V. Krylov. Controlled Diffusion Processes. Springer-Verlag 1980.

....stated below in a simple form. It will not be reproved here even though there seems to be no quotable reference for infinite dimensional problems. However since the value function is continuous and we deal with relaxed controls the proof follows standard arguments, see for instance [32] see also [21]) Proposition 5.3. For every 0#t# #T and x # X 0 we have v(t, x) inf : #A# t, T E T t f (Y(s; t, x, s) ds g(Y(T; t, x, inf : #A# t, E t f (Y(s; t, x, s) ds v( Y( t, x, 26 GOZZI AND S# WIE#CH File: DISTL1 356227 . By:GC . ....

N. V. Krylov, Controlled Diffusion Processes," Springer-Verlag, New York, 1980.

.... t#s (u)v ss (u# S(u) Since Lv 0, g (S t#s (T ) g (S t#s (T ) v (T#S t#s (T ) v(t# s) Z T t Lv(u# S t#s (u) du v s (u# S t#s (u) dS t#s (u) v(t# s) Z T t Y ff#fl t#y (u)dS t#s (u)# 7 in the last step we applied the generalized Ito s formula to v s 2 W 1#2 (see Krylov (1980), Theorem 1onp122forIto s formula with generalized derivatives. By Assumption 4.3, ff# fl) 2 D t . Furthermore, since v solves the variational inequality (4.3) fl(u) for all u 2 [t# T ] By Remark 4.4, v(u# S t#s (u) X ff#fl t#x#s#y (u) 0 with x = v(t# s) Hence (y# ff# fl) 2 A t#s ....

Krylov, N.V. (1980), Controlled Diffusion Processes, Springer-Verlag, New York Heidelberg Berlin.

.... (80) Proof: Since Delta is Lipschitz bounded and linearly bounded, for every x 0 2 K n ; T 0, there exists a unique solution x Delta ( Delta) x Delta ( Delta; x 0 ) 2 L 2 w ( 0; T ] L 2( Omega ; K n ) of (79) satisfying x Delta (0) x 0 with bounded second moments [19]. x Delta ( Delta) is a continuous nonanticipative stochastic process on R satisfying the Ito integral equation x Delta (t) x 0 Z t 0 (Ax Delta (s) B Delta(s; Cx Delta (s) ds Z t 0 [A 0 x Delta (s) B 0 Delta(s; Cx Delta (s) d w 1 (s) w 2 (s) # ; t 0: So x Delta ....

N. V. Krylov. Controlled Diffusion Processes. Springer Verlag 1980.

....the following Bellman principle holds: V (x; t) inf ( i ; i ) i1 E x;t Z t f(X(s) s)ds X i1 c( i )1 f i g V (X( 15) for all (F t ; t 0) stopping times with t T . Such a principle plays an important role in the control of Markov diffusion processes (cf. 8] and [15]) and is made use of by Korn in [13, Theorem 4.6] to prove that the value function of an infinite horizon impulse control problem is a viscosity solution of the relevant qvi. Theorem 3 Suppose (15) holds. The value function V is then a viscosity solution of (6) RISK SENSITIVE IMPULSE CONTROL 9 ....

N. Krylov, Controlled Diffusion Processes, Springer-Verlag, Berlin, 1980.

....property of flows have been discussed by e.g. Kunita [15] Carverhill and Elworthy [4] See Taniguchi [22] for discussions on the strong completeness of a stochastic dynamical system on an open set of R n . For discussions of higher derivatives of solution flows on R n , see Krylov [14] and Norris [20] On a compact manifold, a SDE with C 2 coefficients is strongly complete. In fact the solution flow is C r Gamma1 if the coefficients are C r . Moreover the flow consists of diffeomorphisms. See Kunita [15] Elworthy [9] and Carverhill and Elworthy [4] For discussions in ....

....k(k may depend on c) So sup x2K E e 6p 2 R t 0 c[1 ln(1 jxs j 2 ) ds = sup x2K 1 t Z t 0 e 6p 2 ct (1 jx 0 j 2 )e ks ds 1: The strong completeness follows from theorem 5.1, using (24) and the assumptions on rX and rA. For related estimates on Ejx t j p , see [14]. Note that there is a stochastically complete SDE on R 2 with jrX(x)j jxj but which is not strongly complete: let A j 0, and X(x; y) y 0 0 x 2 2 . See Kunita[16] A different choice of the function f in theorem 5.1 leads to an improvement of a theorem of Taniguchi [22] ....

N. Krylov. Controlled diffusion processes. Springer-Verlag, 1980.

....3J ffi; 0; x; y; Z ffi; Delta) q 3J ffi; 0; x; y; Z ffi; Delta) 3 p 3 : To conclude this section, we present another property of the approximate value function e V ffi; which will be useful in the next section. The following definition is standard (see, e.g. [12] or [10] Definition 4.6. A function : lR n lR is said to be semi concave if there exists a constant C 0, such that the function Phi(x) j (x) Gamma Cjxj 2 is concave on lR n , i.e. Phi(x (1 Gamma )x) Phi(x) 1 Gamma ) Phi(x) 8 2 [0; 1] x; x 2 lR n : A family of ....

....(H3) Functions b, oe, h and g are differentiable in (x; y) with the derivatives being uniformly Lipschitz continuous in (x; y) 2 lR n Theta lR m , uniformly in (t; z) 2 [0; T ] Theta lR m Thetad . The following property of the value functions is a simple modification of those in [10] or [12], we omit the proof here. Proposition 4.7. Let (H1) and (H3) hold. Then e V ffi; s; x; y) is semi concave uniformly in s 2 [0; T ] ffi 2 (0; 1] and 2 [0; 1] In particular, there exists a constant C 0, such that (4:25) Delta y e V ffi; s; x; y) C; 8(s; x; y) 2 [0; T ] Theta lR n ....

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N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, New York, 1980.

....converges to an approximation of some generalized solution (among an infinity) of the HJB equation, thus possibly failing to find the desired value function. Section 2 presents the architecture of the network and the method used ; section 3 derives the gradient descent updating rules ; section 4 proposes a numerical simulation for a highly non linear two dimensional control problem : the Car on the Hill and compares the results to an almost optimal solution obtained with discretization techniques ; and section 5 illustrates the problems encountered on a simple one dimensional example. ....

....factor, we are encouraged to get to the goal as quickly as possible. Figure 3 shows the value function VW obtained by a neural network with 200 hidden units with a learning rate ff = 10 Gamma5 (the fraction of the training points chosen to be on the boundaries is 0:5) As a comparison, figure 4 shows the (almost optimal) value function obtained by discretization methods (based Goal Thrust Gravitation Resistance : Reinforcement R= 1 R= 1 for null velocity R= 1 for max. velocity Figure 2: The Car on the Hill control problem. Figure 3: Value function VW obtained by the neural network. ....

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N.V. Krylov. Controlled Diffusion Processes. Springer-Verlag, New York, 1980.

....a classical solution v of (6.7) On the other hand, this v is the value function of the optimal stochastic control problem similar to (5.7) 5.8) in which a term p =2 dW 0 r is added in the second equation of (5.7) with W 0 being another Brownian motion independent of W . Thus, by [20], we can find a continuous function K(t; x; y) 0, independent of 0, such that (6:9) v yy (t; x; y) K(t; x; y) 8(t; x; y) 2 [0; 1) Theta lR 2 ; 0: Now, we set (6:10) w (t; x) v (t; x; t; x) t; x) 2 [0; 1) Theta lR: Then, similar to (5.22) we have (6:11) w t ....

N. V. Krylov,Controlled Diffusion Processes, Springer-Verlag, 1980.

....(3.3) E sup 0sh fi fi X t;x s Gamma x fi fi k C(1 jxj k )h k 2 (3.4) E fi fi X t;x Gamma X t;y fi fi k Cjx Gamma yj 2 (3. 5) Remark Estimates of the moments for stochastic differential equations are generally proved for deterministic time (see e.g. Krylov 1980 [18], GihmanSkorohod [12] Actually, these results can be generalized for any stopping times, essentially thanks to the optional sampling theorem. Note also that in the diffusion case, estimates of Lemma 3.1 are valid for all orders k, while it is generally not true in the jump diffusion case ....

....if ff ffl t;x;ff for all ff 2 U , we have: v(t; x) sup ff:2U E Z 0 e Gammacs f(s t; X t;x s ; ff s )ds e Gammac v(t ; X t;x ) we omit the dependence of ; ffl in t; x; ff: Remarks 1. Proposition 3.1 is a consequence of Proposition 3. 2 as observed in Krylov ([18] p.135) 2. When the control set U is reduced to a point ff 0 , i.e. there is no control on the process X , we have the well known result that for ffl = 0, 0 is an optimal stopping time for the problem and that f R s 0 e Gammacu f(u t; X t;x u ; ff 0 )du e Gammacs v(s t; X t;x ....

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N.V. Krylov. Controlled Diffusion Processes. Berlin: Springer Verlag, 1980.

.... the existence of value functions which in a sense is opposite to that of [7] We start with solutions of the upper and lower Bellman Isaacs equations which exist by the general theory and prove that they must satisfy certain optimality inequalities (see [18] for the deterministic case and also [6] [16], 17] for the case of stochastic control) which in turn yield that solutions are equal to the value functions. These so called suband superoptimality inequalities of dynamic programming are interesting for their own. The proofs presented here use some ideas from [21] and the proof of dynamic ....

.... W independent of F t and consider a new n 1 n dimensional Wiener process W = W; W ) defined on a product space. W is progressively measurable with respect to a new F t into which F t embeds naturally, and therefore W is also an F t Brownian motion. We refer the reader to [16] and [6] for more on the construction. We will be using P to denote probability on a new space. For fl 0 we define oe fl (x; y; z) to be an n Theta (n 1 n) matrix whose first n 1 columns form the matrix oe(x; y; z) and columns n 1 1; n 1 n form a matrix flI. Matrices oe fl give ....

N.V. Krylov, "Controlled Diffusion Processes", Springer-Verlag, New York, 1980.

....control problem is singular (due to the non compactness of the set of controls) and it is well known that there are many examples where the Bellman equation then fails to hold. In general, that is also going to be the case in this paper. We could have used the normalized Bellman equation as in Krylov (1980) which involves stronger conditions on the model in order to define generalized derivatives of the value function V . The main advantage of the viscosity approach is that it requires weaker conditions on the regularity of the value function V . In fact, we will show that the characterization of ....

Krylov, N.V. (1980) Controlled Diffusion Processes, Springer-Verlag, Berlin.

.... through different versions of the principle of smooth fit [2, 34, 7] or by the compactness arguments 11 for F t measurability of (X Theta Y ) it is required only that X 2 B( 0; t] Theta X ) and Y 2 B(U) 12 based on the first three logical choices in comparison of Sigma and Omega 0 0 [18] in general, is problem specific. This leads to the situation when the quality of the mathematical model is completely defined by its consistency to the real world phenomenon. ffl On the other hand, deterministic processes may not necessarily be convection dominated or purely autonomous (or such ....

Krylov, N.V., Controlled Diffusion Processes, Springer-Verlag, 1980.

....2 WSS (S; t) The final value problem (3.2) is well posed because the partial differential equation is uniformly parabolic. This follows from the properties of Phi listed in Lemma 1. The proof of this Proposition follows the standard procedure for verification theorems in Control Theory (Krylov (1980); Fleming and Soner (1992) It is given in Appendix A. 18 Subscripts indicate partial derivatives; e.g. W t = W= t, etc. W (S; T 0) represents the value of W for t infinitesimally larger than T . This notation is used to be consistent with the way in which the final conditions corresponding ....

Krylov, N. V. (1980). Controlled Diffusion Processes, Springer-Verlag, New York.

.... S(u) t u T; so that, since Lv 0, g (S t;s (T ) g (S t;s (T ) v (T; S t;s (T ) v(t; s) Z T t Lv(u; S t;s (u) du v s (u; S t;s (u) dS t;s (u) v(t; s) Z T t Y ff;fl t;y (u)dS t;s (u) in the last step we applied the generalized Ito s formula to v s 2 W 1;2 (see Krylov (1980), Theorem 1 on p122 for Ito s formula with generalized derivatives. By Assumption 3.2, ff; fl) 2 D t . Furthermore, since v solve the variational inequality (3.2) fl(u) L for all u 2 [t; T ] By Remark 3.3, v(u; S t;s (u) X ff;fl t;x;s;y (u) 0 with x = v(t; s) Hence (y; ff; fl) 2 A ....

Krylov, N.V. (1980), Controlled Diffusion Processes, Springer-Verlag, New York Heidelberg Berlin.

....Bellman equation. Our control problem is singular (due to the non compactness of the set of controls) and it is well known that there are examples where the Bellman equation fails to hold. In general, that is also going to be the case here. We could have used the normalized Bellman equation as in Krylov (1980) which involves stronger conditions on the model in order to define generalized derivatives of the value function V . The main advantage of the viscosity approach is that it requires weaker conditions on the regularity of the value function V . In fact, here, where we work directly with the ....

Krylov, N.V. (1980) Controlled Diffusion Processes, Springer-Verlag, Berlin.

....uN Ax; DuN GH (x; DuN ; D 2 uN ) 0 in XN : 5.19) Equation (5.19) is the one used in the proof of Theorem 3.8 and it is easy to see that it is the equation in XN corresponding to the control problem with evolution given by (5. 17) Therefore, by the finite dimensional theory (see [24, 35, 38] for results and techniques that adapt to our situation to obtain the dynamic programming principle and Theorem 3.8) the function uN (y 0 ) inf ff2U ad (0; 1;U) IE Z 1 0 e Gammat L(A fi 2 y N (t; y 0 ; ff) ff(t) dt (5.20) is the unique viscosity solution of (5.19) in XN and the ....

N.V. KRYLOV, Controlled Diffusion Processes, Springer-Verlag, New York, 1980.

....in ff 2 . It is readily seen that under the assumption (A6) for each ff 2 and any T 0, the adapted solution (X(ff) Y (ff) Z(ff) of (5.1) exists and is unique on [0; T ] Furthermore, let us introduce the notion of L (resp. L(B) continuity and differentiability of N. V. Krylov (cf. [4]) Definition 5.1. An lR valued process f t (ff) t 0g with parameter ff 2 is called L (resp. L(B) continuous at ff 0 2 if (5:2) lim ff ff0 E ( Z T 0 j( t (ff) Gamma t (ff 0 )j 2 dt ) 0: resp. 5:3) lim ff ff0 E ae sup 0tT j t (ff) Gamma t (ff 0 )j 2 oe = ....

.... oe(ff; t; x) oe(ff; t; x; t; x; ff) Let us denote the solution of (5. 7) by X(ff) The assumption of the theorem and the results from the last two steps show that both b and oe are continuous in ff at ff 0 for fixed (t; x) 2 [0; T ] Hence by a continuous dependence theorem (cf. [4]) we have (5:9) L(B) lim ff ff0 X t (ff) X t (ff 0 ) Finally, recall from Theorem 4.1 that the adapted solution of (5.1) must have the form (5:10) Y t (ff) t; X t (ff) ff) Z t (ff) z(t; X t (ff) t; X t (ff) ff) x (t; X t (ff) ff) ff) the conclusion follows immediately from ....

N. V. Krylov, Controlled Diffusion Processes, Springer-Verlag, New York-HeidelergBerlin (1980).

.... 1. In this regime oe 2 ( Gamma) is positive. It is well known that the finalvalue problem (9) 10) admits a unique solution V (S; t) for any given final payoff function f(S) Moreover, the solution is twice continuously differentiable in S and once continuously differentiable in t for t T [10]. Thus, the assumptions of the Proposition are satisfied and a replicating strategy exists for contingent claims with arbitrary payoffs. Notice that V (S; t) is an increasing function of the Leland number A, as one might expect 6 : trading more frequently reduces risk but increases the ....

....strategy for replication which is asymptotically riskless, in the sense that the risk due to hedge slippage diminishes with the size of the rehedging interval and can be made, at least in theory, arbitrarily small. 6 This follows from the Maximum Principle satisfied by equation (9) for A 1 [10]. spread is small or if ffit is large 7 . The case A 1. For convex payoff functions (put, call, etc. V (S; t) reduces to Leland s pricing formula. Notice that the modified volatility oe p 1 A is always positive and hence Leland s formula applies for arbitrary values of A. The situation ....

N. V. Krylov, "Controlled Diffusion Processes", Springer-Verlag, New York, 1980.

....respectively on Omega Theta A and on Precise assumptions on the data will be given later on. We refer the reader interested in stochastic control problems to A. Bensoussan [8] A. Bensoussan and J.L Lions[9, 10] where classical PDE approaches are described and to N. El Karoui[12] N. V Krylov[19] and E.D Sontag[25] where these problems are considered from a probabilistic point of view. The more recent approach by viscosity solutions was first introduced in P.L Lions[20, 21, 22] and is presented in the book of W.H Fleming and H.M Soner[13] According to optimal control theory, it is ....

Krylov N.V: Controlled Diffusion Processes. Springer-Verlag, 1980.

....existence theorem. This is the case of stochastic differential equations where the coefficient is measurable rather than continuous in x, but the determinant of the coefficient is bounded away from zero. This result is from [5] and is an improvement of a weak existence theorem of Krylov [7]. The present proof uses some neocontinuity results from [3] Let us choose a uniform bound k 0 once and for all, and let J be the compact set of all d Theta d matrices A such that the entries of A are bounded by k and det(AA T ) 1=k. We collect the needed facts in a lemma which we state ....

Krylov, N.V., (1980) Controlled Diffusion Processes, Springer-Verlag.

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N. V. Krylov. Controlled diffusion processes. Springer-Verlag, New York, NY, 1980.

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N. V. Krylov. Controlled diffusion processes. SpringerVerlag, New York, NY, 1980.

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N. V. Krylov. Controlled Diffusion Processes. Springer Verlag 1980.

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Krylov, N.V., Controlled diffusion processes, Nauka, Moscow, 1977.

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N. V. Krylov. Controlled Diffusion Processes. Springer Verlag 1980.

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Krylov N.V.: Controlled Diffusion Processes, Springer-Verlag, New York etc., 1980.

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Krylov, N.V. 1980.Controlled diffusion processes. Springer, New York.

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