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the corresponding lifted vector field Fθ,M:eV→L(U,eV)as Fθ,M(s):=tens M(s)◦Fθ,M(s)for all s∈eV, (4.2) with tens Mdefined in Definition 3.2. Note that this definition ensures that M-solutions with respect to Fθ,Mwill not exhibit blow-ups at finite time. However, in order to ensure that the M-solutions are solutions in t... | https://arxiv.org/abs/2505.22646v1 |
We begin by showing that in this setting the theoretical expectation of the rth Picard iteration can also be expressed as a polynomial Prinθdetermined by the expected signature E[X]of the driving signal; see Remark 5.5. Then, we show that our estimator is consistent in Theorem 5.11. We note that the expected signature ... | https://arxiv.org/abs/2505.22646v1 |
A, we denote the set of words of length at most qinAby W(A,q):={(i1, . . . , iℓ)∈ Aℓ:ℓ≤q}. Example 5.3. I= (4, 0, 1 )is a word of length 3 in {0, 1, 2, 3, 4 }. In this case I−= (4, 0)and If= (1). Moreover, if J= (1, 3), then IJ= (4, 1). Theorem 5.4. For all r ,ℓ∈Z≥0, define Q(r,ℓ):= 0 if r=0orℓ=0 2r−1if r≥1andℓ... | https://arxiv.org/abs/2505.22646v1 |
exists a sequence {X(k)}k∈Nof extensions of bounded 1-variation paths such that dp(X(k),X)→0 as k→∞. For all k∈ N, let{Zθ(k,r)}r∈Z≥0be the sequence of the M(∥θ∥)-Picard iterations of the path-dependent differential equation dYt=Fθ,M(∥θ∥)(Yt)dX(k)t, and define Yθ(k,r):=πeV(Zθ(k,r))and Yθ(k,r):=Yθ(k,r)1+1for all r∈Z≥0. B... | https://arxiv.org/abs/2505.22646v1 |
0,θ0)∈Θkn⊕Θunkn. Thus, the known true parameter θkn 0∈Θknwill always be fixed. We will use the notation θ= (θkn 0,θ)∈Θ to denote the full parameter set for some θ∈Θunkn. Now pick ξ∈R+, and define Θξ:={θ∈Θunkn:N(∥θ∥)≥ξ}and Eξ:={σ∈ S :dp(X(σ),0)<ξ}. Note that for every θ∈Θξandσ∈Eξ, we have dp(X(σ),0)<ξ≤ N (∥θ∥), and thus... | https://arxiv.org/abs/2505.22646v1 |
Miranda mapping g:Q(1)→D such that for all k∈[d], g(Qk(1)+) =D+ kand g(Qk(1)−) =D− k. To simplify terminology, we will often call the pair (D,D)a Miranda domain. •For a Miranda domain (D,D), a continuous mapping f= (f1, . . . , fd):D→Rdis said to satisfy the Miranda conditions on(D,D)if fk(θ1)fk(θ2)≤0 for all θ1∈D+ k, ... | https://arxiv.org/abs/2505.22646v1 |
obtain ηk>1 4δfor all k∈[d], where δis defined in (B.10). Therefore, a sufficient condition for r0is 2ρ−r0 (1−ρ−1)β 1 p !C1 pξ<1 4δ=ε 8 (JP(θ0))−1 ∞, and thus we obtain the desired expression in (5.11). □ We now conclude this section by proving that the function Pis in fact, differentiable almost everywhere. PATH-DEP... | https://arxiv.org/abs/2505.22646v1 |
system (6.3), where we omit the term χEξ(σi) (when compared with (5.8)). This also allows us to use E XJ in the coefficents of PIkr, which is computed using the explicit formula in [ 24, Theorem 1]. Our experiments show that even with this assumption, our method is able to effectively estimate parameters. Experiment ... | https://arxiv.org/abs/2505.22646v1 |
of word set influences the performance of the ESMM, and would be an interesting avenue for future work. Figure 3 in Appendix C shows the components of allthe real solutions to the polynomial system (6.3) obtained in each trial and for each set of words. θ1θ2θ3θ4θ5 W3mean 0.3851 4.8549 1.0133 -0.7635 2.9431 std dev 2.72... | https://arxiv.org/abs/2505.22646v1 |
component of ˆθ, and whose diffusion com- ponent is close to the negative of the diffusion component of ˆθ; see Figures 2 to 4 in Appendix C. 7. Non-identifiability of Parameters from the Law The characteristic property of the signatures in Theorem 2.17 ensures that the distribution of the solution Yθof a linear signat... | https://arxiv.org/abs/2505.22646v1 |
t, dY(3) t= −Y(1,2) t−Y(2,1) t dt−1 2Y(2) t◦dW(1) t+1 2Y(1) t◦dW(2) t.(7.5) dY(0) t=dt, dY(1) t= Y(1) t+H(Yt)−Y(3) t dt+dW(1) t, dY(2) t= −Y(2) t+H(Yt)−Y(3) t dt+dW(2) t, dY(3) t= −Y(1,2) t−Y(2,1) t+H(Yt)−Y(3) t dt−1 2Y(2) t◦dW(1) t+1 2Y(1) t◦dW(2) t.(7.6) Using the Stratonovich midpoint sc... | https://arxiv.org/abs/2505.22646v1 |
into known and unknown components 20 Θkn,Θunknsubspaces consisting of known and unknown components of the parameter space 20 Θξ subset of parameters Θξ⊂Θunknallowable with respect to ξ 20 Eξ subset of samples Eξ⊂ S allowable with respect to ξ 20 χEξindicator function on the set Eξ Pr(θ),P(θ) expectation of Picard itera... | https://arxiv.org/abs/2505.22646v1 |
Equation (B.5) proves that Y=πV(Y)as desired. □ Proof of Proposition 4.1. Our first step is to define a control ωfor each X∈GΩp(U)which we will later bound. Recall that for s<t, we define ∆s,t:={(u1,u2):s≤u1<u2≤t}. We claim that ω:∆T→Rdefined by ω(s,t) =C d p(X|∆s,t,0)p is a control for the p-variation of X∈GΩp(U). Ind... | https://arxiv.org/abs/2505.22646v1 |
p (1−ρ−1)β 1 p !N,∥θ∥ ⌊p⌋ . For all θ∈Θ, since F(0,∥θ∥) = 0, there exists N>0 with F(N,∥θ∥)≤1 by continuity. Fix N0>0 and set N(µ):=sup{N∈(0,N0]:F(N,µ)≤1}andM(µ):=1+4C1 p (1−ρ−1)β 1 p !N(µ). Now, consider Xsuch that dp(X,0)≤ N(∥θ∥). Then ω(0,T)bK(M(∥θ∥),∥θ∥)⌊p⌋≤ F(N(∥θ∥),∥θ∥)≤1. In particular, the condition in (B... | https://arxiv.org/abs/2505.22646v1 |
lim θ→0R(θ) ∥θ∥∞=0 since his differentiable at 0. Thus, there exists ε0>0 such that Bε0(θ0)⊂Θξ, and for all θ= (θ1, . . . ,θd)∈eΘξwith∥θ∥∞≤ε0 2∥(JP(θ0))−1∥∞, we have −1 2<Rk(θ) ∥θ∥∞<1 2for all k∈[d]. This implies that −1 2∥θ∥∞+θk<hk(θ)−PIk(θ0)<1 2∥θ∥∞+θkfor all k∈[d]. Now let ε∈(0,ε0]. Set δ:=ε 2 (JP(θ0))−1 ∞. (B.10) M... | https://arxiv.org/abs/2505.22646v1 |
thus, the solution to (7.3). But given the bound on Xand by Proposition 4.1, the M(∥θ2∥)-solution and the solution to (7.3) are the same. So,Yis also the M(∥θ2∥)-solution to (7.3). □ PATH-DEPENDENT SDES: SOLUTIONS AND PARAMETER ESTIMATION 37 Appendix C. Figures for Section 6 This section contains the figures related to... | https://arxiv.org/abs/2505.22646v1 |
stochastic processes. J. Mach. Learn. Res. , 23(176):1–42, 2022. [11] Christa Cuchiero, Philipp Schmocker, and Josef Teichmann. Global universal approximation of functional input maps on weighted spaces. arXiv preprint arXiv:2306.03303 , 2023. [12] Christa Cuchiero, Sara Svaluto-Ferro, and Josef Teichmann. Signature SD... | https://arxiv.org/abs/2505.22646v1 |
Applications of Lie groups to differential equations , volume 107. Springer Science & Business Media, 1993. [35] Anastasia Papavasiliou and Christophe Ladroue. Parameter estimation for rough differential equations. Annals of Statistics , 39(4):2047–2073, 2011. [36] Grigorios A Pavliotis. Stochastic processes and applic... | https://arxiv.org/abs/2505.22646v1 |
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