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71a05e63308e690a0ab44156e0b416a06d566797 | subsection | 4 | 16 | The upper bound | In this section, we assume that t is an integer at least 4 and we reserve \widehat{G} for the graph obtained from a graph G in K_{2,t}-bootstrap process. We will obtain an upper bound on p_c(n; K_{2,t}). More precisely, we will establish thatp_c(n; K_{2,t})=\mathrm {O}\hspace{-2.84526pt}\left(n^{-\tfrac{1}{\eta (t)}}\r... | {
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"raw": "B. Bollobás, Random graphs, in: Combinatorics (Swansea, 1981), London Math. Soc. Lecture Note Ser., vol. 52, Cambridge University Press, Cambridge-New York, 1981, pp.... | 1806.10425 | On $K_{2,t}$-bootstrap percolation | [
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b799137730ce60d3da10a6c6eebc68ff489dc26a | subsection | 5 | 16 | The upper bound | Note that \text{\bf [}A\text{\bf ]} and \text{\bf [}B\text{\bf ]} are not necessary distinct. We show that V(G)=\text{\bf [}A\text{\bf ]}\cup \text{\bf [}B\text{\bf ]} which implies the assertion of the lemma. By contradiction, suppose that V(G)\ne \text{\bf [}A\text{\bf ]}\cup \text{\bf [}B\text{\bf ]}. As G is connec... | {
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8dcbc41097498cefd16e1a22edfdd21fa5c2ce69 | subsection | 6 | 16 | The upper bound | Assume that H is a subgraph of G with minimum possible number of vertices satisfying d(H)=m. We need to prove the following facts about H.Fact 1. The minimum degree of H is 2.Since t\geqslant 4 and G contains a copy of K_{2,t-1}, we find that m>1. For each vertex v\in V(H), it follows from d(H-v)\leqslant d(H) that \de... | {
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"M. R. Bidgoli",
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fc86b87fd4ceb9c14884234112d0715492d4dcf2 | subsection | 7 | 16 | The upper bound | LettingA=\bigcup _{i=1}^rN_G[u_i], \, \, B=\bigcup _{i=1}^sN_G[v_i] \, \, \text{ and } \, \, C=\bigcup _{i=1}^{t-2}N_G[w_i],where r, s are as defined in Definition REF , V(H) is equal to one of the subsets\lbrace u\rbrace \cup A, \lbrace v\rbrace \cup B, \lbrace w\rbrace \cup C, \lbrace v\rbrace \cup A \cup B, \lbrace ... | {
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4f92490aba92ec2651c64fc05064d8a6c1929f5d | subsection | 8 | 16 | The upper bound | Therefore, for any k, |N_{\widehat{H}}(w) \cap N_{\widehat{H}}(w_k)|\geqslant t-1 which implies that N_{\widehat{H}}(w)\setminus \lbrace w_k\rbrace =N_{\widehat{H}}(w_k)\setminus \lbrace w\rbrace by Lemma REF . This shows that \widehat{H}, and in turn \widehat{G}, contains a copy of K_{t-1, t-1}. Since p\gg \log n/n, G... | {
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da2e4cb14dc4aefb4be4e2ae22590782949dbf56 | subsection | 9 | 16 | The upper bound | At the beginning of step i, we set F_i=G[V(L)], {A}_i=\lbrace A\rbrace , {B}_i=B, \ell _i=\ell ^{\prime }_i=0.If there exist two adjacent vertices u, v\in V(H_i)\setminus V(F_i) such that N_G(u)\cap A\ne \varnothing and N_G(v)\cap B\ne \varnothing , then we do the following: First choose a vertex w\in N_G(v)\cap B. The... | {
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"M. R. Bidgoli",
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5f289c3d7295e914be87f671ad100ac2cf227a08 | subsection | 10 | 16 | The upper bound | Since any pair in {P}={\bigcup }_{i\geqslant 0}{A}_i is an independent set in G^{\prime } by Fact 1, the claim concludes that G does not percolate in K_{2,4}-bootstrap process.In order to prove the claim, it is enough to show that there is no pair \lbrace x, y\rbrace \notin {P} with |N_{G^{\prime }}(x)\cap N_{G^{\prime... | {
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"M. R. Bidgoli",
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07d062c81f4bee64affef8c52e1e8c587126a661 | subsection | 11 | 16 | The upper bound | Now, in both cases \alpha =1 and \alpha =2, the structure of Z forces F to be updated to Z during the procedure, a contradiction.We next assume that S\subseteq V(F). From our procedure and Fact 1, we observe that N_F(v)\in {A} for any v\in {B}. This yields S\cap {B}=\varnothing . Hence, there are A_1, A_2, A_3, A_4\in ... | {
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"M. R. Bidgoli",
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46190aa5ef70ff6512c3b707df0a5affd1a6b456 | subsection | 12 | 16 | The lower bound | In this section, we give a lower bound on p_c(n; K_{2,t}). In , Balogh, Bollobás and Morris provided a lower bound on p_c(n; H) for any H. According to their result, p_c(n; K_{2,t})=\Omega (n^{-(t+1)/(2t-2)}). An improvement is given in the following theorem.Theorem 3.1
For any fixed integer t\geqslant 4,p_c(n; K_{2,t... | {
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75d4061bf480b215407f6f85d706ba647988829f | subsection | 13 | 16 | The lower bound | Further, fix a subset S_2\subseteq N_{G^{\prime }}(x)\cap N_{G^{\prime }}(y) such that |S_2|\in \lbrace t-1, t\rbrace and q_i=|\lbrace a_{i1},a_{i2}\rbrace \cap S_2|\in \lbrace 0, 2\rbrace for any i. Put S=S_1\cup S_2 and k=|S|. Assume that&\alpha =|\lbrace i \, | \, p_i=1\rbrace |, \\
&\beta =|\lbrace i \, | \, q_i=2\... | {
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} | 1806.10425 | On $K_{2,t}$-bootstrap percolation | [
"M. R. Bidgoli",
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57c25beca7f0ad98dd56e7237a36a1bf049cc2a7 | subsection | 14 | 16 | The lower bound | It follows from d(H)<(2t-3)/t thatt(\alpha +\beta -\gamma +2\lambda +\mu +2\nu -4)<3(\beta -\gamma +\lambda +\mu +\nu -k),which can be rewritten as(t-3)\big ((\alpha +\beta +\gamma -1)+\mu \big )+(2t-3)\big ((\nu -\gamma )+\lambda \big )+3\big (\alpha +\gamma +\big (k-(t+1)\big )\big )<0.We have reached a contradiction... | {
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0c33d4a744a13685289269d4e86caecd2a837f8d | subsection | 15 | 16 | Concluding remarks | In this paper, we have determined an upper bound for the threshold of K_{2,t}-bootstrap percolation by proposing a subgraph whose existence forces the graph to percolate. Note that if Question REF has an affirmative answer, then implies that K_{2,t}-bootstrap percolation has a coarse threshold. Question REF has been an... | {
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2bd1ddbce6b4fa49357b7cc3c95db2611b695e90 | abstract | 0 | 70 | Abstract | In this paper, we are interested in the following one dimensional forward
stochastic differential equation (SDE) \[ d X_{t}=b(t,X_{t},\omega)d t +\sigma
d B_{t},\quad 0\leq t\leq T,\quad X_{0}=\,x\in \mathbb{R}, \] where the driving
noise $B_{t}$ is a $d$-dimensional Brownian motion. The drift coefficient
$b:[0,T] \tim... | {
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drifts | [
"Olivier Menoukeu-Pamen",
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6a128ac219a679831ebd4cea0f0bdc5d5249ed98 | subsection | 1 | 70 | Introduction | The first main result of the present paper concerns wellposedness of a class of stochastic differential equations of the formdX_t = \left(b_1(t, X_t) + b_2(t,X_t,\omega )\right)\,dt + \sigma \mathrm {d}B_t, \quad 0\le t\le T,\,\, X_0=x\in \mathbb {R}when the drift coefficient b_1:[0,T]\times \mathbb {R}\rightarrow \mat... | {
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drifts | [
"Olivier Menoukeu-Pamen",
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3c68bba022c57fe81a3cfb48ae34aed14a2b9e64 | subsection | 2 | 70 | Introduction | As suggested by an anonymous referee, let us mention however that it seems conceivable that, to some extend, the PDE methods could work when the random part b_2 of the drift is seen as a forcing in the equation, but this remains an open question.
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"raw": "H.-J. Engelbert and W. Schmidt. Strong Markov continuous local martingales and solutions of one-dimensional stochastic differential equations, I, II, III. Math. Nachr., 143, 144, 151(167-184, 241-28... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
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89af8387acee72273f5fc7a0255212f84a72899f | subsection | 3 | 70 | Probabilistic setting | Let T \in (0,\infty ) and d \in \mathbb {N} be fixed and consider a probability space (\Omega , {\cal F}, P) equipped with the completed filtration ({\cal F}_t)_{t\in [0,T]} of a d-dimensional Brownian motion B.
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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637bd506b4b30568dfcf2db822d0b990ac25dd31 | subsection | 4 | 70 | Main results | In this section, we present the main results of the paper.
Refer to the beginning of Section for details regarding Malliavin calculus.
Let us consider the following conditionsIt holds b= b_1 + b_2, where the function b_1:[0,T]\times \mathbb {R}\rightarrow \mathbb {R} is Borel measurable and there is k_1\ge 0 such that ... | {
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"Olivier Menoukeu-Pamen",
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46b3a04f1dc306a34283fc4bddf6924b64fa7fb9 | subsection | 5 | 70 | Main results | Then there exists a unique global strong solution X \in {\cal S}^2(\mathbb {R}) to the SDE\mathrm {d}X_{t}=b(t, X_{t},\omega ) \mathrm {d}t+ \sigma \mathrm {d}B_{t},\,\,\,0\le t\le T,\,\,\,\text{ }X_{0}=\,x\in \mathbb {R}.The proof is given in Sections and .
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-0.004809276200830936,
0.014683356508612633,
-0.0061098020523786545,
0.011014424264431,
0.0... | |
9abbe7334923967c3b461e5488659ca64f71ff5c | subsection | 6 | 70 | Main results | Let us give some examples of drift coefficients satisfying condition REF .The example of a random drift term of the form b_1(t,x) + \varphi (t,x,B_t), where \varphi :[0,T]\times \mathbb {R}\times \mathbb {R}^d\rightarrow \mathbb {R} is a Lipschitz continuous functions (in the second and third variables) seems not to be... | {
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{
"arxiv_id": "",
"doi": "10.1016/j.jfa.2013.12.004",
"end": 1080,
"openalex_id": "https://openalex.org/W2000492429",
"raw": "P. Cheridito and K. Nam. BSDEs with terminal conditions that have bounded Malliavin derivative. J. Funct. Anal., 266(3):1257–1285, 2014.",
... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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-0.0211422014981508... | |
5e039a273ea9ab3ebc077c14ea8fa6f10332cdeb | subsection | 7 | 70 | Some notation | In this section, we prove existence and uniqueness of strong solutions for SDEs.
Since Malliavin calculus will play an important role in our arguments, we briefly introduce the spaces of Malliavin differentiable random variables and stochastic processes
{\cal D}^{1,p}(\mathbb {R}^k) and {\cal L}^{1,p}_a(\mathbb {R}^k),... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/3-540-28329-3",
"end": 403,
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"raw": "D. Nualart. The Malliavin Calculus and Related Topics. Springer Berlin, 2nd edition edition, 2006.",
"source_ref_id": "fbafe24183369f27ec8a... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04260972887277603,
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0.032659318298101425,
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0.0005632397369481623,
0.02... | |
9a62eff5825a031fc1d28bf2664fcbeac0fdd153 | subsection | 8 | 70 | Some notation | Denote by {\cal L}^{1,p}_a(\mathbb {R}^{l}) the space of processes Y \in {\cal H}^2(\mathbb {R}^{l}) such that
Y_t \in {\cal D}^{1,p}(\mathbb {R}^{l}) for all t \in [0,T], the process DY_t admits a square integrable progressively measurable version and\left\Vert Y\right\Vert _{{\cal L}^{1,p}_a(\mathbb {R}^l)}^p := \lef... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04265325888991356,
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0.041890501976013184,
0.02305... | |
547b2948d5419d4edcf964d56b150057e7ab9f47 | subsection | 9 | 70 | Proof of Theorem | In the whole of this section, we assume that conditions REF and REF are satisfied.
The proof of the Theorem REF is given in 5 steps.
In the first step,
we show that there exists a process X^x satisfying the SDE (REF ) in the weak sense.
That is, there is a Brownian motion \tilde{B} such that (X^x_t, \tilde{B}_t) is a w... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 856,
"openalex_id": "",
"raw": "A. Lanconelli and F. Proske. On explicit strong solutions of Itô-SDE's and the Donsker delta function of a diffusion. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 7(3):437–447, 2004.",
"s... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.003397321794182062,
0.006809903774410486,
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0.04108069837093353,
0.010460928082466125,
-0.013802931644022465,
0.004894737619906664,
0.03... | |
622b6fe06b10e6c047dd9aec115b137101dd9b60 | subsection | 10 | 70 | Proof of Theorem | We also obtain from step 2 that E\Big [X^x_t|\mathcal {F}_t\Big ] is Malliavin smooth, see Subsection .In step 4, we prove that X^x_t is \mathcal {F}_t-measurable by showing that
E\Big [X^x_t|\mathcal {F}_t\Big ]=X^x_t . The proof is completed by showing uniqueness.In the last step, we use a pasting argument to show th... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02374516800045967,
0.061743542551994324,
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0.016755908727645874,
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0.03348129615187645,
0.002119286684319377,
0.03137536346912384,
0.024828655645251274,
-0.0... | |
593e0980837139ba9ae0267b7810b5d6c755de6a | subsection | 11 | 70 | 2.2.1. Weak existence. | The following result can be seen as a slight generalization of a result by V.E. Beneš, compare , .
Therein (and throughout the paper) we denote by {\cal E}(\int q\mathrm {d}B) the Doléan-Dade exponential{\cal E}\left(\int q\mathrm {d}B\right)_t := \exp \left(\int _0^t q_u\mathrm {d}B_u - \frac{1}{2} \int _0^t |q_u|^2\m... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1137/0309034",
"end": 98,
"openalex_id": "https://openalex.org/W2094649962",
"raw": "V. E. Benes. Existence of optimal stochastic control laws. SIAM J. Control Optim., 9:446–475, 1971.",
"source_ref_id": "92549d421f49abb9cf9db1aad8... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.039766497910022736,
0.01893715374171734,
-0.011345506645739079,
-0.015824196860194206,
0.011894851922988892,
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0.031587354838848114,
0.039797015488147736,
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0.05707087367773056,
-0.012634942308068275,
0.05813904479146004,
0.02121083252131939,
-0... | |
70dcb85a11597ae5997a8b67c2d14ee6f510a7e7 | subsection | 12 | 70 | 2.2.2. Approximation and compactness. | Let b_n=b_{1,n}+b_2 be such that b_{1,n}: [0,T] \times \mathbb {R} \rightarrow \mathbb {R}, n\ge 1 are smooth coefficients with compact support and converging a.e. to b_1.
Denote by X^{x,n}_{t} the unique strong solution to the SDE (REF ) with drift b_n.
The following result is key to the compactness argument.If T\in (... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03991340473294258,
0.027676954865455627,
-0.03365786373615265,
-0.02860765904188156,
-0.001932924147695303,
-0.02233685925602913,
0.002139852847903967,
0.023603225126862526,
0.03393249586224556,
0.023130245506763458,
0.023313334211707115,
0.0021284096874296665,
-0.007106142118573189,
-0... | |
53e4b8e52d874e9ac7d2cf058369cb45520db339 | subsection | 13 | 70 | 2.2.2. Approximation and compactness. | Moreover,\sup _{0 \le t \le T} E \left[ | D_t X^{x,n}_s |^2 \right] \le {\cal C}(\Vert \tilde{b}_1\Vert _{\infty },|x|^2,b_2^{\text{power}}),where the function {\cal C}(\cdot , \cdot , \cdot ): [0, \infty )^3 \rightarrow [0, \infty ) is continuous and increasing in each components, b_2^{\text{power}} defined in (REF ) ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/3-540-28329-3",
"end": 1378,
"openalex_id": "https://openalex.org/W1554577510",
"raw": "D. Nualart. The Malliavin Calculus and Related Topics. Springer Berlin, 2nd edition edition, 2006.",
"source_ref_id": "fbafe24183369f27ec8... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0803065374493599,
0.05135102570056915,
-0.012967430986464024,
-0.009969665668904781,
0.021556446328759193,
0.003417299361899495,
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0.037773363292217255,
0.009771340526640415,
0.009878131560981274,
-0.021724261343479156,
-0.002765113953500986,
-0.004939065780490637,
... | |
7831de1afac3c3332f6d14d7a28914aac4da72fb | subsection | 14 | 70 | 2.2.2. Approximation and compactness. | Solving (REF ) explicitly givesD^i_tX^{x,n}_s &= e^{\int _t^s \lbrace b_{1,n}^{\prime }(u,X^{x,n}_u) +b_{2}^{\prime }(u,X^{x,n}_u,\omega ) \rbrace \mathrm {d}u}\Big (\int _t^sD^i_tb_2(u,X^{x,n}_u,\omega )e^{-\int _t^u \lbrace b_{1,n}^{\prime }(r,X^{x,n}_r) +b_{2}^{\prime }(r,X^{x,n}_r,\omega ) \rbrace \mathrm {d}r}\mat... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.014433157630264759,
0.006880924105644226,
-0.035365812480449677,
-0.01434161514043808,
-0.01669119857251644,
0.01580629125237465,
0.03298571705818176,
0.010763839818537235,
0.014471299946308136,
0.028118720278143883,
0.029232483357191086,
0.005973129998892546,
-0.02044443041086197,
0.008... | |
d3adf1706e3a4f3c5b6bb62887f385f929ffc041 | subsection | 15 | 70 | 2.2.2. Approximation and compactness. | Using the above representation, we have& D^i_{t^{\prime }} X^{x,n}_s - D^i_t X^{x,n}_s \\
=& e^{\int _{t^{\prime }}^s\lbrace b_{1,n}^{\prime }(u,X^{x,n}_u) +b_{2}^{\prime }(u,X^{x,n}_u,\omega ) \rbrace \mathrm {d}u}\Big (\int _{t^{\prime }}^sD^i_{t^{\prime }}b_2(u,\omega )e^{-\int _{t^{\prime }}^u b_{2}^{\prime }(r,X^{... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.002569461241364479,
0.010239694267511368,
-0.022676879540085793,
-0.04172179475426674,
-0.013299394398927689,
0.002125003607943654,
0.017671484500169754,
0.048894159495830536,
0.011407110840082169,
0.03271818906068802,
-0.01893809251487255,
0.020540429279208183,
-0.004875681363046169,
-... | |
3c2ca074101a349783267fd349d2240447bdd1b6 | subsection | 16 | 70 | 2.2.2. Approximation and compactness. | Next, we wish to use conditions on b_n and thus u_n to show that the second term is finite for T small enough.
Using Hölder inequality, we haveE\left[e^{4\sum _{i=1}^d\int _0^Tu_{i,n}^2(r,x+\sigma \cdot B_r,\omega )\mathrm {d}r}\right]&\le \prod _{i=1}^{d} E\left[e^{12d\int _0^Tu_{i,n}^2(r,x+\sigma \cdot B_r,\omega )\m... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04444541037082672,
0.03934761881828308,
0.0023867276031523943,
0.008074046112596989,
0.0027988245710730553,
0.0019021322950720787,
-0.010638204403221607,
0.008509037084877491,
-0.011676077730953693,
0.008226674050092697,
-0.014621807262301445,
0.014522598125040531,
0.0032567097805440426,
... | |
64406c6c9ebbf535e6052d215315bdebda9667b1 | subsection | 17 | 70 | 2.2.2. Approximation and compactness. | Using the condition on b_n, Hölder inequality successively and the independence of the Brownian motion, we getE\left[e^{12d\int _0^T\frac{\sigma _i^2}{(\sigma _1^2+\cdots +\sigma _d^2)^2 }b_n^2(r,x+\sigma \cdot B_r,\omega )\mathrm {d}r}\right] &\le E\left[e^{24c_{d,\sigma }\int _0^T\left(k^2|1+x+\sigma \cdot B_r|^2 + |... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02299630083143711,
0.022370655089616776,
-0.019089829176664352,
0.028794970363378525,
-0.0029470210429280996,
0.0003764843277167529,
0.0005483938730321825,
0.020158005878329277,
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-0.0213940367102623,
-0.017563864588737488,
0.025376807898283005,
0.0028230363968759775,
... | |
b81a115248411bcf328150606dedccf399b11cd4 | subsection | 18 | 70 | 2.2.2. Approximation and compactness. | Now, using exponential expansion and the Doob maximal inequality, we haveE\left[e^{48c_{d,\sigma }k^2T\sup _{0\le t\le T}|\sigma _i\cdot B^i_t|^2} \right]&= 1 + \sum _{p=1}^\infty \frac{(48c_{d,\sigma }\sigma _i^2k^2T)^p}{p!} E\left[\sup _{0\le t\le T}|B_t|^{2p}\right]\\
&\le 1 + \sum _{p=1}^\infty \frac{(48k^2dT)^p}{p... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.021912869065999985,
0.01739601045846939,
-0.017701204866170883,
-0.027528423815965652,
-0.012398472987115383,
0.06457887589931488,
0.044436126947402954,
0.04559585824608803,
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-0.0039942157454788685,
-0.02081417478621006,
0.03915628418326378,
-0.028169330209493637,
... | |
f90193a575d579b2d7050a5aa82fd2703623728c | subsection | 19 | 70 | 2.2.2. Approximation and compactness. | Using power and exponential expansion, we get by linearity of the expectation and Hölder inequality& E\Big [ \Big (e^{\int _{t^{\prime }}^t b_{1,n}^{\prime }(u,X^{x,n}_u) \mathrm {d}u}e^{\int _{t^{\prime }}^t b_{2}^{\prime }(u,X^{x,n}_u,\omega ) \mathrm {d}u}-1\Big )^6\Big ]\\
= & E\Big [ e^{6\int _{t^{\prime }}^t \lbr... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.027889838442206383,
0.03539629280567169,
-0.025402942672371864,
-0.0198799017816782,
0.00817776471376419,
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0.018705110996961594,
0.014478914439678192,
0.02702018804848194,
0.004088882356882095,
-0.01945270411670208,
0.02973593771457672,
-0.033809561282396317,
-0.00... | |
63e67b297e0dd03677b6bb58eec430565cc2dd2b | subsection | 20 | 70 | 2.2.2. Approximation and compactness. | More specifically, let us focus on J_1 only since the bounds for the other terms follow in a similar way. | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.019032027572393417,
0.05940068140625954,
-0.049785465002059937,
-0.0006996788433752954,
-0.020130909979343414,
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0.03427901491522789,
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0.014697548002004623,
-0.007257200311869383,
-0.00821872241795063,
-0.006303309928625822... | |
1e03e55b400bb45bd0b1577c467e175518a75695 | subsection | 21 | 70 | 2.2.2. Approximation and compactness. | Using dominated convergence theorem, Hölder inequality and Girsanov theorem, we haveJ_1=&\sum _{q=1}^\infty \frac{E\Big [\Big |6\int _{t^{\prime }}^t \lbrace b_{1,n}^{\prime }(u,X^{x,n}_u) + b_{2}^{\prime }(u,X^{x,n}_u,\omega ) \rbrace \mathrm {d}u\Big |^q\Big ]}{q!}\\
\le & \sum _{q=1}^\infty \frac{12^pE\Big [\Big |\i... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03357890620827675,
0.02335260435938835,
-0.0019632212352007627,
-0.01321024727076292,
-0.00960051454603672,
0.011928143911063671,
0.0051207831129431725,
0.025855759158730507,
-0.006555518601089716,
0.016102612018585205,
-0.024833127856254578,
0.026542600244283676,
-0.014515245333313942,
... | |
c9ce762118006319a7a63b820d309096c596a3b0 | subsection | 22 | 70 | 2.2.2. Approximation and compactness. | Now by Proposition ,\sum _{q=1}^\infty \frac{E\Big [\Big (\int _{t^{\prime }}^t \sqrt{12}b_{1,n}^{\prime }(u,x+\sigma \cdot B_u) \mathrm {d}u\Big )^{2q}\Big ]^{\frac{1}{2}}}{q!}
\le & \Big (\sum _{q=1}^{\infty }\frac{C_{\sigma ,k}^q(1+|x|^{q})\sqrt{q!} (t-t^{\prime })^{q/2}
}{q!} \Big )\\
\le &\Big (\sum _{q=1}^{\infty... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.026003776118159294,
0.031161801889538765,
-0.025317056104540825,
-0.03461065888404846,
-0.0009275480988435447,
0.01036183349788189,
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0.0022280230186879635,
0.0008951196796260774,
0.017946267500519753,
-0.004799405578523874,
0.024416690692305565,
-0.01229227799922227... | |
a77763de45c7268ad4443c361db9b8b679a09b8f | subsection | 23 | 70 | 2.2.2. Approximation and compactness. | Therefore there exists a constant C depending on \sigma such thatE[I^2_1] \le \frac{C}{\sqrt{T}}\exp \left\lbrace C_{\sigma ,k} T(1 + |x|)^2 \right\rbrace |t-t^{\prime }|^{1/2}.Repeated application of the Hölder inequality yieldsE[I_2^2]&= E\Big [ \Big (\int _{t^{\prime }}^tD^i_{t^{\prime }}b_2(u,X_u^{n,x},\omega )e^{-... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.05211787298321724,
0.009108422324061394,
-0.04146849364042282,
-0.0027233725413680077,
0.005034806206822395,
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0.03322971984744072,
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0.00994755607098341,
0.008757511153817177,
... | |
24b7d91bff194383b9d74a6e3a892034d8120f7d | subsection | 24 | 70 | 2.2.2. Approximation and compactness. | Moreover, the assumptions on b_2 insure that the two last integral terms on the right hand side of (REF ) are bounded by C. Therefore, there exists \alpha >0 such thatE[I_3^2]\le C \frac{C}{\sqrt{T}} \exp \lbrace C_{\sigma ,k} T(1+|x|)^2\rbrace |t-t^{\prime }|^{\alpha }for 0 \le t^{\prime } \le t \le T_1 with T_1 small... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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8964a5e47ffef11bf2e3a9aa184b3a4fda57cbfd | subsection | 25 | 70 | 2.2.3. Weak convergence to the weak solution. | In this step, we show that for each 0 \le t \le T the above sequence (X^{x,n}_{t} )_{n \ge 1} converges weakly to E\Big [X^x_t|\mathcal {F}_t\Big ] in the space L^2(\Omega ,P;\mathcal {F}_t).Assume b^{\text{exp}}_2<\infty and \Omega is the canonical space.
Choose the sequence b_{1,n}: [0,T] \times \mathbb {R} \rightarr... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b8c5597b79703f503374bf8097361ca5c8325c9f | subsection | 26 | 70 | 2.2.3. Weak convergence to the weak solution. | In fact, using Girsanov transform, Hölder inequality and the fact that (1+|z|^p)e^{-|z|^2 /2s} can be bounded by C_pe^{-\frac{|z|^2}{ 2^{p+1}s}}, where C_p is a constant depending on p, we have\sup _{n} E\left[|h(X^{x,n}_{t})|^2\right] &\le E\left[ e^{2\sum _{i=1}^d\int _0^Tu_{i,n}(r,x+\sigma \cdot B_r,\omega )\mathrm ... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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b9abcaeb30dc6b73ca6990ff4c568a37870b3ded | subsection | 27 | 70 | 2.2.3. Weak convergence to the weak solution. | Here C^1_b([0,T],\mathbb {R}^d) is the space of bounded continuous differentiable functions on [0,T] and with values in \mathbb {R}^d and \dot{\varphi } is the derivative of \varphi . Hence, it is enough to show that \Big (h(X^{x,n}_{t})\mathcal {E}\Big (\int _0^T\dot{\varphi }_r\mathrm {d}B_r\Big )\Big )_{n} converges... | {
"cite_spans": [
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"doi": "10.1007/978-3-662-13225-8",
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"raw": "A. S. Üstünel and M. Zakai. Transformation of Measure on Wiener Space. Springer Monographs in Mathematics. Springer, 2000.",
"source_re... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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bbb8085ef9774a153d67a3ab9ca23158f8bb069a | subsection | 28 | 70 | 2.2.3. Weak convergence to the weak solution. | To see this, let H \in L^2(\Omega , P) and apply (REF ) and the fact that X^{x,n} solves the SDE (REF ) to getE[\tilde{X}^{x,n}_tH] &= E\left[X^{x,n}_tH(\omega - \varphi )\mathcal {E}\left(\int _0^T\dot{\varphi }(u)\mathrm {d}B_u\right)\right]\\
&= E\Big [\Big (x + \int _0^tb_1(u, X^{x,n}_u) + b_2(u, X^{x,n}_u,\omega )... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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5364ca7f1bcac9d8f468be8196d94ac97efdf8b3 | subsection | 29 | 70 | 2.2.3. Weak convergence to the weak solution. | Since X^x satisfies the SDE (without been adapted to the filtration ({\cal F}_t)), with respect to a probability measure Q which is equivalent to P, see the proof of Lemma REF , the above arguments show that \tilde{X}^x(\omega ):= X^x(\omega + \varphi ) satisfiesd\tilde{X}^{x}_t = (b_{1}(t,\tilde{X}^{x}_t) + \tilde{b}_... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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17f1ee6a649ec1fd441695e0b897aa263ffc3c11 | subsection | 30 | 70 | 2.2.3. Weak convergence to the weak solution. | Observe that&\mathcal {E}\Big (\int _0^T\left\lbrace \tilde{u}_n(r,x+\sigma \cdot B_r,\omega )+\dot{\varphi }_r\right\rbrace \mathrm {d}B_r\Big ) = {\cal E}\Big (\int _0^T b_{1,n}^\sigma (r, x +\sigma \cdot B_r,\omega )\mathrm {d}B \Big ){\cal E}\Big (\int _0^T \tilde{b}_2^\sigma \mathrm {d}B \Big ){\cal E}\Big (\int _... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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457a1b5694acb74a7ac31e7e9220e6e09f0c736d | subsection | 31 | 70 | 2.2.3. Weak convergence to the weak solution. | Indeed, by contradiction, suppose that for some t, there exist \epsilon >0 and a subsequence n_l, l\ge 0 such that\Vert X_t^{x,n_l}-E[X^x_t|\mathcal {F}_t]\Vert _{L^2(\Omega ,P)}\ge \epsilon .We also know by the compactness criteria that there exists a further subsequence of n_m, m\ge 0 of n_l, l\ge 0 such thatX_t^{x,n... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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068d7c99cf9f59101a6e51f15c78c03e0183bf47 | subsection | 32 | 70 | 2.2.4. Adaptedness of the weak solution and uniqueness. | Finally, we show that the weak solution X^x_t is (\mathcal {F}_t)_{ t\in [0,T]}-adapted and unique.The weak solution X^x_t to the SDE (REF ) is \mathcal {F}_t-measurable.
Let us first show that X^x_t is \mathcal {F}_t-measurable.
Let h be a globally Lipschitz continuous function.
By Proposition REF , there exists a s... | {
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"source_ref_id": "d4cc84c... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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1297d10f0082ccaaca52ac9aeec9ae2ab07e3926 | subsection | 33 | 70 | 2.2.4. Adaptedness of the weak solution and uniqueness. | Thus, a simple adaptation of the proof of shows that (X^x_t(\omega +\varphi ),B) and (\tilde{X}^x_t(\omega +\varphi ),B) have the same distribution.
Hence, for all t,\varphi , we have E\Big [\tilde{X}^x_t\mathcal {E}\Big (\int _0^T\dot{\varphi }_u\mathrm {d}B_u\Big )\Big ]=E\Big [X^x_t\mathcal {E}\Big (\int _0^T\dot{\v... | {
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"doi": "10.1007/978-1-84628-696-4_13",
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"raw": "I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, New York, 1988.",
"source_ref_id": "d4cc84cb... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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c572e6d95e3a20555a3db8662abd84d73ccb6c9e | subsection | 34 | 70 | 2.2.5. Global existence. | Since the small time T_1 for which the solution exists does not depend on the initial condition (see (REF )) one can use a standard pasting argument to show that the solution exists for all time T>0.
In addition
using the linear growth condition on b_1 and the integrability condition on b_2, it follows from Gronwall le... | {
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"doi": "10.1016/j.jfa.2019.05.010",
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"raw": "O. Menoukeu-Pamen and S. E. A. Mohammed. Flows for singular stochastic differential equations with unbounded drifts. Preprint ArXiv:1704.0368... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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28ebeeb334ab429f353e3d40aba5581d90ee3c0e | subsection | 35 | 70 | Differentiability of the strong solution | In this subsection, we show that the unique strong solution of the SDE (REF ) constructed in the previous subsection is Malliavin differentiable and we derive a representation formula of the Malliavin derivative.
The proof Malliavin differentiability for small time interval follows directly from Lemma REF .
In order to... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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1dd2029cd53c33bb452d70500dc06f6f28e59a71 | subsection | 36 | 70 | Differentiability of the strong solution | \end{array}Successive application of Hölder's inequality to (REF ) yields|X^{t_0,\eta }_{t}|^2 &\le 4|\eta |^2 + 4 \Big | \int _{t_0}^{t} b (u,X^{t_0,\eta }_{u}) \mathrm {d}u \Big |^2 +4 \Big | \int _{t_0}^{t} b_2(u,X^{t_0,\eta }_{u},\omega ) \mathrm {d}u \Big |^2 + 4|\sigma |^2|B_t -B_{t_0}|^2\\
&\le 4|\eta |^2 + 4 \B... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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0a086d3fe69a6258b7d9e8af38153a90d37f05a4 | subsection | 37 | 70 | Differentiability of the strong solution | Then taking exponential on both sides of (REF ), we have\exp \left\lbrace \delta _0\sup _{t_0\le t\le 1} |X^{t_0,\eta }_{t}|^2 \right\rbrace
&\le \exp \left\lbrace 2C_2\delta _0k^2\right\rbrace \times \exp \lbrace \delta _0 C_2|\eta |^2\rbrace \times \exp \left\lbrace C_2\delta _0|\sigma |^2\displaystyle \sup _{t_0 \l... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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7f72a4d736dcdea30383103d0138e9c416aa5d65 | subsection | 38 | 70 | Differentiability of the strong solution | From this (REF ) yieldsE\exp \left\lbrace \delta _0\sup _{t_0\le t\le 1}|X^{n,t_0,\eta }_{t}|^2 \right\rbrace \le C_1 Ee^{C_2\delta _0|\eta |^2}.Note that C_1, C_2 and \delta _0 are independent of \eta and t_0 (but may depend on ||\tilde{b}_1||_\infty and |\sigma |^2). Thus (REF ) is valid for the above choice of \delt... | {
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{
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"doi": "10.1007/3-540-28329-3",
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"source_ref_id": "fbafe24183369f27ec8a... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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8be72cefdd9729eb7d0e48209f00250830379af3 | subsection | 39 | 70 | Differentiability of the strong solution | \end{array}Using the chain-rule for the Malliavin derivatives, we have component wiseD^i_sX^{x_m,n}_t =\left\lbrace \begin{array}{llll}
D^i_sX_{s_m}^{n} + \int _{s_m}^t \Big (\lbrace b_{1,n}^{\prime }(u,X^{x_m,n}_u)+b_{2}^{\prime }(u,X^{x_m,n}_u,\omega )\rbrace D^i_s X^{x_m,n}_u +D^i_sb_2(t,X^{x_m,n}_u,\omega )\Big )\m... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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dd079203d693b7e7a84ac35e0b7d3e17dffd8442 | subsection | 40 | 70 | Differentiability of the strong solution | Let us first focus on D_sX^{x_m,n}_t, when s\le s_m. | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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bae7871f827cefd43a8d88577f34ffe866ff600f | subsection | 41 | 70 | Differentiability of the strong solution | Using Hölder inequality, we haveE\left[|D^i_sX^n_t|^2\right] &\le 2 E\Big [e^{\int _{s_m}^t 2\lbrace b_{1,n}^{\prime }(u,X^{x_m,n}_u)+b_{2}^{\prime }(u,X^{x_m,n}_u,\omega )\rbrace \mathrm {d}u}\Big (\int _{s_m}^tD^i_sb_2(u,X^{x_m,n}_u,\omega )e^{-\int _{s_m}^u\lbrace b_{1,n}^{\prime }(r,X^{x_m,n}_r)+b_{2}^{\prime }(r,X... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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241e46f0c883b178c1aee9ffd97e325981a1bd91 | subsection | 42 | 70 | Differentiability of the strong solution | As before, using Girsanov theorem and Hölder inequality, we have& E\Big [ e^{\int _{s_m}^t 4b_{1,n}^{\prime }(u,X^{x_m,n}_u) \mathrm {d}u}|\mathcal {F}_{s_m}\Big ] \\
&\le E\Big [ \mathcal {E}\Big (\int _{s_m}^tu_n(u,\omega ,X_{s_m}^n+\sigma \cdot (B_u-B_{s_m}))\mathrm {d}B_u\Big )e^{\int _{s_m}^t 4b_{1,n}^{\prime }(u,... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
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9fe82fd0bfce6538ad17df79c863c899eb685c42 | subsection | 43 | 70 | Differentiability of the strong solution | Since b_{1,n} is of spatial linear growth, B_r-B_{s_m} is independent of \mathcal {F}_{s_m} and X_{s_m}^n is \mathcal {F}_{s_m}-measurable, it follows from the Hölder inequality, the exponential expansion and the choice of \tau that there exists a constant C>0 such thatE\left[e^{16\int _{s_m}^{s_{m+1}}u_{i,n}^2(r,X_{s_... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
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c69f3d91fcdbece908004d5dbdd6eb3b6695ec6e | subsection | 44 | 70 | Differentiability of the strong solution | This can be shown as in the proof of Lemma REF using the Gronwall lemma and the probability distribution of the Brownian motion.The case s>s_m is similar (and easier) since the term D^i_sX^m_{s^m} is not involved, it is replaced by the constant \sigma .Since b_2^{\text{power}}<\infty , using the Hölder inequality, the ... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
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94dc6dc73ef8a09975e80d085c04fff302fa8ef5 | subsection | 45 | 70 | Differentiability of the strong solution | Then by Lemma REF , for every n\in \mathbb {N} it holdsE \left[ | D_t X^{n}_s - D_{t^{\prime }} X^{n}_s |^2 \right] \le {\cal C}(\Vert \tilde{b}_1\Vert _{\infty },|x|^2,b_2^{\text{power}}) |t -t^{\prime }|^{\alpha }for 0 \le t^{\prime } \le t \le T_1, \alpha = \alpha (s) > 0 and\sup _{0 \le t \le T_1} E \left[ | D_t X^... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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ea8aedad6db18193912dedb56df05e6bb9c34365 | subsection | 46 | 70 | Representation and moment bounds for the Malliavin derivative | In this subsection we give an explicit representation of the Malliavin derivative DX^x of the solution X^x of the SDE (REF ).
To that end, we will assume that the random part b_2 of the drift does not depend on x.
Such representation can be very useful to derive results concerning DX^x.
The representation we obtain wil... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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f2aae842a524055df8b15cb3e0f3f2d9cbf24721 | subsection | 47 | 70 | Representation and moment bounds for the Malliavin derivative | Then for all t\in [0,T], it holds\int _0^tf^{\prime }(s,X^x_s)\mathrm {d}s=-\int _0^t\int _{\mathbb {R}}f(s,z) L^{X^x}(\mathrm {d}s,\mathrm {d}z).Moreover, the local time-space integral of f \in {\cal H}^0 admits the decomposition (see the proof of )\int _0^t\int _{\mathbb {R}}f(s,z) L^{B^x}(\mathrm {d}s,\mathrm {d}z)&... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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"math.PR"
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34ff8b4cac08dea3666bf031bdb884eb3965ada9 | subsection | 48 | 70 | Representation and moment bounds for the Malliavin derivative | For every 0\le s\le t\le T, the i-th component of the Malliavin derivative of the unique strong solution to the SDE (REF ) admits the following representation:D^i_tX_s^x=e^{\int _t^s\int _{\mathbb {R}} b_{1}(u,z) L^{X^x}(\mathrm {d}s,\mathrm {d}z) }\left(\int _t^sD^i_tb_2(u,\omega )e^{-\int _t^u\int _{\mathbb {R}} b_{1... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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bbf75c11c40dfed4c93981be0b0f5ca55bf65a97 | subsection | 49 | 70 | Representation and moment bounds for the Malliavin derivative | Then, it follows by that b_2 is Malliavin differentiable, andD_sb_2(t) = \alpha _s1_{\lbrace s\le t \rbrace } + \int _s^tD_s\alpha _r\mathrm {d}B_r.Thus, by Burkholder-Davis-Gundy inequality, it holds\sup _{0\le s\le T}E\left[\left(\int _0^T|D_sb_2(t,\omega )|^2\mathrm {d}t\right)^4 \right]\le \Vert \alpha \Vert _{{\ca... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
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b215fd8c25f5106af65195a553e361d4dc8fbd40 | subsection | 50 | 70 | Representation and moment bounds for the Malliavin derivative | It follows from (REF ) and the Hölder inequality that&E\left[\exp \left(k\int _0^t\int _{\mathbb {R}}f(s,z)L^{B^x}(\mathrm {d}s,\mathrm {d}z)\right)\right]\\
& \le E\left[\exp \left(2k\int _0^tf(s,B^x_s)\mathrm {d}B^x_s\right)\right]^{\frac{1}{2}}
\times E\left[\exp \left(4k\int _{T-t}^Tf(T-s,B^x_{T-s})\mathrm {d}\wide... | {
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"s... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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5586b07c08b062bb9598c53d3622f8c073b13afa | subsection | 51 | 70 | Representation and moment bounds for the Malliavin derivative | Indeed, using exponential expansion, and the Hölder inequality, we haveE\left[\exp \left(k\int _{0}^T\frac{|B_{T-s}|^2}{T-s}\mathrm {d}s\right)\right]&=\sum _{n=1}^{\infty }\frac{1}{n!}E\left[\left(k\int _{0}^T\frac{|B_{T-s}|^2}{T-s}\mathrm {d}s\right)^n\right]\le \sum _{n=1}^{\infty }\frac{k^n}{n!}\int _{0}^T\frac{E|B... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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864d998be5d8be3eacfa29dc12362a8633230366 | subsection | 52 | 70 | Representation and moment bounds for the Malliavin derivative | Thus, in order to conclude, we need to show that\left\lbrace e^{-\int _t^s \int _{\mathbb {R}}b_{1,n}(u,z) L^{X^{x,n}}(\mathrm {d}u,\mathrm {d}z)} \left(\int _t^sD^i_tb_2(u,\omega )e^{\int _t^u \int _{\mathbb {R}}b_{1,n}(r,z) L^{X^{x,n}}(\mathrm {d}r,\mathrm {d}z)}\mathrm {d}u + \sigma _i \right)\mathcal {E}\left(\int ... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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98e1587ca82a2c08a19f07cdde34d1e7d61a12bb | subsection | 53 | 70 | Representation and moment bounds for the Malliavin derivative | Using Girsanov theorem and the Cameron-Martin theorem, we haveL= & \Big |E\Big [\mathcal {E}\Big (\int _0^T\dot{\varphi }_r\mathrm {d}B_r\Big )
\Big \lbrace e^{-\int _t^s \int _{\mathbb {R}}b_{1,n}(u,z) L^{X^{x,n}}(\mathrm {d}u,\mathrm {d}z)} \int _t^sD^i_tb_2(u,\omega )e^{\int _t^u \int _{\mathbb {R}}b_{1,n}(r,z) L^{X... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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58404629eea37b59e944677c34f616633a62f080 | subsection | 54 | 70 | Representation and moment bounds for the Malliavin derivative | Repeated use of Hölder inequality, Girsanov transform, the bound on D^i_tb_2(u,\omega +\varphi ) and the fact that |e^x-1|\le |x|(e^x+1) givesI_{1,n}\le &\Big |E\Big [
e^{-2\int _t^s \int _{\mathbb {R}}b_{1,n}(u,z) L^{\tilde{X}^{x,n}}(\mathrm {d}u,\mathrm {d}z)} \Big ]^{1/2}\times E\Big [\Big (\int _t^s(D^i_tb_2(u,\ome... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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659dfa817d3d108e657c87102386b6859bc77d37 | subsection | 55 | 70 | Representation and moment bounds for the Malliavin derivative | Using Cauchy-Schwartz inequality, the Novikov's condition on b_2 and Beneš Theorem, the first term is finite for small time T. Using Lemma REF and enables to conclude that the second term is bounded. Using once more Cauchy-Schwartz inequality, Girsanov transform and Lemma REF , one deduces that the fourth and fifth ter... | {
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... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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952705ee344f1a9ff4623401719c03fe268e3745 | subsection | 56 | 70 | Representation and moment bounds for the Malliavin derivative | By Girsanov transform and Cauchy-Schwartz inequality, we have& E\Big [\int _t^u \int _{\mathbb {R}}b_{1,n}(r,z) L^{\tilde{X}^{x,n}}(\mathrm {d}r,\mathrm {d}z)-\int _t^u \int _{\mathbb {R}}b_{1}(r,z) L^{\tilde{X}^{x}}(\mathrm {d}r,\mathrm {d}z)\Big ]\\
&= E\Big [\mathcal {E}\Big (\int _0^T\Big \lbrace \tilde{u}_n(r,\ome... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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ddf65a8f5643a2bd867f7b878385741d6b8d4494 | subsection | 57 | 70 | Representation and moment bounds for the Malliavin derivative | Using (REF ), Minkowski, Doob maximal inequality and the dominated convergence theorem the second term converges to 0. Similar reasoning as in the proof of Lemma REF enable to conclude that the third term converges to 0. Thus the result follows. | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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8a7c6b16f01b46cefe5a1a37e1f10de7f5ce264a | subsection | 58 | 70 | Stochastic differentiable flow for SDEs with random drifts | The aim of this section is to prove existence of a Sobolev differentiable stochastic flow for the SDE (REF ) with (non-Markovian) random drifts.
Due to the additive decomposition assumption b (t,\omega ,x) = b_1(t,x) + b_2(t, \omega ), the analysis of the flow turns out to be much easier than that of the Malliavin deri... | {
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"raw": "S. E. A. Mohammed, T. Nilssen, and F. Proske. Sobolev differentiable stochastic flows for SDE's with singular coeffcients: Applications to the stochastic transport equation. Ann. Probab., 43(3):1535... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
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c6277faf27a29e2751cfb772b08ebc6f9701dcc0 | subsection | 59 | 70 | Stochastic differentiable flow for SDEs with random drifts | The differentiability of the trajectories of x \mapsto X^{s,x,n}_t follows from the seminal work , from which we further obtain that \partial _xX^{s,x,n}_u:= \frac{\partial }{\partial x}X^{s,x,n}_t satisfies\partial _xX^{s,x,n}_t = 1 + \int _s^t\left(b^{\prime }_{1,n}(u, X^{s,x,n}_u) + b_2^{\prime }(u, X^{s,x,n}_u,\ome... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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c82a91967a5d167f21c6ce588ed33f8c451112f4 | subsection | 60 | 70 | Stochastic differentiable flow for SDEs with random drifts | The solution of this (random) ODE can be explicitly given by\partial _xX^{s,x,n}_t = \exp \left(\int _s^tb_{1,n}^{\prime }(u, X^{s,x,n}_u) + b_2^{\prime }(u, X^{s,x,n}_u,\omega ) \mathrm {d}u \right).Thus, Girsanov's theorem and successive applications of Hölder's inequality give (recall the definition of u_n given in ... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
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f4c683af84739be4be69c7a600c55ad90ad4f62d | subsection | 61 | 70 | Stochastic differentiable flow for SDEs with random drifts | As shown in the proof of Lemma REF , if T is small enough, the sequence I_2^n is bounded, and as shown in (REF ), we haveI^n_3 &=\sum _{q=1}^\infty \frac{E\left[|\int _s^t(4p)b^{\prime }_{1,n}(u, x+\sigma \cdot B_u)\mathrm {d}u|^q \right]}{q!}
\le \sum _{q=1}^\infty \frac{(4p)^{q}E\left[|\int _s^tb^{\prime }_{1,n}(u, x... | {
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} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
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e25e99d87f712b80b8080b5da110a782a9597d55 | subsection | 62 | 70 | Stochastic differentiable flow for SDEs with random drifts | If T is small enough, and M_2 in REF is bounded, then it holdsE\left[ |X^{s_1, x_1}_t - X^{s_2, x_2}_t |^p\right]\le {\cal C}_p(||\tilde{b}_1||_\infty ,|x|^2, T)\left(|s_1 - s_2|^{p/2} + |x_1 - x_2|^{p} \right)for every s_1, s_2, t \in [0,T], x_1, x_2 \in \mathbb {R} and for some continuous function {\cal C}_p increasi... | {
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} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
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] | 2,018 | en | Mathematics | [
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b88295b7c1d722b0e532847a7647405f931081d9 | subsection | 63 | 70 | Stochastic differentiable flow for SDEs with random drifts | Then, for every n \in \mathbb {N}, we haveX^{1, n}_t - X^{2, n}_t & = x_1 - x_2 + \int _{s_1}^tb_{1, n}(u, X^{1,n}_u) + b_2(u,X^{1,n}_u, \omega )\mathrm {d}u + \int _{s_1}^t\sigma \cdot \mathrm {d}B_u\\
&\quad - \int _{s_2}^tb_{1, n}(u, X^{2,n}_u) + b_2(u, X^{2,n}_u,\omega )\mathrm {d}u - \int _{s_2}^t\sigma \cdot \mat... | {
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} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
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16f64f3dd3fbf49d0ecf4dd1927872cb4e6d52f0 | subsection | 64 | 70 | Stochastic differentiable flow for SDEs with random drifts | Since (X^{s_i, x_i, n}_t) converges weakly to the unique solution X^{s_i,x_i}_t of the SDE (REF ) with drift b, (see Lemma REF and Theorem REF ) it follows by convexity and lower-semicontinuity of K\mapsto E[|K|^p] that, taking the limit in (REF ) yields the desired result.We conclude this section with the proof of The... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
"math.PR"
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f6730f8c99c0360a94ced9f504b9f86d13e13c11 | subsection | 65 | 70 | Stochastic differentiable flow for SDEs with random drifts | By weak convergence and (REF ),
it follows thatE\left[\left(\int _\mathbb {R}|\partial _xX^{s,x}_t|^pw(x)\mathrm {d}x \right)^{2/p}\right]<\infty ,where \partial _xX^{s,x}_t denotes the weak derivative of X^{s,x}_t.It remains to show thatE\left[\left(\int _\mathbb {R}|X^{s,x}_t|^pw(x)\mathrm {d}x \right)^{2/p}\right]
<... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
] | [
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d861f35bd060c5bec5a70764588167adbaab4032 | subsection | 66 | 70 | Compactness criteria | The suggested construction of the strong solution for the SDE (REF ) is based on the subsequent relative compactness criteria from Malliavin calculus (see .)Let \left\lbrace \left( \Omega ,\mathcal {A},P\right) ;H\right\rbrace be a Gaussian probability space, that is \left( \Omega ,\mathcal {A},P\right) is a probabilit... | {
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"source_ref_id": "ffc6cb5327ee... | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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07e3b2f05cf1d3ea4492f7bac414c4a1c988eae5 | subsection | 67 | 70 | Compactness criteria | For any 0<\alpha <1/2 define the operator A_{\alpha } on L^{2}([0,1]) byA_{\alpha }v_{s}=2^{k\alpha }v_{s}\text{, if }s=2^{k}+j\text{ }for k\ge 0,0\le j\le 2^{k} andA_{\alpha }1=1.Then for all \beta with \alpha <\beta <(1/2), there exists a constant c_{1} such that\left\Vert A_{\alpha }f\right\Vert \le c_{1}\Big \lbrac... | {
"cite_spans": []
} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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ae781f5f283a1fa90f07db35da07f5fd93bf92ac | subsection | 68 | 70 | An auxiliary result | The following key result generalises to the case of function with spatial polynomial growth.Let B be a d-dimensional Brownian motion starting from the origin and b: [0,1] \times \mathbb {R} \rightarrow \mathbb {R} a compactly supported smooth function such that \Vert \tilde{b}(t,z)\Vert \le k with \tilde{b}(t,z):= \fra... | {
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drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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5a69e93f28b1472f63c47d060fc3784ea13c1809 | subsection | 69 | 70 | An auxiliary result | The result then follows.Olivier Menoukeu-Pamen: University of Liverpool Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences,
L69 7ZL, United Kingdom and African Institute for Mathematical Sciences, Ghana.E-mail address: menoukeu@liverpool.ac.ukFinancial support from the Alexander von
... | {
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} | 1810.01314 | Strong solutions of some one-dimensional SDEs with random and unbounded
drifts | [
"Olivier Menoukeu-Pamen",
"Ludovic Tangpi"
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4bd13ba37d8781d15d000886e694743f1370e102 | abstract | 0 | 24 | Abstract | We present a trajectory generation framework for control of wheeled vehicles
under steering actuator constraints. The motivation is smooth autonomous
driving of heavy vehicles. The key idea is to take into account rate, and
additionally, torque limitations of the steering actuator directly. Previous
methods only take i... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
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37fc586ed10cc8f71997bbab3e2c3ece042f5141 | subsection | 1 | 24 | Background and Motivation | Path planning deals with the generation of paths or trajectories (paths with an associated time law) for a vehicle.
The generated paths/trajectories are used as a reference signal for the controllers implemented in the vehicle.
Planning methods for autonomous vehicles have come a long way from the initial problem of fi... | {
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"source_ref_id": "d2cda4f3656145c8616a97c157... | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
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"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
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61a08e0403a6cbb8ab84d03283392e48a483ffaa | subsection | 2 | 24 | Main Contributions | The contribution of this work comes from the generation of vehicle trajectories that:Directly take into account steering actuator magnitude, rate, and acceleration limitations, generating \mathbf {G}^3 paths;
Ease the controller task and improve passenger comfort;
Can connect arbitrary vehicle configurations;
Have f... | {
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"raw": "T. Fraichard and A. Scheuer, \"From Reeds and Shepp's to continuous-curvature paths\" IEEE Transactions on Robotics 20, no. 6 (2004): 1025-1035.",
"source_ref_id": "852c4e48cc57ab28f5ba70d844a... | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
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"Jonas Mårtensson",
"Bo Wahlberg"
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5b7eda57193595daaf94ce3e2ed0d08ad46fac6b | subsection | 3 | 24 | Outline | Section introduces the vehicle model used and defines the problem we address.
Section presents Sharpness Continuous paths used to solve the stated problem.
Section presents Cubic Curvature paths, a building block of Sharpness Continuous paths.
Section illustrates our simulation results, showing the performance of the m... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
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7e8930589589f4f0b55bb497b30008acb37828cf | subsection | 4 | 24 | Vehicle Model | We start by defining the vehicle model as:\dot{x} = v \cos \theta ,
\qquad \dot{y} = v \sin \theta ,
\qquad \dot{\theta } = v\kappa .(x, y) represents the location of the vehicle rear wheel axle center, \theta its orientation and v is the vehicle velocity.
The curvature \kappa of a vehicle with wheelbase length L is re... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
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c45df1b3e4d341c8d54d54ba42cdc3994390c4a9 | subsection | 5 | 24 | Path Feasibility | Path feasibility depends on the capabilities of the vehicle that executes it and on the path itself.
The limited steering angle amplitude \phi _{\max } imposes a maximum allowed curvature on the path \kappa _{\max }.
This limitation is addressed by generating paths which have a curvature profile |\kappa | \le \kappa _{... | {
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"raw": "L. E. Dubins, \"On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents\" A... | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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124c7ba5b69ee16c7467092cfc8de5dabb75915d | subsection | 6 | 24 | Sharpness Continuous Paths | In this section we present the Sharpness Continuous (SC) paths.
REF introduces the principle behind SC paths.
SC paths are composed of SC turns (detailed in REF ) connected over a line segment.
The process of connecting SC turns over a line segment to form a continuous SC path is detailed in REF .
When generating an SC... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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7d108179770d65c13edc7eb02e53521d803dfeab | subsection | 7 | 24 | Principle | , use a combination of arc circle turns and/or line segments to connect two arbitrary poses.
In , this idea is extended with Curvature Continuous (CC) turns, which replace arc circles by a combination of clothoid and arc circles.
We extend this further, replacing the clothoid segments by cubic curvature paths, achievin... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.2307/2372560",
"end": 93,
"openalex_id": "https://openalex.org/W2313274380",
"raw": "L. E. Dubins, \"On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents\" Am... | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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5af42aa1f8c971ddd85f2041636a170ee01b618b | subsection | 8 | 24 | Sharpness Continuous Turns | We propose Sharpness Continuous (SC) turns, which consist of three segments, an initial cubic curvature path \Gamma _{\mathrm {1,2}}, a circular arc \Gamma _{\mathrm {2,3}}, and a final cubic curvature path \Gamma _{\mathrm {3,4}}.
Figure REF shows an example of an SC turn.
The initial segment \Gamma _{\mathrm {1,2}} s... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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00fa48b31585f442e6a042809843dd7e0445f62d | subsection | 9 | 24 | Sharpness Continuous Turns | The circular arc starts at (x_{\mathrm {2}}, y_{\mathrm {2}}) and has its center at a distance \kappa _{\max }^{-1} perpendicular to the orientation \theta _{\mathrm {2}} at point (x_{\mathrm {2}}, y_{\mathrm {2}}).
Its center is given by(x_\Omega , y_\Omega ) = (x_{\mathrm {2}} - \kappa _{\max }^{-1} \sin \theta _{\ma... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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a13928fbe345fdb58ed810415c70ea2fc8e7f14c | subsection | 10 | 24 | Sharpness Continuous Turns | It is computed using the previous auxiliary circular arc as\mu = \arctan \left( \frac{y_{\mathrm {4}} - \kappa _{\max }^{-1} }{x_{\mathrm {4}}} \right) + \frac{\pi }{2} - \theta _{\mathrm {4}}.Thus, given a certain initial configuration \mathbf {q}_{\mathrm {1}}, the possible positions of the ending configuration \math... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.0266... | |
9e1df8472f3916cd42727ca8933085c6c8f2f896 | subsection | 11 | 24 | Connecting Sharpness Continuous Turns | An SC path between start and goal configurations \mathbf {q}_{\mathrm {s}} and \mathbf {q}_{\mathrm {g}} can be found by connecting two SC turns.
An SC path consists of three elements:an SC turn starting at the start configuration \mathbf {q}_{\mathrm {s}} and ending at a configuration \mathbf {q}_{\mathrm {a}} with nu... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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b5aaaea4bc8b6036ef56a16c62cc44f5f1186c78 | subsection | 12 | 24 | Connecting Sharpness Continuous Turns | \Omega _{\mathrm {a}} is centered at (0, 0) and \Omega _{\mathrm {b}} is located so that \mathbf {q}_{\mathrm {a}} and \mathbf {q}_{\mathrm {b}} are collinear.
We are interested in finding the center of \Omega _{\mathrm {b}}= (x_{\Omega _{\mathrm {b}}} , y_{\Omega _{\mathrm {b}}} ).
From Figure REF (top) it can be seen... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.0... | |
f10b686a1b4c37455c2e21fac45a351be4aad64e | subsection | 13 | 24 | Connecting Sharpness Continuous Turns | An analogous procedure can be used to find the possible departure configurations between any combination of clockwise (right steering) and counter clockwise turns, as shown in Figure REF (bottom).
This procedures are valid if the found tangent configurations \mathbf {q}_{\mathrm {a}} and \mathbf {q}_{\mathrm {b}} do no... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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f81ca2a72bd9de9def699e5ee4ba5f0de5eccc58 | subsection | 14 | 24 | Finding the Shortest SC Path | In order to find the shortest SC path between two configurations \mathbf {q}_{\mathrm {s}} and \mathbf {q}_{\mathrm {g}}, we need to compute all the possible SC turns that can be spanned from these configurations.
The SC turns are then connected, in order to generate possible SC paths.
The process is detailed below.Eac... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.019... | |
a2af22cc68cbfcd65d33d88475c9769deff7a8c6 | subsection | 15 | 24 | Cubic Curvature Paths | Cubic curvature paths are a building block of an SC turn.
They are used as transition paths connecting configurations to and from an arc circle (paths \Gamma _{\mathrm {1,2}} and \Gamma _{\mathrm {3,4}} in REF ), and they guarantee the sharpness continuity of the whole path. | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.0... | |
eb3d80aff105e5192d4a4798bd589938a3c5b1fd | subsection | 16 | 24 | Introduction | Cubic curvature paths are defined as paths with a cubic curvature profile \kappa (s) = a_3 s^3 + a_2 s^2 + a_1 s + a_0, where s is the length along the path.
A cubic curvature profile is the minimum degree polynomial that allows to define arbitrary initial and final curvatures, \kappa _{\mathrm {i}} and \kappa _{\mathr... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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fc1a7c9cf6ddba375f221ae492f007d1cea15dba | subsection | 17 | 24 | Ensuring Steering Rate and Acceleration Constraints | In order to have a feasible path, we need to ensure that a vehicle can follow it while complying with its steering constraints.
If we make both \kappa _{\mathrm {i}} and \kappa _{\mathrm {f}} smaller in magnitude than \kappa _{\max }, we ensure that the path always has a steering angle magnitude smaller than \phi _{\ma... | {
"cite_spans": []
} | 1801.08995 | Trajectory Generation using Sharpness Continuous Dubins-like Paths with
Applications in Control of Heavy Duty Vehicles | [
"Rui Oliveira",
"Pedro F. Lima",
"Marcello Cirillo",
"Jonas Mårtensson",
"Bo Wahlberg"
] | [
"cs.SY",
"cs.RO"
] | 2,018 | en | Computer Science | [
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0.... |
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